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of the constructive and destructive interference of the fundamental wave with all the existing overtones that occur in the string. For example, Figure 8.37 shows the wave trace on an oscilloscope for the sound of a violin. Figure 8.37 The interference of the fundamental frequency with the overtones produced by a bowed string creates the wave form that gives the violin its unique sound. The wavelength of the fundamental frequency is the distance between the tall sharp crests. Chapter 8 Mechanical waves transmit energy in a variety of ways. 423 08-PearsonPhys20-Chap08 7/24/08 2:23 PM Page 424 Tuning a Stringed Instrument Tuning a stringed instrument involves several principles of physics. The universal wave equation, v fλ, indicates that the frequency of a sound wave is directly proportional to the speed of the sound and inversely proportional to its wavelength. The wavelength for the fundamental frequency of the standing wave in a string is fixed at twice the length of the string, but the speed of a wave in a string increases with tension. Thus, if the wavelength does not change, the frequency at which a string vibrates must increase with tension. Changing the tension in the string is known as tuning (Figure 8.38). Wind Instruments Wind instruments produce different musical notes by changing the length of the air columns (Figure 8.39). In 8-7 Inquiry Lab you used a closed pipe and saw that for resonance to occur, a node must be present at the closed end while an antinode is created at the open end. For a closed pipe, the longest wavelength that can resonate is four times as long as the pipe (Figure 8.40). If the pipe is open at both ends, then the wavelengths for which resonance occurs must have antinodes at both ends of the open pipe or open tube (Figure 8.41). The distance from one antinode to the next is one-half a wavelength; thus, the longest wavelength that can resonate in an open pipe is twice as long as the pipe. antinode antinode l λ1 4 l λ1 2 node antinode Figure 8.40 In a closed pipe, the longest possible resonant wavelength is four times the length of the pipe. Figure 8.41 In an open pipe, the longest possible resonant wavelength is twice the length of the pipe. Wind instruments are generally open pipes. The wavelength of the resonant frequency will be decided by the length of the pipe (Figure 8.42). In a clar
inet or oboe, for example, the effective length of the pipe is changed by covering or uncovering holes at various lengths down the side of the pipe. The strongest or most resonant frequency will be the wave whose length is twice the distance from the mouthpiece to the first open hole. Overtones are also generated but the note you hear is that with the longest wavelength. As with stringed instruments, the overtones contribute to the wind instrument’s characteristic sound. If the speed of sound in air never varied, then a given wavelength would always be associated with the same frequency. But the speed of sound changes slightly with air temperature and pressure. Thus, in the case of resonance in a pipe, the length of the pipe must be increased or decreased as the speed of sound increases or decreases to ensure that the frequency is that of the desired note. Figure 8.38 Tuning a guitar e LAB For a probeware activity, go to www.pearsoned.ca/ school/physicssource. Figure 8.39 The trumpeter produces different notes by opening valves to change the instrument’s overall pipe length. Figure 8.42 A variety of wind instruments e WEB To learn how and why wind instruments are affected by temperature, follow the links at www.pearsoned.ca/school/ physicssource. 424 Unit IV Oscillatory Motion and Mechanical Waves 08-PearsonPhys20-Chap08 7/24/08 2:23 PM Page 425 An Interference Pattern from Two In-phase Point Sources Interference patterns carry information about the waves that create them. For this reason, the patterns are often used to determine the properties of the waves. One of the most interesting interference patterns results from waves generated by two point sources that are in phase. Remember that wave sources are in phase if they generate crests at the same time. The ripple tank photograph (Figure 8.43) shows the interference pattern generated by two in-phase point sources that are separated in space. This pattern is the result of constructive and destructive interference as the waves cross. Generally crests appear bright and troughs appear dark. However, in areas where destructive interference occurs, there appear to be fuzzy lines (such as the line indicated by Q1) that seem to radiate approximately from the midpoint between the sources. While the pattern may appear to be complex, its explanation is fairly simple. P1 Q1 R1 S1 P2 S2 Figure 8.43 The interference pattern generated by two
in-phase point sources in a ripple tank. The distance between the sources is 3. Individually, point sources generate waves that are sets of expanding concentric circles. As the crests and troughs from each source move outward, they cross through each other. As with all waves, when the crests from one source overlap crests from the other source (or troughs overlap troughs), constructive interference occurs. In these regions there is increased contrast (as indicated by P1 and R1). At locations where the crests from one source overlap troughs from the other source, destructive interference occurs. In these regions, contrast is reduced. Because the sources oscillate in phase, the locations where constructive and destructive interference occur are at predictable, fixed points. Like standing waves in a spring, the positions of the nodes and antinodes depend on the wavelength and the distance between the sources. Can you identify the regions of constructive and destructive interference in Figure 8.43 above? info BIT Common effects of interference patterns result in the “hot” and “cold” spots for sound in an auditorium. In 2005 renovations were completed for the Jubilee Auditoriums in Edmonton and Calgary. During renovations, the auditoriums were retuned to improve their acoustic properties. PHYSICS INSIGHT An interference pattern for sound can result if two loudspeakers, at an appropriate distance apart, are connected to the same audio frequency generator. When the sound waves diverging from the speakers overlap and interfere, regions of loud sound (maxima) and regions of relative quiet (minima) will be created. interference pattern: a pattern of maxima and minima resulting from the interaction of waves, as crests and troughs overlap while the waves move through each other Chapter 8 Mechanical waves transmit energy in a variety of ways. 425 08-PearsonPhys20-Chap08 7/24/08 2:23 PM Page 426 The pattern in Figure 8.43 can be reproduced by drawing sets of concentric circles about two point sources where each circle represents the crest of a wave front (Figure 8.44). In Figure 8.43 the distance (d) between the sources is equal to three wavelengths (3λ). This can be shown by counting the wavelengths between S1 and S2 in Figure 8.44. P1 Q1 R1 central maximum first order minimum first order maximum minima maxima 5λ 5λ 4 λ1 2 4λ 3λ S1 P2 Q2 R2 S2
Figure 8.44 The interference pattern for two in-phase point sources results from the overlap of two sets of concentric circles. In this diagram, the centres of the circles are three wavelengths apart. Maxima, Minima, and Phase Shifts The central maximum is a line of antinodes. In Figure 8.44, the line P1P2 is the perpendicular bisector of the line S1S2. By definition, every point on P1P2 is equidistant from the points S1 and S2. Thus, crests (or troughs) generated simultaneously at S1 and S2 must arrive at P1P2 at the same time, resulting in constructive interference. Along the line P1P2 only antinodes are created. The line of antinodes along P1P2 is called the central maximum. A nodal line, or minimum, marks locations where waves are exactly out of phase. A little to the right of the central maximum is the line Q1Q2. If you follow this line from one end to the other you will notice that it marks the locations where the crests (lines) from S1 overlap the troughs (spaces) from S2 and vice versa. Waves leave the sources in phase, but all points on Q1Q2 are a one-half wavelength farther from S1 than they are from S2. Thus, at any point on Q1Q2, the crests from S1 arrive one-half a wavelength later than the crests from S2. This means they arrive at the same time as troughs from S2. The greater distance travelled by waves from S1 produces what is called a one-half wavelength phase shift. Waves that began in phase arrive at points on Q1Q2 exactly out of phase. Thus, at every point on Q1Q2 destructive interference occurs. The line, Q1Q2, is known as a nodal line or a minimum. maximum: a line of points linking antinodes that occur as the result of constructive interference between waves minimum: a line of points linking nodes that occur as the result of destructive interference between waves phase shift: the result of waves from one source having to travel farther to reach a particular point in the interference pattern than waves from the other source 426 Unit IV Oscillatory Motion and Mechanical Waves 08-PearsonPhys20-Chap08 7/24/08 2:23 PM Page 427 e WEB To learn more about two-
point interference systems, follow the links at www.pearsoned.ca/school/ physicssource. A first order maximum is the result of a one wavelength phase shift. Moving farther right, another region of constructive interference occurs. To arrive at any point on R1R2, crests from S1 travel exactly one wavelength farther than crests from S2. Crests from S1 arrive at points on R1R2 at the same time as crests from S2 that were generated one cycle later. This one-wavelength phase shift means that all waves arriving at any point on R1R2 are still in phase. The line of antinodes resulting from a one-wavelength shift is known as a first order maximum. An identical first order maximum exists on the left side. The interference pattern is symmetrical about the central maximum. Phase shifts equal to whole wavelengths produce maxima. Moving farther outward from the central maximum, you pass through lines of destructive and constructive interference (minima and maxima). Each region is the result of a phase shift produced when waves travel farther from one source than the other. When the phase shift equals a whole number of wavelengths (0λ, 1λ, 2λ,...), the waves arrive in phase, producing antinodes and resulting in the central, first, second, and third order maxima, etc. In Figure 8.44, since the sources are 3λ apart, the greatest phase shift possible is three wavelengths. This produces the third order maximum directly along the line of S1S2. Phase shifts equal to an odd number of half-wavelengths produce minima. When the phase shift equals an odd number of half-wavelengths 1 λ, 2 λ, … the waves arrive out of phase, producing a nodal line or λ, 5 3 2 2 minimum. In Figure 8.44, the greatest phase shift to produce destructive interference is one-half wavelength less than the three-wavelength λ 5 separation of the sources, or 3λ 1 λ. Because the sources are three 2 2 wavelengths apart, there are exactly three maxima and three minima to the right and to the left of the central maximum. 8-8 Design a Lab 8-8 Design a Lab Interference Patterns and In-phase Sound Sources The Question Do interference patterns exist for two in-phase sound sources? Design and Conduct Your Investigation An audio frequency generator and two speakers can be used to create an interference pattern
for sound. Design a set-up that will enable you to measure the wavelength of sound of known frequencies. If electronic equipment (probeware or waveport) is available, design lab 8-8 to incorporate this equipment. Measure the wavelengths using several maxima and minima to compare measurements. Which type of line gives the best results? How well do the results from this experiment compare with the results from measuring wavelengths using closed-pipe resonance? Chapter 8 Mechanical waves transmit energy in a variety of ways. 427 08-PearsonPhys20-Chap08 7/24/08 2:23 PM Page 428 8.3 Check and Reflect 8.3 Check and Reflect Knowledge 1. What is meant by the term interference? 2. For a standing wave, what is the relationship between the amplitude of an antinode and the amplitude of the waves that combine to create the standing wave? 3. In terms of the length of an air column, what is the longest standing wavelength that can exist in an air column that is (a) closed at one end and (b) open at both ends? 4. An air column is said to be closed if it is closed at one end. Consider a pipe of length (l). For a standing wave in this pipe, what are the lengths of the three longest wavelengths for which an antinode exists at the open end of the pipe? 5. What does it mean to say that two wave generators are in phase? What does it mean to say that two waves are in phase? Applications 6. Two pulses of the same length (l) travel along a spring in opposite directions. The amplitude of the pulse from the right is three units while the amplitude of the pulse from the left is four units. Describe the pulse that would appear at the moment when they exactly overlap if (a) the pulses are on the same side of the spring and (b) the pulses are on opposite sides of the spring. 7. A standing wave is generated in a closed air column by a source that has a frequency of 768 Hz. The speed of sound in air is 325 m/s. What is the shortest column for which resonance will occur at the open end? 8. Draw the interference pattern for two inphase point sources that are 5 apart, as follows. Place two points, S1 and S2, 5 cm apart near the centre of a sheet of paper. Using each of these points as a centre, draw two sets of concentric circles with increasing radii of 1 cm
, 2 cm, 3 cm,..., until you reach the edge of the paper. On the diagram, draw solid lines along maxima and dotted lines along minima. Label the maxima according to their order. Explain why there are five minima on either side of the central maximum. 9. An interference pattern from two in-phase point sources is generated in a ripple tank. On the screen, a point on the second order maximum is measured to be 8.0 cm from one point source and 6.8 cm from the other source. What is the wavelength of this pattern? Extension 10. Do pipe organs, such as those found in churches and concert halls, use closed or open pipes to produce music? What is the advantage of using a real pipe organ as opposed to an electronic organ that synthesizes the sound? e TEST To check your understanding of superposition and interference of pulses and waves, follow the eTest links at www.pearsoned.ca/school/physicssource. 428 Unit IV Oscillatory Motion and Mechanical Waves 08-PearsonPhys20-Chap08 7/24/08 2:23 PM Page 429 info BIT It is common to see police officers using radar to measure the speed of cars on the highway. The radar gun emits waves that are reflected back to it from an oncoming car. A computer in the gun measures the change in frequency and uses that change to calculate the speed of the car. reflected wave λ transmitted RADAR wave moving car λ v radio transmitter Figure 8.45 8.4 The Doppler Effect Have you ever stood at the side of a road and listened to the cars pass? If you listen carefully, you will detect a very interesting phenomenon. At the instant a car passes you, the sound it makes suddenly becomes lower in pitch. This phenomenon was explained by an Austrian physicist named Christian Doppler (1803–1853). Doppler realized that the motion of the source affected the wavelength of the sound. Those waves that moved in the same direction as the source was moving were shortened, making the pitch of the sound higher. Moving in the direction opposite to the motion of the source, the sound waves from the source were lengthened, making the pitch lower. Wavelength and Frequency of a Source at Rest Assume that the frequency of a source is 100 Hz and the speed of sound is 350 m/s (Figure 8.46). According to the universal wave equation, if this source is at rest,
the wavelength of the sound is 3.50 m. v fλ λ v f 350 m s 0 10 s 3.50 m vw 350 m/s λ 3.50 m Figure 8.46 When a wavelength of 3.50 m travels toward you at a speed of 350 m/s, you hear sound that has a frequency of 100 Hz (diagram not to scale). You hear the sound at a frequency of 100 Hz because at a speed of 350 m/s, the time lapse between crests that are 3.50 m apart is 1/100 s. If, however, the wavelengths that travel toward you were 7.0 m long, the time lapse between successive crests would be 1/50 s, a frequency equal to 50 Hz. v f λ 350 m s 7.0 m 50 1 s 50 Hz Chapter 8 Mechanical waves transmit energy in a variety of ways. 429 08-PearsonPhys20-Chap08 7/24/08 2:23 PM Page 430 Wavelength and Frequency of a Moving Source 0.7 m A B C D 4.2 m Figure 8.47 When a sound source moves toward you, the wavelengths in the direction of the motion are decreased. e SIM Research examples of shock waves and the variation of wavelength with a moving source. Go to www.pearsoned.ca/ school/physicssource. direction of motion wave front generated by source at position A by source at position B by source at position C by source at position D Imagine that the source generating the 100-Hz sound is moving toward you at a speed of 70 m/s. Assume that the source is at point A (Figure 8.47) when it generates a crest. While the first crest moves a distance of 3.5 m toward you, the source also moves toward you. The distance the source moves while it generates one wavelength is the distance the source travels in 1/100 s at 70 m/s, or 0.7 m. Because of the motion of the source, the next crest is generated (at point B) only 2.8 m behind the first crest. As long as the source continues at the speed of 70 m/s toward you, the crests travelling in your direction will be only 2.8 m apart. Hence, for a car moving toward you, the sound waves emitted by the car will be “squashed together” and thus reach you more frequently than if the car were stationary. 2.
8 m If waves that are 2.8 m long travel toward you at a speed of 350 m/s, then the frequency of the sound arriving at your ear will be 125 Hz. The pitch of the sound that you hear will have been increased because the source is moving toward you. v fλ v f λ 350 m s 2.80 m 125 1 s 125 Hz At the same time, along a line in the direction opposite to the motion of the source, the wavelengths are increased by the same amount that the waves in front of the source are shortened. For the 100-Hz sound source moving at 70 m/s, the waves behind the source are increased by 0.7 m, to a length of 4.2 m. The time lapse between these crests, which are 4.2 m apart and travelling at 350 m/s, is 0.012 s. Therefore, the perceived frequency in the direction opposite to the motion of the source is about 83 Hz. The pitch of the sound has been lowered. Analysis of the Doppler Effect If the velocity of the sound waves in air is vw, then the wavelength (λ s) that a stationary source(s) with a frequency of fs generates is given by λ s vw fs. The key to this Doppler’s analysis is to calculate the distance the source moves in the time required to generate one wavelength (the period (Ts) of the source). If the source is moving at speed vs, then in the period (Ts) the source moves a distance (ds) that is given by ds vsTs. Since, by definition 430 Unit IV Oscillatory Motion and Mechanical Waves 08-PearsonPhys20-Chap08 7/28/08 9:13 AM Page 431 Ts 1fs, then ds vs fs. Sources Moving Toward You For sources that are moving toward you, ds is the distance by which the wavelengths are shortened. Subtracting ds from λ s gives the lengths of the waves (λ d) that reach the listener. Therefore, ds. s and ds by their equivalent forms gives λ λ d s Replacing v v s w fs f s λ d λ d vs ) (vw fs This is the apparent wavelength (Doppler wavelength) of the sound generated by a source that is moving toward you at a speed vs. Dividing the speed of the waves (vw) by the Doppler
wavelength (λ d) produces the Doppler frequency (fd) of the sound that you hear as the source approaches you. Therefore, v w λ d fd vw vs vw f s f s vw fs vw vs v w vw vs is the Doppler frequency when the source is approaching the listener. Sources Moving Away from You If the source is moving away from the listener, the value of ds is added to the value of λ s, giving λ λ ds. s d s and ds by their equivalent forms and complete the If you replace λ development to find fd, it is easy to see that the Doppler frequency for a sound where the source moves away from the listener is given by fs v w vw vs fd General Form of the Doppler Equation The equations for the Doppler effect are usually written as a single equation of the form fs v w vw vs fd PHYSICS INSIGHT When the distance between you and the source is decreasing, you must subtract to calculate the Doppler effect on frequency and wavelength. Chapter 8 Mechanical waves transmit energy in a variety of ways. 431 08-PearsonPhys20-Chap08 7/28/08 9:17 AM Page 432 Concept Check If you are travelling in your car beside a train that is blowing its whistle, is the pitch that you hear for the whistle higher or lower than the true pitch of the whistle? Explain. Example 8.4 A train is travelling at a speed of 30.0 m/s. Its whistle generates a sound wave with a frequency of 224 Hz. You are standing beside the tracks as the train passes you with its whistle blowing. What change in frequency do you detect for the pitch of the whistle as the train passes, if the speed of sound in air is 330 m/s? Practice Problems 1. You are crossing in a crosswalk when an approaching driver blows his horn. If the true frequency of the horn is 264 Hz and the car is approaching you at a speed of 60.0 km/h, what is the apparent (or Doppler) frequency of the horn? Assume that the speed of sound in air is 340 m/s. 2. An airplane is approaching at a speed of 360 km/h. If you measure the pitch of its approaching engines to be 512 Hz, what must be the actual frequency of the sound of the engines? The speed of sound in air is 345 m
/s. 3. An automobile is travelling toward you at a speed of 25.0 m/s. When you measure the frequency of its horn, you obtain a value of 260 Hz. If the actual frequency of the horn is known to be 240 Hz, calculate vw, the speed of sound in air. 4. As a train moves away from you, the frequency of its whistle is determined to be 475 Hz. If the actual frequency of the whistle is 500 Hz and the speed of sound in air is 350 m/s, what is the train’s speed? Answers 1. 278 Hz 2. 364 Hz 3. 325 m/s 4. 18.4 m/s Given fs vw vs 224 Hz 330 m/s 30.0 m/s Required (a) Doppler frequency for the whistle as the train approaches (b) Doppler frequency for the whistle as the train moves away (c) change in frequency Analysis and Solution Use the equations for Doppler shifts to find the Doppler frequencies of the whistle. (a) For the approaching whistle, (b) For the receding whistle, fs v w vs vw fd fd m 330 s m m 30.0 330 s s fs v w vs fd fd 224 Hz vw m 330 s m m 30.0 330 s s 224 Hz 224 Hz 330 m s 300 m s 246.4 Hz 246 Hz 224 Hz 330 m s 360 m s 205.3 Hz 205 Hz (c) The change in pitch is the difference in the two frequencies. Therefore, the pitch change is f 246.4 Hz 205.3 Hz 41.1 Hz Paraphrase and Verify As the train passes, the pitch of its whistle is lowered by a frequency of about 41.1 Hz. 432 Unit IV Oscillatory Motion and Mechanical Waves 08-PearsonPhys20-Chap08 7/24/08 2:23 PM Page 433 e WEB To learn more about possible health effects of sonic booms at close range, and the recent concerns of the Innu Nation, follow the links at www.pearsoned.ca/ school/physicssource. The Sound Barrier Jet planes are not allowed to break or exceed the sound barrier in the airspace over most cities. When an object travels at speeds at, or greater than, the speed of sound, it creates a sonic boom. The boom is the result of the shock wave created by the motion of the object. Bow Waves A
boat moving through water produces a bow wave. The crest of the wave moves sideways away from the object, producing the wave’s characteristic V-shape. For an airplane moving through the fluid medium of the atmosphere, a V-shaped bow wave, or pressure wave, travels outward at the speed of sound (Figure 8.48). If the speed of the airplane is less than the speed of sound, the bow wave produced at any instant lags behind the bow wave produced just an instant earlier. The bow wave carries energy away from the plane in a continuous stream (Figure 8.49(a)). Sonic Boom However, for an airplane travelling at the speed of sound, the bow wave and the airplane travel at the same speed. Instant by instant, crests of the bow wave are produced at the same location as the crest of the bow wave produced by the plane an instant earlier (Figure 8.49). The energy stored in the bow wave becomes very intense. To the ear of an observer, crests of successively produced bow waves arrive simultaneously in what is known as a sonic boom. In early attempts to surpass the speed of sound, many airplanes were damaged. At the speed of sound, there is a marked increase in drag and turbulence. This effect damaged planes not designed to withstand it. A reporter assumed the increased drag acted like a barrier to travelling faster than sound and coined the term sound barrier. Mathematically, from the arguments presented above, the Doppler wavelength is given by λ d vs) (vw. f s Figure 8.48 When conditions are right, the change in pressure produced by the airplane’s wings can cause sufficient cooling of the atmosphere so that a cloud forms. The extreme conditions present when a jet is travelling near the speed of sound often result in the type of cloud seen in this photo. (a) Slower than speed of sound: Pressure waves move out around plane. (b) At speed of sound: Pressure waves at nose form a shock wave. (c) At supersonic speed: Shock waves form a cone, resulting in a sonic boom. Figure 8.49 As an airplane accelerates from subsonic to supersonic speeds, the changing relationship between the plane and the bow waves or pressure waves results in a sonic boom. Chapter 8 Mechanical waves transmit energy in a variety of ways. 433 08-PearsonPhys20-Chap08 7/24/08 2:23 PM Page 434 vw, which If a plane is travelling at the
speed of sound, then vs means that for any sound produced by the jet, the Doppler wavelength in the direction of the jet’s motion is zero. Even if the plane’s speed is greater than that of sound, the bow waves still combine to form a shock wave. In this way a sonic boom can be heard for any object, such as a rifle bullet, that has a supersonic speed. THEN, NOW, AND FUTURE Ultrasound While impressive given the technology available at the time, the results of early attempts at using ultrasound in medicine were of poor quality. Initially, ultrasound images from within a body were very blurry and two-dimensional. By today’s standards, the technology was extremely crude and there was virtually no scientific understanding of how the sound would behave when it encountered different types of tissue. Today, computers have made it possible to form three-dimensional images that can be rotated so that you can see all sides. Doppler ultrasound is used to detect blood flow through an organ. Today, 4-D ultrasound (time is the 4th dimension) is a real-time 3-D image that moves. 1. What are the advantages and disadvantages of ultrasound imaging compared with other imaging techniques such as CT scans and MRI? Figure 8.50 A 3-D ultrasound picture of a developing fetus 8.4 Check and Reflect 8.4 Check and Reflect Knowledge 1. What causes the Doppler effect? 2. Two sound sources have the same frequency when at rest. If they are both moving away from you, how could you tell if one was travelling faster than the other? 3. Explain the cause of a sonic boom. Applications 4. The siren of a police car has a frequency of 660 Hz. If the car is travelling toward you at 40.0 m/s, what do you perceive to be the frequency of the siren? The speed of sound in air is 340 m/s. 5. A police car siren has a frequency of 850 Hz. If you hear this siren to have a frequency that is 40.0 Hz greater than its true frequency, what was the speed of the car? The speed of sound is 350 m/s. 6. A jet, travelling at the speed of sound (Mach 1), emits a sound wave with a frequency of 1000 Hz. Use the Doppler effect equations to calculate the frequency of this sound as the jet first approaches you, then moves away from you. Explain what
.2) (a) What affects the speed of a water wave? (b) What is the nature of the motion of the medium when a longitudinal wave moves through it? (c) Describe how the speed of a wave affects its wavelength and its amplitude. (d) Explain why waves are considered a form of Simple Harmonic Motion. (e) If speed is constant, how does wavelength vary with frequency? 3. (8.3) (a) Describe the conditions required to produce constructive and destructive interference in waves. (b) Describe how the principle of superposition applies to what happens when two pulses of identical length and amplitude interfere to produce no apparent pulse. (c) Define node, antinode, and standing wave. (d) In terms of the wavelength of the waves that have combined to form a standing wave, describe the position of the nodes and antinodes as you move away from the fixed end of a spring. (e) Why can a standing wave be generated only by what is defined as resonant frequency? 4. (8.4) (a) Does the Doppler effect apply only to sound or can it apply to any form of wave motion? Explain. (b) How are the waves in the direction of a source’s motion affected as the speed of the source increases? Applications 5. The speed of a wave in a spring is 15.0 m/s. If the length of a pulse moving in the spring is 2.00 m, how long did it take to generate the pulse? Why don’t we talk about the frequency for a pulse? 436 Unit IV Oscillatory Motion and Mechanical Waves 6. Waves are generated by a straight wave generator. The waves move toward and reflect from a straight barrier. The angle between the wave front and the barrier is 30°. Draw a diagram that shows what you would observe if this occurred in a ripple tank. Use a line drawn across the middle of a blank sheet of paper to represent the barrier. Draw a series of wave fronts about 1 cm apart intersecting the barrier at 30°. Use a protractor to make sure the angle is correct. Now draw the reflected waves. Draw a ray to indicate the motion of the incident wave front and continue this ray to indicate the motion of the reflected wave front. Hint: Draw the reflected ray as if it had a new source. 7. A ripple tank is set up so that the water in it is 0.7 cm deep
. In half of the tank, a glass plate is placed on the bottom to make the water in that half shallower. The glass plate is 0.5 cm thick. Thus, the tank has a deep section (0.7 cm) and a shallow section (0.2 cm). In the deep section the wave velocity is 15.0 cm/s while in the shallow section the velocity is 10.0 cm/s. Straight waves, parallel to the edge of the glass, move toward the line between the deep and shallow sections. If the waves have a frequency of 12.0 Hz, what changes in wavelength would you observe as they enter the shallow section? What would happen to the direction of the motion? 8. A ripple tank is set up as described in question 7. For this ripple tank you measure the speed of the waves to be 12.0 cm/s and 9.0 cm/s in the deep and shallow sections, respectively. If waves in the deep section that are 11.5 cm long cross over to the shallow section, what would be the wavelength in the shallow section? 9. The term ultrasound means the frequency is higher than those that our ears can detect (about 20 kHz). Animals can often hear sounds that, to our ears, are ultrasound. For example, a dog whistle has a frequency of 22 kHz. If the speed of sound in air is 350 m/s, what is the wavelength of the sound generated by this whistle? 10. A spring is stretched to a length of 7.0 m. A frequency of 2.0 Hz generates a standing wave in the spring that has six nodes. (a) Sketch the standing wave pattern for the spring. (b) Calculate the velocity of the wave. 08-PearsonPhys20-Chap08 7/24/08 2:23 PM Page 437 11. The figure shows two waves that occupy the same point in space. Copy the sketch onto a sheet of paper using the dimensions indicated. Draw the wave that results from the interference of these two waves. 20. When a police car is at rest, the wavelength of the sound from its siren is 0.550 m. If the car is moving toward you at a speed of 120 km/h, what is the frequency at which you hear the siren? Assume that the speed of sound is 345 m/s. 3.5 cm 5.0 cm A 1.5 cm A A 10 cm A A 12. If a frequency of 1
.5 Hz generates a standing wave in a spring that has three antinodes, (a) what frequency generates a standing wave with five antinodes in the same spring, and (b) what is the fundamental frequency for this spring? 13. A violin string is 33.0 cm long. The thinnest string on the violin is tuned to vibrate at a frequency of 659 Hz. (a) What is the wave velocity in the string? (b) If you place your finger on the string so that its length is shortened to 28.0 cm, what is the frequency of the note that the string produces? 14. (a) What is the shortest closed pipe for which resonance is heard when a tuning fork with a frequency of 426 Hz is held at the open end of the pipe? The speed of sound in air is 335 m/s. (b) What is the length of the next longest pipe that produces resonance? 21. If the speed of sound in air is 350 m/s, how fast must a sound source move toward you if the frequency that you hear is twice the true frequency of the sound? What frequency would you hear if this sound source had been moving away from you? Extensions 22. Describe an arrangement that you might use if you wanted to create an interference pattern similar to the one in Figure 8.44 on page 426 by using sound waves that have a frequency of 512 Hz. Except for standing waves in strings or pipes, why do you think that we do not often find interference patterns in nature? 23. Explain why the number of maxima and minima in the interference pattern generated by two inphase point sources depends on the ratio of the distance between the sources to the wavelength. e TEST To check your understanding of waves and wave motion, follow the eTest links at www.pearsoned.ca/ school/physicssource. Consolidate Your Understanding 15. Draw the interference pattern generated by two inphase point sources that are four wavelengths apart. Answer each of the following questions in your own words. Provide examples to illustrate your explanation. 16. In the interference pattern for two in-phase point sources, a point on a second order maximum is 2.8 cm farther from one source than the other. What is the wavelength generated by these sources? 17. The horn on a car has a frequency of 290 Hz. If the speed of sound in air is 340 m/s and the car is moving toward you at a speed of 72.0 km/
h, what is the apparent frequency of the sound? 18. How fast is a sound source moving toward you if you hear the frequency to be 580 Hz when the true frequency is 540 Hz? The speed of sound in air is 350 m/s. Express your answer in km/h. 19. If the speed of sound in air is 350 m/s, how fast would a sound source need to travel away from you if the frequency that you hear is to be onehalf the true frequency? What would you hear if this sound source had been moving toward you? 1. What are the advantages and disadvantages of using a spring as a model for wave motion? 2. What are the conditions for which a standing wave pattern is generated? Why are standing waves not often seen in nature? 3. Explain how the energy in a wave is transmitted from one place to another. 4. Describe what is meant by the principle of superposition. How does this principle explain standing waves? 5. What is meant by resonance? Think About It Review your answers to the Think About It questions on page 393. How would you answer each question now? Chapter 8 Mechanical waves transmit energy in a variety of ways. 437 08-PearsonPhys20-Chap08 7/24/08 2:23 PM Page 438 UNIT IV PROJECT Earthquakes Scenario The tsunami that swept coastal regions of the Indian Ocean on December 26, 2004, was set off by an earthquake centred off the coast of the island of Sumatra in Indonesia. Seismographs around the world identified the location and strength of the earthquake. It was determined that the earthquake rated about 9.0–9.3 on the Richter scale. You have been asked by your government to make a presentation on the seismology of earthquakes. Your challenge is threefold. • First: you are to explain the nature of earthquake shock waves, their movement through Earth, and how the location of the earthquake epicentre is identified by seismographs around the world. • Second: you are to explain how the intensity of earthquakes is measured. This means that you must explain what the Richter scale is, and how it is used to rate earthquake intensity. • Third: you are to demonstrate the operation of a seismograph. Planning Your team should consist of three to five members. Choose a team manager and a record keeper. Assign other tasks as they arise. The first task is to decide the structure of your presentation and the research questions you will need to
investigate. Questions you will need to consider are: How will you present the information to your audience? What are the resources at your disposal? Do you have access to computers and presentation programs such as PowerPoint®? Which team members will design, build, and demonstrate the model seismograph? Brainstorm strategies for research and create a schedule for meeting the deadlines for all phases of the project. Where is your team going to look for the information necessary to complete the project? What types of graphics will be most effective to assist your presentation? How will you best demonstrate the function of your seismograph? Your final report should include written, graphic, and photographic analyses of your presentation. Assessing Results Assess the success of your project based on a rubric* designed in class that considers: research strategies thoroughness of the experimental design effectiveness of the experimental technique effectiveness of the team’s public presentation Materials • materials, as needed, for the construction of your model seismograph Procedure 1 Research the nature of the shock waves set off by an earthquake. Be alert to Internet sites that may contain unreliable or inaccurate information. Make sure that you evaluate the reliability of the sources of information that you use for your research. If you gather information from the Internet, make sure you identify who sponsors the site and decide whether or not it is a reputable source of information. Maintain a list of your references and include it as an appendix to your report. Use graphics to explain how the shock waves move through Earth, and how seismologists locate the epicentre of an earthquake. 2 Research the history of the Richter scale and its use in identifying the intensity of an earthquake. 3 Design and build your model of a seismograph. Decide how you will demonstrate its use in your presentation. 4 Prepare an audio-visual presentation that would inform your audience on the nature of earthquakes and how they are detected. Thinking Further Write a short appendix (three or four paragraphs) to your report to suggest steps that governments might take to make buildings safer in earthquake zones. Answer questions such as: What types of structures are least susceptible to damage by earthquakes? What types are most susceptible? *Note: Your instructor will assess the project using a similar assessment rubric. 438 Unit IV Oscillatory Motion and Mechanical Waves 08-PearsonPhys20-Chap08 7/24/08 2:23 PM Page 439 UNIT IV SUMMARY Unit Concepts and Skills: Quick Reference Concepts Chapter 7 Period Frequency Spring constant Summary Resources and Skill Building Oscillatory motion requires a set
of conditions. 7.1 Period and Frequency Period is the time for one complete cycle, measured in seconds (s). If the period of each cycle remains constant, the object is moving with oscillatory motion. Frequency is the number of cycles per second, measured in Hertz (Hz). 7.2 Simple Harmonic Motion The spring constant is the amount of force needed to stretch or compress the spring 1 m and is measured in N/m. It can also be thought of as the stiffness of a spring. QuickLab 7-1; Inquiry Lab 7-2; Minds On; Figures 7.4, 7.5 QuickLab 7-1; Inquiry Lab 7-2; Example 7.1; Figure 7.5 QuickLab 7-3; Examples 7.2–7.4 Hooke’s law Hooke’s law states that the deformation of an object is proportional to the force causing it. Figures 7.9–7.16 Simple harmonic motion SHM refers to anything that moves with uniform oscillatory motion and conforms to Hooke’s law. Figures 7.19–7.23; eSIM Pendulum motion The pendulum is a simple harmonic oscillator for angles less than 15°. Figures 7.25–7.27; Example 7.5; Inquiry Lab 7-4; Table 7.5 Figures 7.44, 7.45; Then, Now, and Future QuickLab 8-1; Inquiry Lab 8-2; Inquiry Lab 8-3 Acceleration of a mass-spring system Relationship between acceleration and velocity of a mass-spring system Period of a mass-spring system Period of a pendulum Resonance Forced frequency Resonance effects on buildings and bridges 7.3 Position, Velocity, Acceleration, and Time Relationships The acceleration of a mass-spring system depends on displacement, mass, and the spring constant, and it varies throughout the motion of the mass-spring system. Figure 7.28 The acceleration and velocity of a mass-spring system are continually changing. The velocity of a mass-spring system is determined by its displacement, spring constant, and mass. Figures 7.29–7.33; Example 7.6 The period of a mass-spring oscillator is determined by its mass and spring constant, but not its amplitude. Figures 7.35–7.37; Example 7.7 A pendulum’s period is determined by its length and the gravitational field strength, but not the mass of the bob.
Figures 7.39, 7.40; eSIM; Example 7.8 7.4 Applications of Simple Harmonic Motion Resonance is the natural frequency of vibration of an object. Figure 7.41; QuickLab 7-5 Forced frequency is the frequency at which an external force is applied to an oscillating object. Figure 7.41; QuickLab 7-5 Bridges and buildings can resonate due to the force of the wind. Chapter 8 Mechanical waves transmit energy in a variety of ways. Wave properties may be qualitative or quantitative. 8.1 The Properties of Waves Waves have many properties that can be used to analyze the nature of the wave and the way it behaves as it moves through a medium. Some of these properties are qualitative (crest, trough, wave front, medium, incident wave, reflected wave, wave train) while others are quantitative (amplitude, wavelength, frequency, wave velocity). Universal wave equation 8.2 Transverse and Longitudinal Waves Waves can move through a medium either as transverse or longitudinal waves. The relationship among the frequency, wavelength, and wave velocity is given by the universal wave equation. Inquiry Lab 8-4; Inquiry Lab 8-5; Example 8.1; Example 8.2 Interference patterns may result when more than one wave moves through a medium. 8.3 Superposition and Interference When two or more waves travel in different directions through the same point in space, their amplitudes combine according to the principle of superposition. Depending on the properties of the waves, they may form an interference pattern. Interference patterns can often be used to determine the properties of the waves from which they are formed. Inquiry Lab 8-6; Example 8.3; Inquiry Lab 8-7; Design a Lab 8-8 Doppler effect Sonic boom 8.4 The Doppler Effect When a sound source moves either toward or away from a sensor (ear or microphone), the frequency of the sound that is detected will be different from the frequency emitted by the source. When an object is travelling at the speed of sound, it creates a shock wave known as a sonic boom. Example 8.4 Figure 8.49 Unit IV Oscillatory Motion and Mechanical Waves 439 08-PearsonPhys20-Chap08 7/24/08 2:23 PM Page 440 UNIT IV REVIEW Vocabulary 1. Use your own words to define the following terms, concepts, principles, or laws. Give examples where appropriate. amplitude antinodes closed-pipe air column
constructive interference crest destructive interference diverging Doppler effect equilibrium forced frequency frequency fundamental frequency Hooke’s law incident wave in phase interference longitudinal wave maximum mechanical resonance medium minimum nodes or nodal points open-pipe air column oscillation oscillatory motion overtone period phase shift principle of superposition pulse ray reflected wave resonance resonant frequency restoring force shock wave simple harmonic motion simple harmonic oscillator sonic boom sound barrier spring constant standing waves transverse wave trough two-point-source interference pattern 440 Unit IV Oscillatory Motion and Mechanical Waves wave wave front wave train wave velocity wavelength Knowledge CHAPTER 7 2. How are the units of frequency and period similar? How are they different? 3. The SI unit for frequency is Hz. What are two other accepted units? 4. For any simple harmonic oscillator, in what position is (a) the velocity zero? (b) the restoring force the greatest? 5. Why doesn’t a pendulum act like a simple harmonic oscillator for large amplitudes? 6. The equation for Hooke’s law uses a negative sign (F kx). Why is this sign necessary? 7. Aboriginal bows used for hunting were made from wood. Assuming the wood deforms according to Hooke’s law, explain how you would go about measuring the spring constant of the wood. 8. Suppose the same pendulum was tested in both Calgary and Jasper. In which location would you expect the pendulum to oscillate more slowly? Explain. 9. Explain why the sound from one tuning fork can make a second tuning fork hum. What conditions must be necessary for this to happen? 10. A pendulum in a clock oscillates with a resonant frequency that depends on several factors. From the list below, indicate what effect (if any) the following variables have on the pendulum’s resonant frequency. (a) length of pendulum arm (b) latitude of clock’s position (c) longitude of clock’s position (d) elevation (e) restoring force 08-PearsonPhys20-Chap08 7/24/08 2:23 PM Page 441 CHAPTER 8 11. The diagram below shows waves in two springs. For each of the springs, how many wavelengths are shown? 12. Sound waves, travelling through air, are reflected from the wall of a building. Describe how the reflection affects the speed, (a) the wavelength, (b) (c) the amplitude, and (d)
the direction of a wave train. 13. Points of zero displacement on a transverse wave have the greatest kinetic energy. Which points on a longitudinal wave have the greatest kinetic energy? 14. How is the shape of a circular wave front changed when it reflects from a straight barrier? 15. What aspect of a pulse determines the amount of energy it transfers? 16. When water waves enter a region where they travel slower, what happens to the (a) frequency, (b) wavelength, and (c) direction of the waves? 17. In the interference pattern from two in-phase point sources, what name is given to a line along which destructive interference occurs? 18. What determines the speed at which a wave travels through a spring? 19. What causes a standing wave in a spring? 20. Draw a transverse wave train that consists of two wavelengths. On your diagram, label the equilibrium position for the medium, a crest, a trough, the amplitude, a wavelength, and the direction of the wave velocity. Along the wavelength that you identified above, draw several vector arrows to indicate the direction of the motion of the medium. 21. Why does moving your finger along the string of a violin alter the note that it produces? 22. What property of the sound produced by a tuning fork is affected by striking the tuning fork with different forces? What does that tell you about the relationship between the properties of the sound and the sound wave created by striking the tuning fork? 23. When two in-phase point sources generate an interference pattern, what conditions are required to create (a) the central maximum and (b) a second order maximum? 24. In terms of the length of an open pipe, what is the longest wavelength for which resonance can occur? 25. You are walking north along a street when a police car with its siren on comes down a side street (travelling east) and turns northward on the street in front of you. Describe what you would hear, in terms of frequency of the sound of its siren, before and after the police car turns. 26. What is the relationship between frequency, wavelength, and wave velocity? 27. Why does the frequency of a sound source that is moving toward you seem to be higher than it would be if the source were at rest? Applications 28. Determine the force necessary to stretch a spring (k 2.55 N/m) to a distance of 1.20 m. 29. A musician plucks a guitar string. The string has
a frequency of 400.0 Hz and a spring constant of 5.0 104 N/m. What is the mass of the string? 30. When a pendulum is displaced 90.0° from the vertical, what proportion of the force of gravity is the restoring force? 31. While performing a demonstration to determine the spring constant of an elastic band, a student pulls an elastic band to different displacements and measures the applied force. The observations were recorded in the table below. Plot the graph of this data. Can the spring constant be determined? Why or why not? Displacement (m) 0.1 0.2 0.3 0.4 0.5 0.6 Force (N) 0.38 1.52 3.42 6.08 9.5 13.68 32. A force of 40.0 N is required to move a 10.0-kg horizontal mass-spring system through a displacement of 80.0 cm. Determine the acceleration of the mass when its displacement is 25.0 cm. Unit IV Oscillatory Motion and Mechanical Waves 441 08-PearsonPhys20-Chap08 7/24/08 2:23 PM Page 442 33. Use the following table to determine the spring constant of a spring. Displacement (cm) Force (mN) 2.5 5.0 7.5 10.0 12.5 10.0 21.0 31.0 39.0 49.0 34. A 50.0-g mass oscillates on the end of a vertical mass-spring system (k 25.0 N/m) with a maximum acceleration of 50.0 m/s2. (a) What is its amplitude of vibration? (b) What is the maximum velocity of the mass? 35. A bee’s wing has a mass of 1.0 105 kg and makes one complete oscillation in 4.5 103 s. What is the maximum wing speed if the amplitude of its motion is 1.10 cm? 36. A skyscraper begins resonating in a strong wind. A tuned mass damper (m 10.0 t) at the top of the building moves through a maximum displacement of 1.50 m in the opposite direction to dampen the oscillations. If the mass damper is attached to a horizontal spring and has a maximum speed of 1.40 m/s, what is the period of its oscillations? 37. A branch at the top of a tree sways with
simple harmonic motion. The amplitude of motion is 0.80 m and its speed is 1.5 m/s in the equilibrium position. What is the speed of the branch at the displacement of 0.60 m? 38. A tuned mass damper at the top of a skyscraper is a mass suspended from a thick cable. If the building sways with a frequency of 0.125 Hz, what length must the cable supporting the weight be to create a resonance in the damper? 39. When a wave slows down, what property of the wave is not affected? What effect does this have on the other properties of the wave? Explain. 40. Explain how a wave can transmit energy through a medium without actually transmitting any matter. 41. A light wave is transmitted through space at 3.00 108 m/s. If visible light has wavelengths ranging from about 4.30 107 m to 7.50 107 m long, what range of frequencies are we able to see? 42. Radio waves travel at the speed of light waves (3.00 108 m/s). If your radio is tuned to a station broadcasting at 1250 kHz, what is the length of the waves arriving at the radio antenna? 442 Unit IV Oscillatory Motion and Mechanical Waves 43. A pendulum oscillates with a period of 0.350 s. Attached to the pendulum is a pen that marks a strip of paper on the table below the pendulum as it oscillates. When the strip of paper is pulled sideways at a steady speed, the pen draws a sine curve on the paper. What will be the wavelength of the sine curve if the speed of the paper is 0.840 m/s? 44. A submarine sends out a sonar wave that has a frequency of 545 Hz. If the wavelength of the sound is 2.60 m, how long does it take for the echo to return when the sound is reflected from a submarine that is 5.50 km away? 45. A wire is stretched between two points that are 3.00 m apart. A generator oscillating at 480 Hz sets up a standing wave in the wire that consists of 24 antinodes. What is the velocity at which waves move in this wire? 46. A spring is stretched to a length of 5.40 m. At that length the speed of waves in the spring is 3.00 m/s. If a standing wave with a frequency 2.50 Hz (a) were generated in this spring, how many
nodes and antinodes would there be along the spring? (b) What is the next lower frequency for which a standing wave pattern could exist in this spring? 47. The second string on a violin is tuned to the note D with a frequency of 293 Hz. This is the fundamental frequency for the open string, which is 33.0 cm long. (a) What is the speed of the waves in the string? If you press on the string with your finger so (b) that the oscillating portion of the string is 2/3 the length of the open string, what is the frequency of the note that is created? 48. An audio frequency generator set at 154 Hz is used to generate a standing wave in a closed-pipe resonator, where the speed of sound is 340 m/s. (a) What is the shortest air column for which resonance is heard? (b) What is the next longer column length for which resonance is heard? 49. A submarine’s sonar emits a sound with a frequency of 875 Hz. The speed of sound in seawater is about 1500 m/s. If you measure the frequency of the sound to be 870 Hz, what is the velocity of the submarine? 50. A police car is travelling at a speed of 144 km/h. It has a siren with a frequency of 1120 Hz. Assume that the speed of sound in air is 320 m/s. (a) If the car is moving toward you, what frequency will you hear for the siren? If the car had been moving away from you at the same speed, what frequency would you have heard? (b) 08-PearsonPhys20-Chap08 7/24/08 2:23 PM Page 443 Extensions 51. What generalization can be made about the frequency of vibration with regard to the mass for a mass-spring system? (Assume all other qualities remain constant.) 52. An alien crash-lands its spaceship on a planet in our solar system. Unfortunately, it is unable to tell which planet it is. From the wreckage of the spaceship the alien constructs a 1.0-m-long pendulum from a piece of wire with four metal nuts on the end. If this pendulum swings with a period of 3.27 s, on which planet did the alien land: Mercury, Venus, or Earth? 53. Use a compass to draw a simulation of the wave pattern generated by two in-phase point source generators that
are 3.5 wavelengths apart. Near the middle of the page, place two points (S1 and S2) 3.5 cm apart to represent the positions of the sources. Draw wavelengths 1.0 cm long by drawing concentric circles that increase in radii by 1.0-cm increments. Locate on the diagram all the maxima and minima that are generated by this set-up and draw lines to indicate their positions. How does this pattern differ from the one in Figure 8.44 on page 426? Explain why these differences occur. 59. Outline a procedure that you could use to determine the mass of a horizontal mass-spring system without measuring the mass on a scale. 60. A student wants to determine the mass of the bob on a pendulum but only has access to a stopwatch and a ruler. She decides to pull the pendulum bob back through a displacement of 10° and time 20 complete oscillations. Will it be possible to determine the mass from the data gathered? Explain. 61. Construct a concept map for the simple harmonic motion of a pendulum. Include the following terms: period, displacement, restoring force, velocity, length, and gravitational field strength. 62. In a paragraph, explain why Huygens’s pendulum clock was a revolution in clock making and what the limitations were in its design. Be sure to use these terms: pendulum length, resonant frequency, forced frequency, and gravitational field strength. 63. Research the term “red shift” as used in astronomy. Prepare a report on the importance of red shift to our understanding of the nature of the universe. 54. In a stereo system, there are two speakers set at some distance apart. Why does a stereo system not result in an interference pattern? 64. Describe how to use springs to explore what happens to pulses transmitted from one medium to another in which the wave speed is different. 55. If a sound source is at rest, the frequency you hear and the actual frequency are equal. Their 1). If the sound source ratio equals one (fd/fs moves toward you at an ever-increasing speed, this frequency ratio also increases. Plot a graph for the ratio of the frequencies vs. the speed of the sound source as the speed of the source increases from zero to Mach 1. What is the value of the ratio when the speed of the source is Mach 1? Skills Practice 56. Use a graphing calculator or another suitable means to plot a graph of period against
frequency. What type of relationship is this? 57. Outline an experimental procedure that you could perform to determine the spring constant of a vertical mass-spring system. 58. Sketch a diagram of a horizontal mass-spring system in three positions: at both extremes of its motion, and in its equilibrium position. In each diagram, draw vector arrows representing the restoring force, velocity, and acceleration. State whether these are at a maximum or a minimum. 65. Explain to someone who has not studied physics the differences in the ways objects and waves transport energy between points on Earth. Self-assessment 66. Identify a concept or issue that you studied in this unit and would like to learn more about. 67. Learning often requires that we change the way we think about things. Which concept in this unit required the greatest change in your thinking about it? Explain how your thinking changed. 68. Which of the concepts in this unit was most helpful in explaining to you how objects interact? e TEST To check your understanding of oscillatory motion and waves, follow the eTest links at www.pearsoned.ca/school/physicssource. Unit IV Oscillatory Motion and Mechanical Waves 443 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 444 U N I T V Momentum Momentum and Impulse and Impulse Many situations and activities in the real world, such as snowboarding, involve an object gaining speed and momentum as it moves. Sometimes two or more objects collide, such as a hockey stick hitting a puck across the ice. What physics principles apply to the motion of colliding objects? How does the combination of the net force during impact and the interaction time affect an object during a collision? 444 Unit V 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 445 Unit at a Glance C H A P T E R 9 The momentum of an isolated system of interacting objects is conserved. 9.1 Momentum Is Mass Times Velocity 9.2 Impulse Is Equivalent to a Change in Momentum 9.3 Collisions in One Dimension 9.4 Collisions in Two Dimensions Unit Themes and Emphases • Change and Systems • Science and Technology Focussing Questions In this study of momentum and impulse, you will investigate the motion of objects that interact, how the velocity of a system of objects is related before and after collision, and how safety devices incorporate the concepts of momentum and impulse. As
you study this unit, consider these questions: • What characteristics of an object affect its momentum? • How are momentum and impulse related? Unit Project An Impulsive Water Balloon • By the time you complete this unit, you will have the skills to design a model of an amusement ride that is suitable for a diverse group of people. You will first need to consider acceptable accelerations that most people can tolerate. To test your model, you will drop a water balloon from a height of 2.4 m to see if it will remain intact. e WEB Research the physics concepts that apply to collisions in sports. How do athletes apply these concepts when trying to score goals for their team? How do they apply these concepts to minimize injury? Write a summary of your findings. Begin your search at www.pearsoned.ca/school/physicssource. Unit V Momentum and Impulse 445 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 446 C H A P T E R 9 Key Concepts In this chapter, you will learn about: impulse momentum Newton’s laws of motion elastic and inelastic collisions Learning Outcomes When you have completed this chapter, you will be able to: Knowledge define momentum as a vector quantity explain impulse and momentum using Newton’s laws of motion explain that momentum is conserved in an isolated system explain that momentum is conserved in one- and twodimensional interactions compare and contrast elastic and inelastic collisions Science, Technology, and Society explain that technological problems lend themselves to multiple solutions 446 Unit V The momentum of an isolated system of interacting objects is conserved. Most sports involve objects colliding during the play. Hockey checks, curling takeouts, football tackles, skeet shooting, lacrosse catches, and interceptions of the ball in soccer are examples of collisions in sports action. Players, such as Randy Ferbey, who are able to accurately predict the resulting motion of colliding objects have a better chance of helping their team win (Figure 9.1). When objects interact during a short period of time, they may experience very large forces. Evidence of these forces is the distortion in shape of an object at the moment of impact. In hockey, the boards become distorted for an instant when a player collides with them. Another evidence of these forces is a change in the motion of an object. If a goalie gloves a shot aimed at the net, you can see how the impact of the puck affects the motion of the
goalie’s hand. In this chapter, you will examine how the net force on an object and the time interval during which the force acts affect the motion of the object. Designers of safety equipment for sports and vehicles use this type of analysis when developing new safety devices. In a system of objects, you will also investigate how their respective velocities change when the objects interact with each other. Figure 9.1 Sports such as curling involve applying physics principles to change the score. Randy Ferbey, originally from Edmonton, won the Brier (Canadian) Curling Championship six times, and the World Curling Championship four times. 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 447 9-1 QuickLab 9-1 QuickLab Predicting Angles After Collisions Problem How do the masses of two objects affect the angle between their paths after they collide off centre? Materials pennies and nickels with smooth, circular outer edges marking devices for the paths (paper, tape, pencil, ruler) stack of books protractor (optional) Procedure 1 Set up the books and paper as shown in Figure 9.2. Open the cover of the book at the top of the stack for backing. Tape the paper securely to the lab bench. 2 Position one penny at the bottom of the ramp. Mark its initial position by drawing an outline on the paper. 3 Place the incoming penny at the top of the ramp as shown in Figure 9.2. Mark its initial position. 4 Predict the path each coin will take after they collide off centre. Lightly mark the predicted paths. 5 Send the coin down the ramp and mark the position of each coin after collision. Observe the relative velocities of the coins to each other both before and after collision. initial position of penny tape stack of books for ramp paper taped in position identical penny at bottom of ramp tape Figure 9.2 Think About It before after direction of motion angle between two coins after collision Figure 9.3 6 Determine if the angle between the paths after collision is less than 90, 90, or greater than 90 (Figure 9.3). 7 Repeat steps 5 and 6, but have the incoming coin collide at a different contact point with the coin at the bottom of the ramp. 8 Repeat steps 2 to 7 using a penny as the incoming coin and a nickel at the bottom of the ramp. 9 Repeat steps 2 to 7 using a nickel as the incoming coin and a penny at the bottom of the
ramp. Questions 1. What was the approximate angle formed by the paths of the two coins after collision when the coins were (a) the same mass? (b) of different mass? 2. Describe how the speeds of the two coins changed before and after collision. 3. How can you predict which coin will move faster after collision? 1. Under what circumstances could an object initially at rest be struck and move at a greater speed after collision than the incoming object? 2. Under what circumstances could a coin in 9-1 QuickLab rebound toward the ramp after collision? Discuss your answers in a small group and record them for later reference. As you complete each section of this chapter, review your answers to these questions. Note any changes to your ideas. Chapter 9 The momentum of an isolated system of interacting objects is conserved. 447 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 448 info BIT A tragic avalanche occurred during the New Year’s Eve party in the Inuit community of Kangiqsualujjuaq, formerly in Quebec and now part of Nunavut. At 1:30 a.m. on January 1, 1999, snow from the nearby 365-m mountain slope came cascading down, knocking out a wall and swamping those inside the gymnasium at the party. The snow on the mountain was initially about 1 m thick. After the avalanche was over, the school was covered with up to 3 m of snow. 9.1 Momentum Is Mass Times Velocity Snow avalanches sliding down mountains involve large masses in motion. They can be both spectacular and catastrophic (Figure 9.4). Unbalanced forces affect the motion of all objects. A mass of snow on the side of a mountain experiences many forces, such as wind, friction between the snow and the mountain, a normal force exerted by the mountain on the snow, and gravity acting vertically downward. Skiers and animals moving along the mountain slope also apply forces on the mass of snow. When a large mass of snow becomes dislodged and slides down a mountain slope due to gravity, it not only gains speed but also more mass as additional snow becomes dislodged along the downward path. info BIT Most avalanches occur on slopes that form an angle of 30 to 45 with the horizontal, although they can occur on any slope if the right conditions exist. In North America, a large avalanche may release about 230 000 m3 of snow. Figure 9.4 When the risk
of an avalanche seems imminent, ski patrols reduce the mass of snow along a mountain slope by forcing an avalanche to take place. They do this by targeting large masses of snow with guns or explosives to dislodge the snow. 448 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 449 momentum: product of the mass of an object and its velocity Momentum Is a Vector Quantity All objects have mass. The momentum, p, of an object is defined as the product of the mass of the object and its velocity. Since momentum is the product of a scalar (mass) and a vector (velocity), momentum is a vector quantity that has the same direction as the velocity. p mv Momentum has units of kilogram-metres per second (kgm/s). When you compare the momenta of two objects, you need to consider both the mass and the velocity of each object (Figure 9.5). Although two identical bowling balls, A and B, have the same mass, they do not necessarily have the same momentum. If ball A is moving very slowly, it has a very small momentum. If ball B is moving much faster than ball A, ball B’s momentum will have a greater magnitude than ball A’s because of its greater speed. Figure 9.5 The bowling ball in both photos is the same. However, the bowling ball on the left has less momentum than the ball on the right. What evidence suggests this? In real life, almost no object in motion has constant momentum because its velocity is usually not constant. Friction opposes the motion of all objects and eventually slows them down. In most instances, it is more accurate to state the instantaneous momentum of an object if you can measure its instantaneous velocity and mass. Concept Check How would the momentum of an object change if (a) the mass is doubled but the velocity remains the same? (b) the velocity is reduced to 1 of its original magnitude? 3 (c) the direction of the velocity changes from [E] to [W]? e WEB Switzerland has a long history of studying avalanches. Find out what causes an avalanche. What physical variables do avalanche experts monitor? What models are scientists working on to better predict the likelihood and severity of avalanches? Begin your search at www.pearsoned.ca/school/ physicssource. Chapter 9 The momentum of an isolated system of interacting objects is conserved. 4
49 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 450 Relating Momentum to Newton’s Second Law e WEB Research how momentum applies to cycling and other sports. Write a brief report of your findings. Begin your search at www.pearsoned.ca/school/ physicssource. The concept of momentum can be used to restate Newton’s second law. From Unit II, Newton’s second law states that an external non-zero net force acting on an object is equal to the product of the mass of the object ma. Acceleration is defined as the rate of and its acceleration, F net v change of velocity. For constant acceleration, a i. If you or t t i for a in Newton’s second law, you get substitute t v v v v f f F net f ma v v i m t mv mv i t f The quantity mv is momentum. So the equation can be written as F net p p i t f p where F net is constant t Written this way, Newton’s second law relates the net force acting on an object to its rate of change of momentum. It is interesting to note that Newton stated his second law of motion in terms of the rate of change of momentum. It may be worded as: An external non-zero net force acting on an object is equal to the rate of change of momentum of the object. F net p where F net is constant t e SIM For a given net force, learn how the mass of an object affects its momentum and its final velocity. Follow the eSim links at www.pearsoned.ca/school/ physicssource. This form of Newton’s law has some major advantages over the way it was ma only applies to situations where written in Unit II. The equation F net the mass is constant. However, by using the concept of momentum, it is possible to derive another form for Newton’s second law that applies to situations where the mass, the velocity, or both the mass and velocity are changing, such as an accelerating rocket where the mass is decreasing as fuel is being burned, while the velocity is increasing. In situations where the net force changes over a time interval, the average net force is equal to the rate of change of momentum of the object. F netave p t 450 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/
24/08 2:43 PM Page 451 In Example 9.1, a person in a bumper car is moving at constant velocity. Since both the person and the car move together as a unit, both objects form a system. The momentum of the system is equal to the total mass of the system times the velocity of the system. Example 9.1 W E v Figure 9.6 A 180-kg bumper car carrying a 70-kg driver has a constant velocity of 3.0 m/s [E]. Calculate the momentum of the cardriver system. Draw both the velocity vector and the momentum vector. Given mc 180 kg md 70 kg v 3.0 m/s [E] Required momentum of system ( p) velocity and momentum vector diagrams Analysis and Solution The driver and bumper car are a system because they move together as a unit. Find the total mass of the system. mT mc md 180 kg 70 kg 250 kg The momentum of the system is in the direction of the velocity of the system. So use the scalar form of p mv to find the magnitude of the momentum. p mTv (250 kg)(3.0 m/s) 7.5 102 kgm/s Draw the velocity vector to scale (Figure 9.7). W E 1.0 m/s v 3.0 m/s Figure 9.7 Practice Problems 1. A 65-kg girl is driving a 535-kg snowmobile at a constant velocity of 11.5 m/s [60.0 N of E]. (a) Calculate the momentum of the girl-snowmobile system. (b) Draw the momentum vector for this situation. 2. The combined mass of a bobsled and two riders is 390 kg. The sled-rider system has a constant momentum of 4.68 103 kgm/s [W]. Calculate the velocity of the sled. Answers 1. (a) 6.90 103 kgm/s [60.0 N of E] (b) N 3 kgm/s p 6.90 10 60.0° 2. 12.0 m/s [W] 2000 kgm/s E Chapter 9 The momentum of an isolated system of interacting objects is conserved. 451 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 452 Draw the momentum vector to scale (Figure 9.8). W E 100 kgm/s p 7.5 102 kgm/
s Figure 9.8 Paraphrase The momentum of the car-driver system is 7.5 102 kgm/s [E]. Using Proportionalities to Solve Momentum Problems Example 9.2 demonstrates how to solve momentum problems using proportionalities. In this example, both the mass and velocity of an object change. Example 9.2 An object has a constant momentum of 2.45 102 kgm/s [N]. Determine the momentum of the object if its mass decreases to 1 of its original value 3 and an applied force causes the speed to increase by exactly four times. The direction of the velocity remains the same. Explain your reasoning. Practice Problems 1. Many modern rifles use bullets that have less mass and reach higher speeds than bullets for older rifles, resulting in increased accuracy over longer distances. The momentum of a bullet is initially 8.25 kgm/s [W]. What is the momentum if the speed of the bullet increases by a factor of 3 and 2 its mass decreases by a factor of 3? 4 2. During one part of the liftoff of a model rocket, its momentum increases by a factor of four while its mass is halved. The velocity of the rocket is initially 8.5 m/s [up]. What is the final velocity during that time interval? Answers 1. 9.28 kgm/s [W] 2. 68 m/s [up] 452 Unit V Momentum and Impulse Analysis and Solution From the equation p mv, p m and p v. Figure 9.9 represents the situation of the problem. before after N S v 4 v Figure 9.9 p 2.45 102 kgm/s [N] p? p 1 m 3 and p 4v Calculate the factor change of p. 4 4 1 3 3 Calculate p. p 4 4 (2.45 102 kgm/s) 3 3 3.27 102 kgm/s The new momentum will be 3.27 102 kgm/s [N]. 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 453 9.1 Check and Reflect 9.1 Check and Reflect Knowledge 1. (a) Explain, in your own words, the concept of momentum. (b) State the SI units of momentum. 2. Explain why momentum is a vector quantity. 3. How is momentum related to Newton’s second law? 4. Explain why stating Newton’s second law
in terms of momentum is more useful than stating it in terms of acceleration. 5. Explain, in your own words, the difference between momentum and inertia. 6. Provide three examples of situations in which (a) velocity is the dominant factor affecting the momentum of an object (b) mass is the dominant factor affecting the momentum of an object. 12. Draw a momentum vector diagram to represent a 425-g soccer ball flying at 18.6 m/s [214]. 13. At what velocity does a 0.046-kg golf ball leave the tee if the club has given the ball a momentum of 3.45 kgm/s [S]? 14. (a) A jet flies west at 190 m/s. What is the momentum of the jet if its total mass is 2250 kg? (b) What would be the momentum of the jet if the mass was 3 of its original 4 value and the speed increased to 6 of 5 its original value? 15. The blue whale is the largest mammal ever to inhabit Earth. Calculate the mass of a female blue whale if, when alarmed, it swims at a velocity of 57.0 km/h [E] and has a momentum of 2.15 106 kgm/s [E]. Applications Extensions 7. A Mexican jumping bean moves because an insect larva inside the shell wiggles. Would it increase the motion to have the mass of the insect greater or to have the mass of the shell greater? Explain. 8. What is the momentum of a 6.0-kg bowling ball with a velocity of 2.2 m/s [S]? 9. The momentum of a 75-g bullet is 9.00 kgm/s [N]. What is the velocity of the bullet? 10. (a) Draw a momentum vector diagram for a 4.6-kg Canada goose flying with a velocity of 8.5 m/s [210]. (b) A 10.0-kg bicycle and a 54.0-kg rider both have a velocity of 4.2 m/s [40.0 N of E]. Draw momentum vectors for each mass and for the bicycle-rider system. 11. A hockey puck has a momentum of 3.8 kgm/s [E]. If its speed is 24 m/s, what is the mass of the puck? 16. A loaded transport truck with a mass of 38 000 kg is travelling at 1.20 m/s [W]. What will be the
velocity of a 1400-kg car if it has the same momentum? 17. Summarize the concepts and ideas associated with momentum using a graphic organizer of your choice. See Student References 4: Using Graphic Organizers on pp. 869–871 for examples of different graphic organizers. Make sure that the concepts and ideas are clearly presented and appropriately linked. e TEST To check your understanding of momentum, follow the eTest links at www.pearsoned.ca/school/ physicssource. Chapter 9 The momentum of an isolated system of interacting objects is conserved. 453 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 454 info BIT Legendary stunt person Dar Robinson broke nine world records and made 21 “world firsts” during his career. One world record was a cable jump from the CN Tower in 1980 for the film The World’s Most Spectacular Stuntman. While tied to a 3-mm steel cable, Robinson jumped more than 366 m and stopped only a short distance above the ground. 9.2 Impulse Is Equivalent to a Change in Momentum Stunt people take the saying, “It isn’t the fall that hurts, it’s the sudden stop at the end,” very seriously. During the filming of a movie, when a stunt person jumps out of a building, the fall can be very dangerous. To minimize injury, stunt people avoid a sudden stop when landing by using different techniques to slow down more gradually out of sight of the cameras. These techniques involve reducing the peak force required to change their momentum. Sometimes stunt people jump and land on a net. Other times, they may roll when they land. For more extreme jumps, such as from the roof of a tall building, a huge oversized, but slightly under-inflated, air mattress may be used (Figure 9.10). A hidden parachute may even be used to slow the jumper to a safer speed before impact with the surface below. Despite all these precautions, injuries occur as stunt people push the limits of what is possible in their profession. Designers of safety equipment know that a cushioned surface can reduce the severity of an impact. Find out how the properties of a landing surface affect the shape of a putty ball that is dropped from a height of 1 m by doing 9-2 QuickLab. Figure 9.10 The thick mattress on the ground provides a protective cushion for the stunt person when he lands. Why
do you think the hardness of a surface affects the extent of injury upon impact? 454 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 455 9-2 QuickLab 9-2 QuickLab Softening the Hit Problem How is the change in the shape of a putty ball upon impact related to the structure of the landing surface? Materials putty-type material closed cell foam or felt pad urethane foam pad or pillow waxed paper or plastic wrap metre-stick Procedure 1 Choose three surfaces of varying softness onto which to drop a putty ball. One of the surfaces should be either a lab bench or the floor. Cover each surface with some waxed paper or plastic wrap to protect it. 2 Knead or work the putty until you can form three pliable balls of equal size. 3 Measure a height of 1 m above the top of each surface. Then drop the balls, one for each surface (Figure 9.11). 4 Draw a side-view sketch of each ball after impact. 1 m 1 m Figure 9.11 Questions 1. Describe any differences in the shape of the putty balls after impact. 2. How does the amount of cushioning affect the deformation of the putty? 3. Discuss how the softness of the landing surface might be related to the time required for the putty ball to come to a stop. Justify your answer with an analysis involving the kinematics equations. Force and Time Affect Momentum In 9-2 QuickLab, you found that the softer the landing surface, the less the shape of the putty ball changed upon impact. The more cushioned the surface, the more the surface became indented when the putty ball collided with it. In other words, the softer and more cushioned landing surface provided a greater stopping distance for the putty ball. Suppose you label the speed of the putty ball at the instant it touches the landing surface vi, and the speed of the putty ball after the impact vf. 0. So the greater the For all the landing surfaces, vi was the same and vf stopping distance, the longer the time required for the putty ball to stop (Figure 9.12). In other words, the deformation of an object is less when the stopping time is increased. vf 0 for surfaces A and B viA viB tB tA viA dA viB dB harder landing surface (
A) more cushioned landing surface (B) Figure 9.12 The stopping distance of the putty ball was greater for the more cushioned landing surface (B). So the time interval of interaction was greater on surface B. Chapter 9 The momentum of an isolated system of interacting objects is conserved. 455 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 456 PHYSICS INSIGHT To visualize the effect of how Fnet and t can vary but p remains the same, consider the effect of changing two numbers being multiplied together to give the same product. 3 12 36 6 6 36 18 2 36 As the first number increases, the second number decreases in order to get the same product. Project LINK How will the net force and time interval affect the water balloon when it is brought to a stop? What types of protective material will you use to surround the water balloon and why? Apart from the different stopping times, what other differences were there between the drops that would have affected the shape of the putty ball upon impact? The answer to this question requires looking at Newton’s second law written in terms of momentum. From the previous section, the general form of Newton’s second law states that the rate of change of momentum is equal to the net force acting on an object. p t F net If you multiply both sides of the equation by t, you get F net t p For all the landing surfaces, since m, vi (at the first instant of impact), and vf (after the impact is over) of the putty ball were the same, pi was the t p, so the 0. So p was the same for all drops. But F same and pf net product of net force and stopping time was the same for all drops. From Figure 9.12 on page 455, the stopping time varied depending on the amount of cushioning provided by the landing surface. If the stopping time was short, as on a hard landing surface, the magnitude of the net force acting on the putty ball was large. Similarly, if the stopping time was long, as on a very cushioned landing surface, the magnitude of the net force acting on the putty ball was small. This analysis can be used to explain why the putty ball became more deformed when it landed on a hard surface. To minimize changes to the shape of an object being dropped, it is important to minimize Fnet required to stop the object, and this happens when
you maximize t (Figure 9.13). It is also important to note where net acts on a large area, the result of the impact will have a F net acts. If F net acts on only one different effect on the shape of the object than if F small part on the surface of the object. (a) direction of motion (b) direction of motion concrete p Fnet t p Fnet t bed of straw Figure 9.13 Identical eggs are dropped from a height of 2 m onto a concrete floor or a pile of straw. Although p is the same in both situations, the magnitude of the net force acting on the egg determines whether or not the egg will break. Concept Check In 9-2 QuickLab, was the momentum of the putty ball at the first instant of impact zero? Explain your reasoning. 456 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 457 Impulse Is the Product of Net Force and Interaction Time impulse: product of the net force on an object and the time interval during an interaction. Impulse causes a change in the momentum of the object. e SIM Learn how the mass and acceleration of an object affect its change in momentum. Follow the eSim links at www.pearsoned.ca/school/ physicssource. t p, the product of net force and interaction time In the equation F net is called impulse. Impulse is equivalent to the change in momentum that an object experiences during an interaction. Every time a net force acts on an object, the object is provided with an impulse because the force is applied for a specific length of time. F net If you substitute the definition of momentum, p mv, the equation t p becomes t (mv) F net If m is constant, then the only quantity changing on the right-hand side of the equation is v. So the equation becomes F net t mv So impulse can be calculated using either equation: F net t p or F net t mv The unit of impulse is the newton-second (Ns). From Unit II, a newton is defined as 1 N 1 kgm/s2. If you substitute the definition of a newton in the unit newton-seconds, you get (s) 1 Ns 1 m kg 2 s m kg 1 s which are the units for momentum. So the units on both sides of the impulse equation are equivalent. Since force
is a vector quantity, impulse is also a vector quantity, and the impulse is in the same direction as the net force. To better understand how net force and interaction time affect the change in momentum of an object, do 9-3 Design a Lab. 9-3 Design a Lab 9-3 Design a Lab Providing Impulse The Question What is the effect of varying either the net force or the interaction time on the momentum of an object? Design and Conduct Your Investigation State a hypothesis to answer the question using an “if/then” statement. Then design an experiment to measure the change in momentum of an object. First vary Fnet, then repeat the experiment and vary t instead. List the materials you will use, as well as a detailed procedure. Check the procedure with your teacher and then do the investigation. To find the net force, you may need to find the force of friction and add it, using vectors, to the applied force. The force of kinetic friction is the minimum force needed to keep an object moving at constant velocity once the object is in motion. Analyze your data and form conclusions. How well did your results agree with your hypothesis? Compare your results with those of other groups in your class. Account for any discrepancies. e LAB For a probeware activity, go to www.pearsoned.ca/school/physicssource. Chapter 9 The momentum of an isolated system of interacting objects is conserved. 457 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 458 Example 9.3 demonstrates how, for the same impulse, varying the interaction time affects the average net force on a car during a front-end collision (the net force on the car is not constant). Example 9.3 W E Practice Problems 1. Two people push a car for 3.64 s with a combined net force of 200 N [S]. (a) Calculate the impulse provided to the car. (b) If the car has a mass of 1100 kg, what will be its change in velocity? 2. A dog team pulls a 400-kg sled that has begun to slide backward. In 4.20 s, the velocity of the sled changes from 0.200 m/s [backward] to 1.80 m/s [forward]. Calculate the average net force the dog team exerts on the sled. Answers 1. (a) 728 Ns [S], (b) 0.662 m/s
[S] 2. 190 N [forward] v Figure 9.14 To improve the safety of motorists, modern cars are built so the front end crumples upon impact. A 1200-kg car is travelling at a constant velocity of 8.0 m/s [E] (Figure 9.14). It hits an immovable wall and comes to a complete stop in 0.25 s. (a) Calculate the impulse provided to the car. (b) What is the average net force exerted on the car? (c) For the same impulse, what would be the average net force exerted on the car if it had a rigid bumper and frame that stopped the car in 0.040 s? Given m 1200 kg (a) and (b) t 0.25 s (c) t 0.040 s v i 8.0 m/s [E] info BIT Some early cars were built with spring bumpers that tended to bounce off whatever they hit. These bumpers were used at a time when people generally travelled at much slower speeds. For safety reasons, cars today are built to crumple upon impact, not bounce. This results in a smaller change in momentum and a reduced average net force on motorists. The crushing also increases the time interval during the impulse, further decreasing the net force on motorists. Required (a) impulse provided to car (b) and (c) average net force on car ( F netave) Analysis and Solution When the car hits the wall, the final velocity of the car is zero. v f 0 m/s During each collision with the wall, the net force on the car is not constant, but the mass of the car remains constant. (a) Use the equation of impulse to calculate the impulse provided to the car. f F netave v i) t mv m(v (1200 kg)[0 (8.0 m/s)] (1200 kg)(8.0 m/s) 9.6 103 kgm/s impulse 9.6 103 Ns [W] 458 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 459 netave. For (b) and (c), substitute the impulse from part (a) and solve for F F netave t 9.6 103 Ns 03 Ns 9.6 1 t F netave (b) F netave 9.6 0.25 3 Ns s 10 (
c) F netave m kg 3.8 104 2 s 3.8 104 N 9.6 04 0. 3 Ns 1 0 0 s m kg 2.4 105 2 s 2.4 105 N PHYSICS INSIGHT netave is in the opposite F direction to the initial momentum of the car, because from Newton’s third law, the wall is exerting a force directed west on the car. F netave 3.8 104 N [W] F netave 2.4 105 N [W] Paraphrase and Verify (a) The impulse provided to the car is 9.6 103 Ns [W]. The average net force exerted by the wall on the car is (b) 3.8 104 N [W] when it crumples, and (c) 2.4 105 N [W] when it is rigid. The change in momentum is the same in parts (b) and (c), but the time intervals are different. So the average net force is different in both netave on the car with the rigid frame is situations. The magnitude of F more than 6 times greater than when the car crumples. Impulse Can Be Calculated Using a Net Force-Time Graph One way to calculate the impulse provided to an object is to graph the net force acting on the object as a function of the interaction time. Suppose a net force of magnitude 30 N acts on a model rocket for 0.60 s during liftoff (Figure 9.15). From the net force-time graph in Figure 9.16, the product t is equal to the magnitude of the impulse. But this product is also Fnet the area under the graph. Magnitude of Net Force vs. Interaction Time for a Model Rocket Fnet ) 50 40 30 20 10 0 0 0.10 0.20 0.30 Time t (s) 0.40 0.50 0.60 Figure 9.15 What forces act on the rocket during liftoff? Figure 9.16 Magnitude of net force as a function of interaction time for a model rocket. The area under the graph is equal to the magnitude of the impulse provided to the rocket. Chapter 9 The momentum of an isolated system of interacting objects is conserved. 459 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 460 The magnitude of the impulse provided to the rocket is magnitude of impulse Fnet t (30 N)(0.60 s)
18 Ns In other words, the area under a net force-time graph gives the magnitude of the impulse. Note that a net force acting over a period of time causes a change in momentum. When Fnet is not constant, you can still calculate the impulse by finding the area under a net force-time graph. Figure 9.17 shows the magnitude of the net force exerted by a bow on an arrow during the first part of its release. The magnitude of the net force is greatest at the beginning and decreases linearly with time Magnitude of Net Force vs. Interaction Time for an Arrow Shot with a Bow 200 150 100 50 0 0 10 20 30 Time t (ms) 40 50 Figure 9.17 Magnitude of net force as a function of interaction time for an arrow shot with a bow. In this case, the area under the graph could be divided into a rectangle and a triangle or left as a trapezoid (Figure 9.18). So the magnitude of the impulse provided to the arrow is 1 (a b)(h) magnitude of impulse 2 1 (100 N 200 N)(0.050 s) 2 7.5 Ns Sometimes two net force-time graphs may appear different but the magnitude of the impulse is the same in both cases. Figure 9.19 (a) shows a graph where Fnet is much smaller than in Figure 9.19 (b). The value of t is different in each case, but the area under both graphs is the same. So the magnitude of the impulse is the same in both situations. (a 30 25 20 15 10 5 0 0 Magnitude of Net Force vs. Interaction Time (b) 2.00 4.00 6.00 Time t (s 30 25 20 15 10 5 0 0 Magnitude of Net Force vs. Interaction Time 2.00 4.00 6.00 Time t (s) Figure 9.19 What other graph could you draw that has the same magnitude of impulse? info BIT The area of a trapezoid is equal to 1 (a b)(h). 2 a h b Figure 9.18 460 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 461 Effect of a Non-linear Net Force on Momentum In real life, many interactions occur during very short time intervals (Figure 9.20). If you tried to accurately measure the net force, you would find that it is difficult, if not impossible,
to do. In addition, the relationship between Fnet and t is usually non-linear, because Fnet increases from zero to a very large value in a short period of time (Figure 9.21). Magnitude of Net Force vs. Interaction Time Fnetave ) Time t (s) Figure 9.21 The average net force gives some idea of the maximum instantaneous net force that an object actually experienced during impact. Figure 9.20 When a baseball bat hits a ball, what evidence demonstrates that the force during the interaction is very large? What evidence demonstrates that the force on the ball changes at the instant the ball and bat separate? From a practical point of view, it is much easier to measure the interaction time and the overall change in momentum of an object during an interaction, rather than Fnet. In this case, the equation of Newton’s second law expressed in terms of momentum is F netave p t and the equation of impulse is F netave t p or F netave t mv In all the above equations, F netave on the object during the interaction. represents the average net force that acted In Example 9.4, a golf club strikes a golf ball and an approximation of the net force-time graph is used to simplify the calculations for impulse. In reality, the net force-time graph for such a situation would be similar to that shown in Figure 9.21. info BIT The fastest recorded speed for a golf ball hit by a golf club is 273 km/h. Chapter 9 The momentum of an isolated system of interacting objects is conserved. 461 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 462 Example 9.4 A golfer hits a long drive sending a 45.9-g golf ball due east. Figure 9.22 shows an approximation of the net force as a function of time for the collision between the golf club and the ball. (a) What is the impulse provided to the ball? (b) What is the velocity of the ball Magnitude of Net Force vs. Interaction Time for a Golf Ball Being Hit by a Golf Club 6000 5000 4000 3000 2000 1000 ).2 0.6 0.4 Time t (ms) 0.8 1.0 1.2 Figure 9.22 W E at the moment the golf club and ball separate? Given m 45.9 g ti 1.1 ms tf Fneti 5000 N Fnetf Fnetmax 0.1
ms 0 N 0 N Required (a) impulse provided to ball (b) velocity of ball after impact (v f ) Analysis and Solution The impulse and velocity after impact are in the east direction since the golfer hits the ball due east. (a) t tf ti 1.1 ms 0.1 ms 1.0 ms or 1.0 103 s Fnet Figure 9.23 magnitude of impulse area under net force-time graph 1 (t)(Fnetmax) 2 1 (1.0 103 s)(5000 N) 2 2.5 Ns impulse 2.5 Ns [E] (b) Impulse is numerically equal to mv or m(v v i). f i But v 0 m/s So, impulse m (v 2.5 Ns mv f 0) v f f Ns 2.5 m m kg s 2.50 2 s g (45.9 g) k 1 g 0 0 10 54 m/s Paraphrase (a) The impulse provided to the ball is 2.5 Ns [E]. (b) The velocity of the ball after impact is 54 m/s [E]. Practice Problems 1. (a) Draw a graph of net force as a function of time for a 0.650-kg basketball being shot. During the first 0.15 s, Fnet increases linearly from 0 N to 22 N. During the next 0.25 s, Fnet decreases linearly to 0 N. (b) Using the graph, calculate the magnitude of the impulse provided to the basketball. (c) What is the speed of the ball when it leaves the shooter’s hands? 2. (a) A soccer player heads the ball with an average net force of 21 N [W] for 0.12 s. Draw a graph of the average net force on the ball as a function of time. Assume that Fnetave is constant during the interaction. (b) Calculate the impulse provided to the soccer ball. (c) The impulse changes the velocity of the ball from 4.0 m/s [E] to 2.0 m/s [W]. What is the mass of the ball? Answers 1. (a 25 20 15 10 5 0 Magnitude of Net Force vs. Interaction Time for a Basketball Being Shot 0 0.10 0.20 0.30 Time t (s) 0.40 0.50 (b) 4.4 Ns, (c
) 6.8 m/s 2. (a 30 25 20 15 10 5 0 Magnitude of Average Net Force vs. Interaction Time for a Soccer Ball Being Hit 0 0.03 0.06 Time t (s) 0.09 0.12 (b) 2.5 Ns [W], (c) 0.42 kg 462 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 463 The Design of Safety Devices Involves Varying Fnetave and t Many safety devices are based on varying both the average net force acting on an object and the interaction time for a given impulse. Suppose you attached a sled to a snowmobile with a rope hitch. As long as the sled is accelerating along a horizontal surface or is being pulled uphill, there is tension in the rope because the snowmobile applies a force on the sled (Figure 9.24). If the driver in Figure 9.24 (a) brakes suddenly to slow down, the momentum of the snowmobile changes suddenly. However, the sled continues to move in a straight line until friction eventually slows it down to a stop. In other words, the only way that the momentum of the f acts for a long enough period of time. sled changes noticeably is if F (a) (b FT 0 a FT 0 θ Figure 9.24 (a) A snowmobile accelerating along a horizontal surface, and (b) the same snowmobile either moving at constant speed or accelerating uphill. In both (a) and (b), the tension in the rope is not zero. Suppose the snowmobile driver is heading downhill and applies the brakes suddenly as in Figure 9.25 (a). F g will cause the sled to accelerate downhill as shown in Figure 9.25 (b). The speed of the sled could become large enough to overtake the snowmobile, bump into it, or tangle the rope. (a) n o f c ti o n o ti o m d ir e FT 0 θ (b) a Ff on m1 m1 Fg θ a m2 Ff on m2 Fg θ Figure 9.25 (a) The snowmobile is braking rapidly, and the tension in the rope is zero. (b) The free-body diagrams for the snowmobile and sled only show forces parallel to the incline. Chapter 9 The momentum of an isolated system of interacting objects is conserved. 463
09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 464 The driver can change the momentum of the snowmobile suddenly by using the brakes. But, as before, the only way that the momentum of the f acts for a long enough time interval. sled can eventually become zero is if F With experience, a driver learns to slow down gradually so that a towed sled remains in its proper position. Some sleds are attached to snowmobiles using a metal tow bar, which alleviates this problem (Figure 9.26). Since the tow bar can never become slack like a rope, the sled always remains a fixed distance from the snowmobile. Tow bars usually have a spring mechanism that increases the time during which a force can be exerted. So if the driver brakes or changes direction suddenly, the force exerted by the snowmobile on the sled acts for a longer period of time. Compared to a towrope, the spring mechanism in the tow bar can safely cause the momentum of the sled to decrease in a shorter period of time. Figure 9.26 A rigid tow bar with a spring mechanism provides the impulse necessary to increase or decrease the momentum of a towed sled. e WEB During takeoff, the magnitude of Earth’s gravitational field changes as a rocket moves farther away from Earth’s surface. The mass of a rocket also changes because it is burning fuel to move upward. Research how impulse and momentum apply to the design and function of rockets and thrust systems. Write a brief report of your findings, including diagrams where appropriate. Begin your search at www.pearsoned.ca/ school/physicssource. Safety devices in vehicles are designed so that, for a given impulse such as in a collision, the interaction time is increased, thereby reducing the average net force. This is achieved by providing motorists with a greater distance to travel, which increases the time interval required to stop the motion of the motorist. Three methods are used to provide this extra distance and time: • The dashboard is padded and the front end of the vehicle is designed to crumple. • The steering column telescopes to collapse, providing an additional 15–20 cm of distance for the driver to travel forward. • The airbag is designed to leak after inflation so that the fully inflated bag can decrease in thickness over time from about 30 cm to about 10 cm. In fact, an inflated airbag distributes the net force experienced during a collision over the motorist’s chest and head
. By spreading the force over a greater area, the magnitude of the average net force at any one point on the motorist’s body is reduced, lowering the risk of a major injury. 464 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 465 A similar reasoning applies to the cushioning in running shoes and the padding in helmets and body pads used in sports (Figure 9.27). For a given impulse, all these pieces of equipment increase the interaction time and decrease the average net force. (a) (b) Figure 9.27 Padding in sports equipment helps reduce the risk of major injuries, because for a given impulse, the interaction time is increased and the average net force on the body part is reduced. (a) Team Canada in the World Women’s hockey tournament in Sweden, 2005. (b) Calgary Stampeders (in red) playing against the B.C. Lions in 2005. The effect of varying the average net force and the interaction time can be seen with projectiles. A bullet fired from a pistol with a short barrel does not gain the same momentum as another identical bullet fired from a rifle with a long barrel, assuming that each bullet experiences the same average net force (Figure 9.28). In the gun with the shorter barrel, the force from the expanding gases acts for a shorter period of time. So the change in momentum of the bullet is less. t p t p Figure 9.28 For the same average net force on a bullet, a gun with a longer barrel increases the time during which this force acts. So the change in momentum is greater for a bullet fired from a long-barrelled gun. Chapter 9 The momentum of an isolated system of interacting objects is conserved. 465 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 466 Improved Sports Performance Involves Varying Fnetave and t In baseball, a skilled pitcher knows how to vary both the net force acting on the ball and the interaction time, so that the ball acquires maximum velocity before it leaves the pitcher’s hand (Figure 9.29). To exert the maximum possible force on the ball, a pitcher uses his arms, torso, and legs to propel the ball forward. To maximize the time he can exert that force, the pitcher leans back using a windup and then takes a long step forward. This way, his hand
can be in contact with the ball for a longer period of time. The combination of the greater net force and the longer interaction time increases the change in momentum of the ball. Figure 9.29 When a pitcher exerts a force on the ball during a longer time interval, the momentum of a fastball increases even more. In sports such as hockey, golf, and tennis, coaches emphasize proper “follow through.” The reason is that it increases the time during which the puck or ball is in contact with the player’s stick, club, or racquet. So the change in momentum of the object being propelled increases. A similar reasoning applies when a person catches a ball. In this case, a baseball catcher should decrease the net force on the ball so that the ball doesn’t cause injury and is easier to hold onto. Players soon learn to do this by letting their hands move with the ball. For the same impulse, the extra movement with the hands results in an increased interaction time, which reduces the net force. This intentional flexibility when catching is sometimes referred to as having “soft hands,” and it is a great compliment to a football receiver. Hockey goalies allow their glove hand to fly back when snagging a puck to reduce the impact and allow them a better chance of keeping the puck in their glove. Boxers are also taught to “roll with the punch,” because if they move backward when hit, it increases the interaction time and decreases the average net force of an opponent’s blow. 466 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 467 9.2 Check and Reflect 9.2 Check and Reflect Knowledge 1. (a) What quantities are used to calculate impulse? (b) State the units of impulse. 2. How are impulse and momentum related? 3. What graph could you use to determine the impulse provided to an object? Explain how to calculate the impulse using the graph. 4. What is the effect on impulse if (a) the time interval is doubled? (b) the net force is reduced to 1 of its 3 original magnitude? 5. Even though your mass is much greater than that of a curling stone, it is dangerous for a moving stone to hit your feet. Explain why. Applications 6. Using the concept of impulse, explain how a karate expert can break a board. 7. (a) From the graph
below, what is the magnitude of the impulse provided to a 48-g tennis ball that is served due south? (b) What is the velocity of the ball when the racquet and ball separate? Magnitude of Net Force vs. Interaction Time for a Tennis Ball Being Hit by a Racquet ) 1000 900 800 700 600 500 400 300 200 100 0 0.0 1.0 2.0 3.0 Time t (ms) 4.0 5.0 6.0 9. During competitive world-class events, a four-person bobsled experiences an average net force of magnitude 1390 N during the first 5.0 s of a run. (a) What will be the magnitude of the impulse provided to the bobsled? (b) If the sled has the maximum mass of 630 kg, what will be the speed of the sled? 10. An advertisement for a battery-powered 25-kg skateboard says that it can carry an 80-kg person at a speed of 8.5 m/s. If the skateboard motor can exert a net force of magnitude 75 N, how long will it take to attain that speed? 11. Whiplash occurs when a car is rear-ended and either there is no headrest or the headrest is not properly adjusted. The torso of the motorist is accelerated by the seat, but the head is jerked forward only by the neck, causing injury to the joints and soft tissue. What is the average net force on a motorist’s neck if the torso is accelerated from 0 to 14.0 m/s [W] in 0.135 s? The mass of the motorist’s head is 5.40 kg. Assume that the force acting on the head is the same magnitude as the force on the torso. 12. What will be the change in momentum of a shoulder-launched rocket that experiences a thrust of 2.67 kN [W] for 0.204 s? Extensions 13. Experienced curlers know how to safely stop a moving stone. What do they do and why? 14. Research one safety device used in sports that applies the concept of varying Fnetave and t for a given impulse to prevent injury. Explain how the variables that affect impulse are changed by using this device. Begin your search at www.pearsoned.ca/school/physicssource. 8. What will be the magnitude of the impulse generated by a slapshot when an average
net force of magnitude 520 N is applied to a puck for 0.012 s? e TEST To check your understanding of impulse, follow the eTest links at www.pearsoned.ca/school/ physicssource. Chapter 9 The momentum of an isolated system of interacting objects is conserved. 467 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 468 info BIT The horns of a bighorn ram can account for more than 10% of its mass, which is about 125 kg. Rams collide at about 9 m/s, and average about 5 collisions per hour. Mating contests between any two rams may last for more than 24 h in total. 9.3 Collisions in One Dimension During mating season each fall, adult bighorn rams compete for supremacy in an interesting contest. Two rams will face each other, rear up, and then charge, leaping into the air to butt heads with tremendous force (Figure 9.30). Without being consciously aware of it, each ram attempts to achieve maximum momentum before the collision, because herd structure is determined by the outcome of the contest. Often, rams will repeat the head-butting interaction until a clear winner is determined. While most other mammals would be permanently injured by the force experienced during such a collision, the skull and brain structure of bighorn sheep enables them to emerge relatively undamaged from such interactions. In the previous section, many situations involved an object experiencing a change in momentum, or impulse, because of a collision with another object. When two objects such as bighorn rams collide, what relationship exists among the momenta of the objects both before and after collision? In order to answer this question, first consider one-dimensional collisions involving spheres in 9-4 QuickLab. Figure 9.30 By lunging toward each other, these bighorn rams will eventually collide head-on. During the collision, each ram will be provided with an impulse. 468 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 469 9-4 QuickLab 9-4 QuickLab Observing Collinear Collisions Problem What happens when spheres collide in one dimension? before direction of motion after Materials one set of four identical ball bearings or marbles (set A) a second set of four identical ball bearings or marbles of double the mass (set B) a third set of
four identical ball bearings or marbles of half the mass (set C) 1-m length of an I-beam curtain rod or two metre-sticks with smooth edges masking tape Procedure 1 Lay the curtain rod flat on a bench to provide a horizontal track for the spheres. Tape the ends of the rod securely. If you are using metre-sticks, tape them 5 mm apart to form a uniform straight horizontal track. 2 Using set A, place three of the spheres tightly together at the centre of the track. 3 Predict what will happen when one sphere of set A moves along the track and collides with the three stationary spheres. Figure 9.31 4 Test your prediction. Ensure that the spheres remain on the track after collision. Record your observations using diagrams similar to Figure 9.31. 5 Repeat steps 2 to 4, but this time use set B, spheres of greater mass. 6 Repeat steps 2 to 4, but this time use set C, spheres of lesser mass. 7 Repeat steps 2 to 4 using different numbers of stationary spheres. The stationary spheres should all be the same mass, but the moving sphere should be of a different mass in some of the trials. Questions 1. Describe the motion of the spheres in steps 4 to 6. 2. Explain what happened when (a) a sphere of lesser mass collided with a number of spheres of greater mass, and (b) a sphere of greater mass collided with a number of spheres of lesser mass. In 9-4 QuickLab, for each set of spheres A to C, when one sphere hit a row of three stationary ones from the same set, the last sphere in the row moved outward at about the same speed as the incoming sphere. But when one sphere from set A hit a row of spheres from set B, the last sphere in the row moved outward at a much slower speed than the incoming sphere, and the incoming sphere may even have rebounded. When one sphere from set A hit a row of spheres from set C, the last sphere in the row moved outward at a greater speed than the incoming sphere, and the incoming sphere continued moving forward. To analyze these observations, it is important to first understand what a collision is. A collision is an interaction between two objects in which a force acts on each object for a period of time. In other words, the collision provides an impulse to each object. e MATH Explore how the masses of two colliding objects affect their velocities just after collision. Follow the eMath links at www.pears
oned.ca/school/ physicssource. collision: an interaction between two objects where each receives an impulse Chapter 9 The momentum of an isolated system of interacting objects is conserved. 469 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 470 system: two or more objects that interact with each other Systems of Objects in Collisions Each trial in 9-4 QuickLab involved two or more spheres colliding with each other. A group of two or more objects that interact is called a system. You encountered the concept of a system in Unit III in the context of energy. For each system in 9-4 QuickLab, the total mass remained constant because the mass of each sphere did not change as a result of the interaction. However, friction was an external force that acted on the system. For example, in steps 4 to 6 of 9-4 QuickLab, you likely observed that the speed of the sphere moving outward was a little less than the speed of the incoming sphere. In real life, a system of colliding objects is provided with two impulses: one due to external friction and the other due to the actual collision (Figure 9.32). External friction acts before, during, and after collision. The second impulse is only present during the actual collision. Since the actual collision time is very short, the impulse due to external friction during the collision is relatively small. before during after viA A 0 viB B Ff on A Ff on B 0 vfA A vfB B Ff on A Ff on B FB on A A B FA on B Ff on A tc ts Figure 9.32 External friction acts throughout the entire time interval of the interaction ts. But the action-reaction forces due to the objects only exist when the objects actually collide, and these forces only act for time interval tc. If you apply the form of Newton’s second law that relates net force to momentum to analyze the motion of a system of objects, you get net)sys (F p sys where p t sys is the momentum of the system The momentum of a system is defined as the sum of the momenta of all the objects in the system. So if objects A, B, and C form a system, the momentum of the system is p sys p A p B p C In the context of momentum, when the mass of a system is constant and no external net force acts on the system, the system is isolated. So 0.
In 9-5 Inquiry Lab, a nearly isolated system of objects is net)sys (F involved in a one-dimensional collision. Find a quantitative relationship for the momentum of such a system in terms of momenta before and after collision. 470 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 471 9-5 Inquiry Lab 9-5 Inquiry Lab Relating Momentum Just Before and Just After a Collision Required Skills Initiating and Planning Performing and Recording Analyzing and Interpreting Communication and Teamwork Question How does the momentum of a system consisting of two objects compare just before and just after a collision? Hypothesis State a hypothesis relating the momentum of a system immediately before and immediately after collision, where objects combine after impact. Remember to write an “if/then” statement. Variables Read the procedure carefully and identify the manipulated variables, the responding variables, and the controlled variables. Materials and Equipment one of these set-ups: air track, dynamics carts, Fletcher’s trolley, bead table or air table with linear guides colliding objects for the set-up chosen: gliders, carts, discs, blocks, etc. objects of different mass fastening material (Velcro™ strips, tape, Plasticine™, magnets, etc.) balance timing device (stopwatch, spark-timer, ticker-tape timer, electronic speed-timing device, or time-lapse camera) metre-sticks Procedure 1 Copy Tables 9.1 and 9.2 on page 472 into your notebook. 2 Set up the equipment in such a way that friction is minimized and the two colliding objects travel in the same straight line. 5 Set up the timing device to measure the velocity of object 1 just before and just after collision. Object 2 will be stationary before collision. The velocities of both objects will be the same after collision because they will stick together. 6 Send object 1 at a moderate speed on a collision course with object 2. Ensure that both objects will stick together and that the timing device is working properly. Make adjustments if needed. 7 Send object 1 at a moderate speed on a collision course with stationary object 2, recording the relevant observations and the masses as trial 1. 8 Send object 1 at a different speed on a collision course with stationary object 2, recording the relevant observations and the masses as trial 2. 9 Change the mass of one of the objects and again send
object 1 at a moderate speed on a collision course with stationary object 2, recording the relevant observations and the masses as trial 3. 10 If you can simultaneously measure the speed of two objects, run trials where both objects are in motion before the collision. Do one trial in which they begin moving toward each other and stick together upon impact, and another trial where they move apart after impact. If you remove the fastening material, you will have to remeasure the masses of the objects. Include the direction of motion for both objects before and after collision. 11 If you did not do step 10, do two more trials, changing the mass of one of the objects each time. Include the direction of motion for both objects before and after collision. 3 Attach some fastening material to the colliding objects, so that the two objects remain together after impact. e LAB 4 Measure and record the masses of the two objects. If necessary, change the mass of one object so that the two objects have significantly different masses. For a probeware activity, go to www.pearsoned.ca/school/physicssource. Chapter 9 The momentum of an isolated system of interacting objects is conserved. 471 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 472 Analysis 1. Determine the velocities for each colliding object in each trial, and record them in your data table. Show your calculations. 2. For each trial, calculate the momentum of each object just before and just after collision. Show your calculations. Record the values in your data table. 3. Calculate the momentum of the system just before and just after collision for each trial. Show your calculations. Record the values in your data table. 4. Calculate the difference between the momentum of the system just before and just after collision. Show your calculations. Record the values in your data table. 5. What is the relationship between the momentum of the system just before and just after collision? Does this relationship agree with your hypothesis? 6. What effect did friction have on your results? Explain. 7. Check your results with other groups. Account for any discrepancies. Table 9.1 Mass and Velocity Before and After for Object 1 Before and After for Object 2 Initial Velocity v 1i (m/s) Final Velocity v 1f (m/s) Mass m2 (g) Initial Velocity v 2i (m/s) Trial Mass m1 (g) 1 2 3 4
5 Table 9.2 Momentum Final Velocity v 2f (m/s) Change in Momentum of System sys g m s p Before and After for Object 1 Before and After for Object 2 Before and After for System Initial Final Momentum Momentum of System sysi g m p s of System sysf g m p s Initial Trial Momentum 1i g p m s Final Momentum 1f g p m s Initial Momentum 2i g p m s Final Momentum 2f g p m s 1 2 3 4 5 472 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 473 Momentum Is Conserved in One-dimensional Collisions In 9-5 Inquiry Lab, you discovered that, in one-dimensional collisions, the momentum of a system immediately before collision is about the same as the momentum of the system immediately after collision. If the external force of friction acting on the system is negligible, the momentum of the system is constant. This result is true no matter how many objects are in the system, how many of those objects collide, how massive the objects are, or how fast they are moving. The general form of Newton’s second law for a system is net)sys (F p sys t In an isolated system, the external net force on the system is zero, net)sys (F 0. So p sys 0 t p sys to be zero, the change in momentum of the system In order for t must be zero. p sysf sys p p sysi p sysi 0 0 p sysf In other words, p sys constant. This is a statement of the law of conservation of momentum. In Unit III, you encountered another conservation law, that is, in an isolated system the total energy of the system is conserved. Conservation laws always have one quantity that remains unchanged. In the law of conservation of momentum, it is momentum that remains unchanged. law of conservation of momentum: momentum of an isolated system is constant When no external net force acts on a system, the momentum of the system remains constant. p sysi p net)sys sysf where (F 0 Concept Check Why did cannons on 16th- to 19th-century warships need a rope around the back, tying them to the side of the ship (Figure 9.33)? Figure 9.33 Chapter 9 The momentum of an isolated system of interacting objects is conserved. 473 09-Phys20-Ch
ap09.qxd 7/24/08 2:43 PM Page 474 Freaction FB on A A B Faction FA on B Figure 9.34 The action-reaction forces when two objects collide e SIM Learn how the momentum of a system just before and just after a one- dimensional collision are related. Vary the ratio of the mass of two pucks. Follow the eSim links at www.pearsoned.ca/school/ physicssource. Writing the Conservation of Momentum in Terms of Mass and Velocity Suppose a system consists of two objects, A and B. If the system is isolated, 0. Consider the internal forces of the system. At collision time, (F net)sys object A exerts a force on object B and object B exerts a force on object A (Figure 9.34). From Newton’s third law, these action-reaction forces are related by the equation F A on B F B on A Objects A and B interact for the same time interval t. If you multiply both sides of the equation by t, you get an equation in terms of impulse: F A on B t F B on A t Since impulse is equivalent to a change in momentum, the equation can be rewritten in terms of the momenta of each object: B B p p p Bi p Bi p 0 A 0 p Af p Bf A p p Bf p Ai p Af p Ai If the mass of each object is constant during the interaction, the equation can be written in terms of m and v: mBv Bi mAv Af mBv Bf mAv Ai This equation is the law of conservation of momentum in terms of the momenta of objects A and B. So if two bighorn rams head-butt each other, the sum of the momenta of both rams is constant during the collision, even though the momentum of each ram changes. The law of conservation of momentum has no known exceptions, and holds even when particles are travelling close to the speed of light, or when the mass of the colliding particles is very small, as in the case of electrons. In real life, when objects collide, external friction acts on nearly all systems and the instantaneous forces acting on each object are usually not known (Figure 9.35). Often, the details of the interaction are also unknown. However, you do not require such information to apply the law of conservation of momentum. Instead, it is the mass and instantaneous velocity of the objects immediately before
and immediately after collision that are important, so that the effects of external friction are minimal, and do not significantly affect the outcome. Figure 9.35 During a vehicle collision, many forces cause a change in the velocity and shape of each vehicle. 474 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 475 Conservation of Momentum Applied to Rockets In Unit II, the motion of a rocket was explained using Newton’s third law. However, the conservation of momentum can be used to explain why a rocket can accelerate even in a vacuum. When the engines of a rocket burn fuel, the escaping exhaust gas has mass and considerable speed. When a rocket is in outer space, external friction is negligible. So the rocket-exhaust gas system is an isolated system. For a two-object system, the equation for the conservation of momentum is gas gas rocket rocket p p p 0 p where, during time interval t, p rocket is the change in momentum of the rocket including any unspent fuel and p gas is the change in momentum of the fuel that is expelled in the form of exhaust gas. It is the change in momentum of the exhaust gas that enables a rocket to accelerate (Figure 9.36). In the case of a very large rocket, such as a Saturn V, the magnitude of p would be very large (Figure 9.37). gas procket change in momentum of rocket pgas change in momentum of exhaust gas Figure 9.37 From the law of conservation of momentum, the magnitude of p gas is equal to the magnitude of p rocket. That is why a rocket can accelerate on Earth or in outer space. Figure 9.36 With a height of about 112 m, the Saturn V rocket was the largest and most powerful rocket ever built. info BIT None of the 32 Saturn rockets that were launched ever failed. Altogether 15 Saturn V rockets were built. Three Saturn V rockets are on display, one at each of these locations: the Johnson Space Center, the Kennedy Space Center, and the Alabama Space and Rocket Center. Of these three, only the rocket at the Johnson Space Center is made up entirely of former flight-ready, although mismatched, parts. info BIT Design of the Saturn V began in the 1950s with the intent to send astronauts to the Moon. In the early 1970s, this type of rocket was used to launch the Skylab space station. The rocket engines in the first stage burned a combination of
kerosene and liquid oxygen, producing a total thrust of magnitude 3.34 107 N. The rocket engines in the second and third stages burned a combination of liquid hydrogen and liquid oxygen. The magnitude of the total thrust produced by the second-stage engines was 5.56 106 N, and the third-stage engine produced 1.11 106 N of thrust. Concept Check (a) Refer to the second infoBIT on this page. Why is less thrust needed by the second-stage engines of a rocket? (b) Why is even less thrust needed by the third-stage engine? In Example 9.5 on the next page, the conservation of momentum is applied to a system of objects that are initially stationary. This type of interaction is called an explosion. Chapter 9 The momentum of an isolated system of interacting objects is conserved. 475 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 476 Example 9.5 A 75-kg hunter in a stationary kayak throws a 0.72-kg harpoon at 12 m/s [right]. The mass of the kayak is 10 kg. What will be the velocity of the kayak and hunter immediately after the harpoon is released? Given mp v pi 75 kg 0 m/s mk v ki 10 kg 0 m/s mh v hi v hf 0.72 kg 0 m/s 12 m/s [right] before left right after? vTf 12 m/s vhf Figure 9.38 Practice Problems 1. A 110-kg astronaut and a 4000-kg spacecraft are attached by a tethering cable. Both masses are motionless relative to an observer a slight distance away from the spacecraft. The astronaut wants to return to the spacecraft, so he pulls on the cable until his velocity changes to 0.80 m/s [toward the spacecraft] relative to the observer. What will be the change in velocity of the spacecraft? 2. A student is standing on a stationary 2.3-kg skateboard. If the student jumps at a velocity of 0.37 m/s [forward], the velocity of the skateboard becomes 8.9 m/s [backward]. What is the mass of the student? Answers 1. 0.022 m/s [toward the astronaut] 2. 55 kg Required final velocity of hunter and kayak Analysis and Solution Choose the kayak, hunter, and harpoon as an isolated system. The hunter and kayak move
together as a unit after the harpoon is released. So find the total mass of the hunter and kayak. mT mp mk 75 kg 10 kg 85 kg The hunter, kayak, and harpoon each have an initial velocity of zero. So the system has an initial momentum of zero. p sysi 0 Apply the law of conservation of momentum. sysf p p sysi p p p hf Tf sysi mhv 0 mTv hf Tf m hv m T 0.72 kg 85 kg v Tf hf (12 m/s) 0.10 m/s 0.10 m/s [left] v Tf Paraphrase and Verify The velocity of the kayak and hunter will be 0.10 m/s [left] immediately after the harpoon is released. Since the harpoon is thrown right, from Newton’s third law, you would expect the hunter and kayak to move left after the throw. So the answer is reasonable. 476 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/24/08 2:43 PM Page 477 In Example 9.6, a dart is fired at a stationary block sitting on a glider. This situation involves two objects (dart and block) that join together and move as a unit after interaction. This type of interaction is called a hit-and-stick interaction. Example 9.6 A wooden block attached to a glider has a combined mass of 0.200 kg. Both the block and glider are at rest on a frictionless air track. A dart gun shoots a 0.012-kg dart into the block. The velocity of the block-dart system after collision is 0.78 m/s [right]. What was the velocity of the dart just before it hit the block? Given mb v bi 0.200 kg 0 m/s before left? vdi 0.012 kg 0.78 m/s [right] f md v right after vf 0.78 m/s Figure 9.39 Required initial velocity of dart (v di) Analysis and Solution Choose the block, glider, and dart as an isolated system. The dart, block, and glider move together as a unit after collision. The block-glider unit has an initial velocity of zero. So its initial momentum is zero. p bi 0 Apply the law of conservation of momentum. sysf p p sysf md
)v (mb mdv mb m d f f 0.200 kg 0.012 kg 0.012 kg (0.78 m/s) (0.78 m/s) 0.212 kg 0.012 kg 14 m/s 14 m/s [right] p sysi p p di bi 0 mdv di v di v di Paraphrase Practice Problems 1. A student on a skateboard, with a combined mass of 78.2 kg, is moving east at 1.60 m/s. As he goes by, the student skilfully scoops his 6.4-kg backpack from the bench where he had left it. What will be the velocity of the student immediately after the pickup? 2. A 1050-kg car at an intersection has a velocity of 2.65 m/s [N]. The car hits the rear of a stationary truck, and their bumpers lock together. The velocity of the cartruck system immediately after collision is 0.78 m/s [N]. What is the mass of the truck? Answers 1. 1.5 m/s [E] 2. 2.5 103 kg The dart had a velocity of 14 m/s [right] just before it hit the block. Chapter 9 The momentum of an isolated system of interacting objects is conserved. 477 09-Phys20-Chap09.qxd 7/24/08 2:44 PM Page 478 Example 9.7 involves a basketball player, initially moving with some velocity, colliding with a stationary player. After the interaction, both players move in different directions. Example 9.7 A basketball player and her wheelchair (player A) have a combined mass of 58 kg. She moves at 0.60 m/s [E] and pushes off a stationary player (player B) while jockeying for a position near the basket. Player A ends up moving at 0.20 m/s [W]. The combined mass of player B and her wheelchair is 85 kg. What will be player B’s velocity immediately after the interaction? Given mA v Ai v Af 58 kg 0.60 m/s [E] 0.20 m/s [W] before 0.60 m/s vAi mB v Bi 85 kg 0 m/s W E after 0.20 m/s vAf? vBf Figure 9.40 Player A Player B Player A Player B Practice Problems 1. A 0.25-
kg volleyball is flying west at 2.0 m/s when it strikes a stationary 0.58-kg basketball dead centre. The volleyball rebounds east at 0.79 m/s. What will be the velocity of the basketball immediately after impact? 2. A 9500-kg rail flatcar moving forward at 0.70 m/s strikes a stationary 18 000-kg boxcar, causing it to move forward at 0.42 m/s. What will be the velocity of the flatcar immediately after collision if they fail to connect? Answers 1. 1.2 m/s [W] 2. 0.096 m/s [backward] Required final velocity of player B (v Bf) Analysis and Solution Choose players A and B as an isolated system. Player B has an initial velocity of zero. So her initial momentum is zero. p Bi 0 Apply the law of conservation of momentum. p Ai mAv Ai p sysi p Bi sysf p p Af 0 mAv Af m (v v A Bf m B 58 kg 85 kg p Bf mBv Bf v Ai Af) [0.60 m/s (0.20 m/s)] (0.60 m/s 0.20 m/s) 5 8 8 5 0.55 m/s 0.55 m/s [E] v Bf Paraphrase Player B’s velocity is 0.55 m/s [E] just after collision. 478 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/24/08 2:44 PM Page 479 In Example 9.8, two football players in motion collide with each other. After the interaction, the players bounce apart. Example 9.8 A 110-kg Stampeders football fullback moving east at 1.80 m/s on a snowy playing field is struck by a 140-kg Eskimos defensive lineman moving west at 1.50 m/s. The fullback is bounced west at 0.250 m/s. What will be the velocity of the Eskimos defensive lineman just after impact? Given mS v Si v Sf 110 kg 1.80 m/s [E] 0.250 m/s [W] before vSi 1.80 m/s W m E v Ei 140 kg 1.50 m/s [W] E 1.50 m/s vEi after vSf 0.250 m
/s? vEf Figure 9.41 Required final velocity of Eskimos lineman (v Ef) Analysis and Solution Choose the fullback and lineman as an isolated system. Apply the law of conservation of momentum. p sysi p p Ei Si mEv Ei mEv Ef mSv Si v Ef v Ef Sf Si sysf v v Ei mSv Sf m Sv m E p Ef mEv Ef mEv Ei p p Sf mSv Sf mSv Si m S v m E m S(v Sf) v Si Ei m E 110 kg [(1.80 m/s) 140 kg (0.250 m/s)] (1.50 m/s) (1.80 m/s 0.250 m/s) 110 140 1.50 m/s 0.111 m/s 0.111 m/s [E] Practice Problems 1. A 72-kg snowboarder gliding at 1.6 m/s [E] bounces west at 0.84 m/s immediately after colliding with an 87-kg skier travelling at 1.4 m/s [W]. What will be the velocity of the skier just after impact? 2. A 125-kg bighorn ram butts heads with a younger 122-kg ram during mating season. The older ram is rushing north at 8.50 m/s immediately before collision, and bounces back at 0.11 m/s [S]. If the younger ram moves at 0.22 m/s [N] immediately after collision, what was its velocity just before impact? Answers 1. 0.62 m/s [E] 2. 8.6 m/s [S] Paraphrase The velocity of the Eskimos defensive lineman immediately after impact is 0.111 m/s [E]. Chapter 9 The momentum of an isolated system of interacting objects is conserved. 479 09-Phys20-Chap09.qxd 7/24/08 2:44 PM Page 480 PHYSICS INSIGHT The law of conservation of energy states that the total energy of an isolated system remains constant. The energy may change into several different forms. This law has no known exceptions. e SIM Predict the speed of two pucks just after a onedimensional collision using momentum and energy concepts. Follow the eSim links at www.pearsoned.ca/school/ physicssource.
Elastic and Inelastic Collisions in One Dimension In Examples 9.3 to 9.8, some of the collisions involved hard objects, such as the golf club hitting the golf ball. Other collisions, such as the block and dart, involved a dart that became embedded in a softer material (a block of wood). In all these collisions, it was possible to choose an isolated system so that the total momentum of the system was conserved. When objects collide, they sometimes deform, make a sound, give off light, or heat up a little at the moment of impact. Any of these observations indicate that the kinetic energy of the system before collision is not the same as after collision. However, the total energy of the system is constant. Concept Check (a) Is it possible for an object to have energy and no momentum? Explain, using an example. (b) Is it possible for an object to have momentum and no energy? Explain, using an example. Elastic Collisions Suppose you hit a stationary pool ball dead centre with another pool ball so that the collision is collinear and the balls move without spinning immediately after impact. What will be the resulting motion of both balls (Figure 9.42)? The ball that was initially moving will become stationary upon impact, while the other ball will start moving in the same direction as the incoming ball. If you measure the speed of both balls just before and just after collision, you will find that the speed of the incoming 1 mv2, 2 ball is almost the same as that of the outgoing ball. Since Ek the final kinetic energy of the system is almost the same as the initial kinetic energy of the system. Figure 9.42 Many collisions take place during a game of pool. What evidence suggests that momentum is conserved during the collision shown in the photo? What evidence suggests that energy is conserved? 480 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/24/08 2:44 PM Page 481 If the initial kinetic energy of a system is equal to the final kinetic energy of the system after collision, the collision is elastic. In an elastic collision, the total kinetic energy of the system is conserved. elastic collision: a collision Ekf in which Eki Eki Ekf Most macroscopic interactions in the real world involve some of the initial kinetic energy of the system being converted to sound, light, or deformation (Figure 9.43). When deformation occurs, some of the initial kinetic energy of
the system is converted to heat because friction acts on objects in almost all situations. These factors make it difficult to achieve an elastic collision. Even if two colliding objects are hard and do not appear to deform, energy is still lost in the form of sound, light, and/or heat due to friction. Usually, the measured speed of an object after interaction is a little less than the predicted speed, which indicates that the collision is inelastic. Example 9.9 demonstrates how to determine if the collision between a billiard ball and a snooker ball is elastic. Project LINK How will you apply the concepts of conservation of momentum and conservation of energy to the design of the water balloon protection? info BIT A steel sphere will bounce as high on a steel anvil as a rubber ball will on concrete. However, when a steel sphere is dropped on linoleum or hardwood, even more kinetic energy is lost and the sphere hardly bounces at all. The kinetic energy of the sphere is converted to sound, heat, and the deformation of the floor surface. To try this, use flooring samples. Do not try this on floors at home or at school. Figure 9.43 Is the collision shown in this photo elastic? What evidence do you have to support your answer? Chapter 9 The momentum of an isolated system of interacting objects is conserved. 481 09-Phys20-Chap09.qxd 7/24/08 2:44 PM Page 482 Example 9.9 Practice Problems 1. A 45.9-g golf ball is stationary on the green when a 185-g golf club face travelling horizontally at 1.24 m/s [E] strikes it. After impact, the club face continues moving at 0.760 m/s [E] while the ball moves at 1.94 m/s [E]. Assume that the club face is vertical at the moment of impact so that the ball does not spin. Determine if the collision is elastic. 2. An argon atom with a mass of 6.63 1026 kg travels at 17 m/s [right] and strikes another identical argon atom dead centre travelling at 20 m/s [left]. The first atom rebounds at 20 m/s [left], while the second atom moves at 17 m/s [right]. Determine if the collision is elastic. Answers 1. inelastic 2. elastic A 0.160-kg billiard ball travelling at 0.500 m/s [N] strikes a
stationary 0.180-kg snooker ball and rebounds at 0.0230 m/s [S]. The snooker ball moves off at 0.465 m/s [N]. Ignore possible rotational effects. Determine if the collision is elastic. 0.160 kg 0.180 kg 0.500 m/s [N] Given mb ms v bi v 0 m/s si v bf v 0.465 m/s [N] sf 0.0230 m/s [S] before after N S vsf 0.465 m/s Required determine if the collision is elastic snooker ball vbi 0.500 m/s snooker ball billiard ball vbf 0.0230 m/s Figure 9.44 billiard ball Analysis and Solution Choose the billiard ball and the snooker ball as an isolated system. Calculate the total initial kinetic energy and the total final kinetic energy of the system. Eki mb(vbi)2 1 1 ms(vsi)2 2 2 1 (0.160 kg)(0.500 m/s)2 0 2 Ekf 0.0200 kgm2/s2 0.0200 J mb(vbf)2 1 1 ms(vsf)2 2 2 1 (0.160 kg)(0.0230 m/s)2 2 1 (0.180 kg)(0.465 m/s)2 2 0.0195 kgm2/s2 0.0195 J Since Eki Ekf, the collision is inelastic. Paraphrase The collision between the billiard ball and the snooker ball is inelastic. 482 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/24/08 2:44 PM Page 483 inelastic collision: a collision in which Eki Ekf e WEB Research examples of elastic and inelastic one- dimensional collisions. Then analyze how the momentum and energy change in those collisions. Begin your search at www.pearsoned.ca/ school/physicssource. Inelastic Collisions In 9-2 QuickLab on page 455, after the putty ball collided with a hard surface, the putty ball became stationary and had no kinetic energy. Upon impact, the putty ball deformed and the kinetic energy of the putty ball was converted mostly to thermal energy. Although the total energy of the system was
conserved, the total initial kinetic energy of the system was not equal to the total final kinetic energy of the system after collision. This type of collision is inelastic. In an inelastic collision, the total kinetic energy of the system is not conserved. Eki Ekf One type of inelastic collision occurs when two objects stick together after colliding. However, this type of interaction does not necessarily mean that the final kinetic energy of the system is zero. For example, consider a ballistic pendulum, a type of pendulum used to determine the speed of bullets before electronic timing devices were invented (Figure 9.45). suspension wire ceiling pistol v bullet block height of swing Figure 9.45 When a bullet is fired into the block, both the block and bullet move together as a unit after impact. The pendulum consists of a stationary block of wood suspended from the ceiling by light ropes or cables. When a bullet is fired at the block, the bullet becomes embedded in the wood upon impact. The kinetic energy of the bullet is converted to sound, thermal energy, deformation of the wood and bullet, and the kinetic energy of the pendulum-bullet system. The initial momentum of the bullet causes the pendulum to move upon impact, but since the pendulum is suspended by cables, it swings upward just after the bullet becomes embedded in the block. As the pendulum-bullet system swings upward, its kinetic energy is converted to gravitational potential energy. Example 9.10 involves a ballistic pendulum. By using the conservation of energy, it is possible to determine the speed of the pendulum-bullet system immediately after impact. By applying the conservation of momentum to the collision, it is possible to determine the initial speed of the bullet. Chapter 9 The momentum of an isolated system of interacting objects is conserved. 483 09-Phys20-Chap09.qxd 7/24/08 2:44 PM Page 484 Example 9.10 A 0.0149-kg bullet from a pistol strikes a 2.0000-kg ballistic pendulum. Upon impact, the pendulum swings forward and rises to a height of 0.219 m. What was the velocity of the bullet immediately before impact? backward before after forward vbi? h Given mb 0.0149 kg mp 2.0000 kg h 0.219 m Required initial velocity of bullet (v bi ) Ek 0 Ep 0 vf? Ek 0 Ep (mb mp)gh Figure 9.46 Analysis and Solution Choose the
pendulum and the bullet as an isolated system. Since the pendulum is stationary before impact, its initial velocity is zero. So its initial momentum is zero. p pi 0 Immediately after collision, the bullet and pendulum move together as a unit. The kinetic energy of the pendulum-bullet system just after impact is converted to gravitational potential energy. Ek Ep Practice Problems 1. A 2.59-g bullet strikes a stationary 1.00-kg ballistic pendulum, causing the pendulum to swing up to 5.20 cm from its initial position. What was the speed of the bullet immediately before impact? 2. A 7.75-g bullet travels at 351 m/s before striking a stationary 2.5-kg ballistic pendulum. How high will the pendulum swing? Answers 1. 391 m/s 2. 6.0 cm Apply the law of conservation of energy to find the speed of the pendulum-bullet system just after impact. 1 (mb 2 Ep Ek mp) (vf)2 (mb mp) g(h) (vf)2 2g(h) vf v f (0.219 m) 2g(h) 29.81 m s2 2.073 m/s 2.073 m/s [forward] Apply the law of conservation of momentum to find the initial velocity of the bullet. p bi mbv bi p sysi p pi sysf p p sysf mp)v 0 (mb mpv mb m b v bi f f 0.0149 kg 2.0000 kg 0.0149 kg (2.073 m/s) 484 Unit V Momentum and Impulse v bi (2.073 m/s) 2.0149 kg 0.0149 kg 280 m/s 280 m/s [forward] 09-Phys20-Chap09.qxd 7/24/08 2:44 PM Page 485 Paraphrase The initial velocity of the bullet immediately before impact was 280 m/s [forward]. Example 9.11 demonstrates how to determine if the collision in Example 9.10 is elastic or inelastic by comparing the kinetic energy of the system just before and just after collision. Example 9.11 Determine if the collision in Example 9.10 is elastic or inelastic. Given mb mp 0.0149 kg 2.0000 kg v bi v f 280 m/s [forward] from Example 9.10 2.
073 m/s [forward] from Example 9.10 Required initial and final kinetic energies (Eki and Ekf) to find if the collision is elastic Analysis and Solution Choose the pendulum and the bullet as an isolated system. Calculate the total initial kinetic energy and the total final kinetic energy of the system. Ekf Eki mb(vbi)2 1 1 mp(vpi)2 2 2 1 (0.0149 kg)(280 m/s)2 0 2 585 kgm2/s2 585 J mp)(vf)2 1 (mb 2 1 (0.0149 kg 2.0000 kg)(2.073 m/s)2 2 4.33 kgm2/s2 4.33 J Since Eki Ekf, the collision is inelastic. Paraphrase and Verify Since the kinetic energy of the system just before impact is much greater than the kinetic energy of the system just after impact, the collision is inelastic. This result makes sense since the bullet became embedded in the pendulum upon impact. Practice Problems 1. In Example 9.6 on page 477, how much kinetic energy is lost immediately after the interaction? 2. (a) Determine if the interaction in Example 9.8 on page 479 is elastic. (b) What percent of kinetic energy is lost? Answers 1. 1.1 J 2. (a) inelastic (b) 98.7% Chapter 9 The momentum of an isolated system of interacting objects is conserved. 485 09-Phys20-Chap09.qxd 7/24/08 2:44 PM Page 486 9.3 Check and Reflect 9.3 Check and Reflect Knowledge 1. In your own words, state the law of conservation of momentum. 2. (a) In the context of momentum, what is an isolated system? (b) Why is it necessary to choose an isolated system when solving a momentum problem? 3. Explain the difference between an elastic and an inelastic collision. Include an example of each type of collision in your answer. 8. A 60.0-kg student on a 4.2-kg skateboard is travelling south at 1.35 m/s. A friend throws a 0.585-kg basketball to him with a velocity of 12.6 m/s [N]. What will be the velocity of the student and skateboard immediately after he catches the ball? 9. A hockey forward with
a mass of 95 kg skates in front of the net at 2.3 m/s [E]. He is met by a 104-kg defenceman skating at 1.2 m/s [W]. What will be the velocity of the resulting tangle of players if they stay together immediately after impact? 4. What evidence suggests that a collision is 10. A 75.6-kg volleyball player leaps toward the (a) elastic? (b) inelastic? Applications 5. Give two examples, other than those in the text, of possible collinear collisions between two identical masses. Include a sketch of each situation showing the velocity of each object immediately before and immediately after collision. 6. A student is sitting in a chair with nearly frictionless rollers. Her homework bag is in an identical chair right beside her. The chair and bag have a combined mass of 20 kg. The student and her chair have a combined mass of 65 kg. If she pushes her homework bag away from her at 0.060 m/s relative to the floor, what will be the student’s velocity immediately after the interaction? 7. At liftoff, a space shuttle has a mass of 2.04 106 kg. The rocket engines expel 3.7 103 kg of exhaust gas during the first second of liftoff, giving the rocket a velocity of 5.7 m/s [up]. At what velocity is the exhaust gas leaving the rocket engines? Ignore the change in mass due to the fuel being consumed. The exhaust gas needed to counteract the force of gravity has already been accounted for and should not be part of this calculation. net to block the ball. At the top of his leap, he has a horizontal velocity of 1.18 m/s [right], and blocks a 0.275-kg volleyball moving at 12.5 m/s [left]. The volleyball rebounds at 6.85 m/s [right]. (a) What will be the horizontal velocity of the player immediately after the block? (b) Determine if the collision is elastic. 11. A 220-kg bumper car (A) going north at 0.565 m/s hits another bumper car (B) and rebounds at 0.482 m/s [S]. Bumper car B was initially travelling south at 0.447 m/s, and after collision moved north at 0.395 m/s. (a) What is the mass of bumper car B? (b) Determine if the collision is
elastic. Extension 12. Summarize the concepts and ideas associated with one-dimensional collisions using a graphic organizer of your choice. See Student References 4: Using Graphic Organizers on pp. 869–871 for examples of different graphic organizers. Make sure that the concepts and ideas are clearly presented and appropriately linked. e TEST To check your understanding of the conservation of momentum and one-dimensional collisions, follow the eTest links at www.pearsoned.ca/school/physicssource. 486 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/24/08 2:44 PM Page 487 9.4 Collisions in Two Dimensions Many interactions in the universe involve collisions. Comets, asteroids, and meteors sometimes collide with celestial bodies. Molecules and atoms are constantly colliding during chemical reactions throughout the universe: in stars, in Earth’s atmosphere, and even within your body. An interesting collision in recent history occurred on June 30, 1908, at Tunguska, Siberia, between a cosmic object and Earth (Figure 9.47). Eyewitnesses reported seeing a giant fireball that moved rapidly across the sky and eventually collided with the ground. Upon impact, a tremendous explosion occurred producing an atmospheric shock wave that circled Earth twice. About 2000 km2 of forest were levelled and thousands of trees were burned. In fact, there was so much fine dust in the atmosphere that people in London, England, could read a newspaper at night just from the scattered light. info BIT Scientists speculate that the cosmic object that hit Tunguska was about 100 m across and had a mass of about 1 106 t. The estimated speed of the object was about 30 km/s, which is 1.1 105 km/h. After the collision at Tunguska, a large number of diamonds were found scattered all over the impact site. So the cosmic object contained diamonds as well as other materials. Figure 9.47 The levelled trees and charred remnants of a forest at Tunguska, Siberia, after a cosmic object collided with Earth in 1908. Although the chance that a similar collision with Earth during your lifetime may seem remote, such collisions have happened throughout Earth’s history. In real life, most collisions occur in three dimensions. Only in certain situations, such as those you studied in section 9.3, does the motion of the interacting objects lie along a straight line. In this section, you will examine collisions that occur in two
dimensions. These interactions occur when objects in a plane collide off centre. In 9-1 QuickLab on page 447, you found that when two coins collide off centre, the resulting path of each coin is in a different direction from its initial path. You may have noticed that certain soccer or hockey players seem to be at the right place at the right time whenever there is a rebound from the goalie. How do these players know where to position themselves so that they can score on the rebound? Find out by doing 9-6 Inquiry Lab. Chapter 9 The momentum of an isolated system of interacting objects is conserved. 487 09-Phys20-Chap09.qxd 7/24/08 2:44 PM Page 488 9-6 Inquiry Lab 9-6 Inquiry Lab Analyzing Collisions in Two Dimensions Required Skills Initiating and Planning Performing and Recording Analyzing and Interpreting Communication and Teamwork Question How does the momentum of a two-body system in the x and y directions compare just before and just after a collision? Hypothesis State a hypothesis relating the momentum of a system in each direction immediately before and immediately after collision. Remember to write an “if/then” statement. Variables Read the procedure and identify the controlled, manipulated, and responding variables in the experiment. Materials and Equipment air table or bead table pucks spark-timer or camera set-up to measure velocities rulers or metre-sticks protractors Procedure 1 Copy Tables 9.3 and 9.4 on page 489 into your notebook. 2 Label the pucks as “puck 1” and “puck 2” respectively. Measure the mass of each puck and record it in Table 9.3. 3 Set up the apparatus so that puck 2 is at rest near the centre of the table. 4 Have each person in your group do one trial. Each time, send puck 1 aimed at the left side of puck 2, recording the paths of both pucks. Make sure the recording tracks of both pucks can be used to accurately measure their velocities before and after collision. 5 Have each person in your group measure and analyze one trial. Help each other as needed to ensure the measurements and calculations for each trial are accurate. 6 Find a suitable point on the recorded tracks to be the impact location. 7 On the path of puck 1 before collision, choose an interval where the speed is constant. Choose the positive x-axis to be in the initial direction of puck 1.
8 Using either the spark dots, the physical centre of the puck, or the leading or trailing edge of the puck, measure the distance and the time interval. Record those values in Table 9.3. 9 On the path of each puck after collision, choose an interval where the speed is constant. Measure the distance, direction of motion relative to the positive x-axis, and time interval. Record those values in Table 9.3. Analysis 1. Calculate the initial velocity and initial momentum of puck 1. Record the values in Table 9.4. 2. Calculate the velocity of puck 1 after collision. Resolve the velocity into x and y components. Record the values in Table 9.4. 3. Use the results of question 2 to calculate the x and y components of the final momentum of puck 1. Record the values in Table 9.4. 4. Repeat questions 2 and 3 but this time use the data for puck 2. Explain why the y component of the momentum of puck 2 is negative. 5. Record the calculated values from each member of your group as a different trial in Table 9.4. 6. For each trial, state the relationship between the initial momentum of the system in the x direction and the final momentum of the system in the x direction. Remember to consider measurement errors. Write this result as a mathematical statement. 7. The initial momentum of the system in the y direction was zero. For each trial, what was the final momentum of the system in the y direction? Remember to consider measurement errors. Write this result as a mathematical statement. 8. Compare your answers to questions 6 and 7 with other groups. Does this relationship agree with your hypothesis? Account for any discrepancies. e LAB For a probeware activity, go to www.pearsoned.ca/school/physicssource. 488 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/24/08 2:44 PM Page 489 Table 9.3 Mass, Distance, Time Elapsed, and Angle Before and After for Puck 1 After for Puck 2 Trial Mass m1 (g) Initial Distance d1i (m) Initial Time Elapsed t1i (s) Final Distance d1f (m) Final Time Elapsed t1f (s) Final Angle () 1f Mass m2 (g) Final Distance d2f (m) Final Time Elapsed t2f (s) Final Angle () 2f
1 2 3 4 5 Table 9.4 Velocity and Momentum Before and After for Puck 1 After for Puck 2 Initial x Final x Initial x Velocity Momentum Velocity Final y Final x Velocity Momentum Momentum Velocity Final y Final x (m/s) p1ix gm v1fx v1ix (m/s) v1fy (m/s) p1fx gm p1fy gm v2fx (m/s) v2fy s s Final x Final y Final y Velocity Momentum Momentum p2fy gm s (m/s) p2fx gm s s Trial 1 2 3 4 5 Momentum Is Conserved in Two-dimensional Collisions In 9-6 Inquiry Lab, you found that along each direction, x and y, the momentum of the system before collision is about the same as the momentum of the system immediately after collision. In other words, momentum is conserved in two-dimensional collisions. This result agrees with what you saw in 9-5 Inquiry Lab, where only one-dimensional collisions were examined. As in one-dimensional collisions, the law of conservation of momentum is valid only when no external net force acts on the system. In two dimensions, the motion of each object in the system must be analyzed in terms of two perpendicular axes. To do this, you can either use a vector addition diagram drawn to scale or vector components. The law of conservation of momentum can be stated using components in the x and y directions. In two-dimensional collisions where no external net force acts on the system, the momentum of the system in both the x and y directions remains constant. psysix psysfx and psysiy psysfy where (F net)sys 0 e SIM Apply the law of conservation of momentum to twodimensional collisions. Go to www.pearsoned.ca/school/ physicssource. Chapter 9 The momentum of an isolated system of interacting objects is conserved. 489 09-Phys20-Chap09.qxd 7/24/08 2:44 PM Page 490 info BIT In championship curling, rebound angles and conservation of momentum are crucial for placing stones in counting position behind guards. Just nudging a stone several centimetres can make all the difference. Concept Check (a) Will the magnitude of the momentum of an object always increase if a non-zero net force acts on it? Explain, using an example. (b) How can the momentum of an object change
but its speed remain the same? Explain, using an example. Example 9.12 involves a curling stone colliding off centre with an identical stone that is at rest. The momentum of each stone is analyzed in two perpendicular directions. Example 9.12 A 19.6-kg curling stone (A) moving at 1.20 m/s [N] strikes another identical stationary stone (B) off centre, and moves off with a velocity of 1.17 m/s [12.0° E of N]. What will be the velocity of stone B after the collision? Ignore frictional and rotational effects. Given mA v Bi 19.6 kg 0 m/s before N mB v Af 19.6 kg 1.17 m/s [12.0 E of N] v Ai 1.20 m/s [N] after N vAf 1.17 m/s vBf? 12.0° E W Figure 9.48 S N W vBi 0 m/s E vAi 1.20 m/s S Required final velocity of stone B (v Bf) Analysis and Solution Choose both curling stones as an isolated system. Stone B has an initial velocity of zero. So its initial momentum is zero. p Bi 0 Resolve all velocities into east and north components (Figure 9.49). vAf Vector v Ai v Bi v Af East component North component 0 1.20 m/s 0 (1.17 m/s)(sin 12.0) 0 (1.17 m/s)(cos 12.0) 12.0° E Figure 9.49 490 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/28/08 9:26 AM Page 491 Apply the law of conservation of momentum to the system in the east and north directions. E direction psysiE pBiE pBfE pAiE pBfE pBiE psysfE pAfE pAiE mAvAiE 0 0 (19.6 kg)(1.17 m/s)(sin 12.0) 4.768 kgm/s pAfE mBvBiE mAvAfE The negative sign indicates the vector is directed west. N direction psysiN pBiN pBfN pAiN pBfN pBiN psysfN pAfN pAfN pAiN
mAvAiN mAvAfN mBvBiN (19.6 kg)(1.20 m/s) 0 (19.6 kg)(1.17 m/s)(cos 12.0) 1.089 kgm/s Draw a vector diagram of the components of the final momentum of stone B and find the magnitude of the resultant p Pythagorean theorem. Bf using the N 1.089 kg m/s W Figure 9.50 p Bf θ 4.768 kg m/s E pBf (4.768 kgm/s)2 (1.089 kgm/s)2 4.8906 kgm/s Use the magnitude of the momentum and the mass of stone B to find its final speed. mBvBf pBf mB pBf vBf vBf 4.8906 kgm/s 19.6 kg 0.250 m/s Use the tangent function to find the direction of the final momentum. tan pBfN pBfW tan–1 12.9 1.089 kgm/s 4.768 kgm/s The final velocity will be in the same direction as the final momentum. Paraphrase The velocity of stone B after the collision is 0.250 m/s [12.9° N of W]. Practice Problems 1. A 97.0-kg hockey centre stops momentarily in front of the net. He is checked from the side by a 104-kg defenceman skating at 1.82 m/s [E], and bounces at 0.940 m/s [18.5 S of E]. What is the velocity of the defenceman immediately after the check? 2. A 1200-kg car, attempting to run a red light, strikes a stationary 1350-kg vehicle waiting to make a turn. Skid marks show that after the collision, the 1350-kg vehicle moved at 8.30 m/s [55.2 E of N], and the other vehicle at 12.8 m/s [36.8 W of N]. What was the velocity of the 1200-kg vehicle just before collision? Note this type of calculation is part of many vehicle collision investigations where charges may be pending. Answers 1. 1.03 m/s [15.7 N of E] 2. 15.6 m/s [N] Chapter 9 The momentum of an isolated system of interacting objects is conserved. 491 09
-Phys20-Chap09.qxd 7/24/08 2:44 PM Page 492 centre of mass: point where the total mass of an object can be assumed to be concentrated info BIT When an object is symmetric and has uniform density, its centre of mass is in the same location as the physical centre of the object. Example 9.13 involves a football tackle with two players. Each player has an initial velocity, but when they collide, both players move together as a unit. To analyze the motion, the centre of mass of the combination of both players must be used. The centre of mass is a point that serves as an average location of the total mass of an object or system. Depending on the distribution of mass, the centre of mass may be located even outside the object. Generally, momentum calculations are made using the centre of mass of an object. No matter where any external forces are acting on an object, whether the object is rotating or not, or whether the object is deformable or rigid, the translational motion of the object can be easily analyzed in terms of its centre of mass. Example 9.13 A 90-kg quarterback moving at 7.0 m/s [270] is tackled by a 110-kg linebacker running at 8.0 m/s [0]. What will be the velocity of the centre of mass of the combination of the two players immediately after impact? Practice Problems 1. A 2000-kg car travelling at 20.0 m/s [90.0] is struck at an intersection by a 2500-kg pickup truck travelling at 14.0 m/s [180]. If the vehicles stick together upon impact, what will be the velocity of the centre of mass of the cartruck combination immediately after the collision? 2. A 100-kg hockey centre is moving at 1.50 m/s [W] in front of the net. He is checked by a 108-kg defenceman skating at 4.20 m/s [S]. Both players move off together after collision. What will be the velocity of the centre of mass of the combination of the two players immediately after the check? Answers 1. 11.8 m/s [131] 2. 2.30 m/s [71.7 S of W] Given mq v qi 90 kg 7.0 m/s [270] before 8.0 m/s vli y 110 kg 8.0 m/s [0] ml v li after y x 7.
0 m/s vqi x vf? Figure 9.51 Required final velocity of centre of mass of the two players (v f) Analysis and Solution Choose the quarterback and the linebacker as an isolated system. The linebacker tackled the quarterback. So both players have the same final velocity. Resolve all velocities into x and y components. Vector v qi v li x component y component 0 7.0 m/s 8.0 m/s 0 Apply the law of conservation of momentum to the system in the x and y directions. 492 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/28/08 9:30 AM Page 493 x direction pqix psysix plix psysfx plix psysfx psysfx pqix mqvqix mlvlix 0 (110 kg)(8.0 m/s) 880 kgm/s y direction psysiy pliy psysfy pqiy pliy psysfy psysfy pqiy mqvqiy mlvliy (90 kg)(7.0 m/s) 0 630 kgm/s Draw a vector diagram of the components of the final momentum of the two players and find the magnitude of the resultant p Pythagorean theorem. sysf using the y 324º 880 kgm/s θ x 630 kgm/s psysf Figure 9.52 psysf (880kgm/s)2 (630 kgm/s)2 1082 kgm/s Use the magnitude of the momentum and combined masses of the two football players to find their final speed. psysf vf (mq ml)vf psysf ml) (mq 1082 kgm/s (90 kg 110 kg) = 5.4 m/s Use the tangent function to find the direction of the final momentum. tan psysfy psysfx tan1 630 kgm/s 880 kgm/s = 35.6° The final velocity will be in the same direction as the final momentum. From Figure 9.52, is below the positive x-axis. So the direction of v measured counterclockwise from the positive x-axis is 360 35.6 324.4. f v f 5.4 m/s [324] Paraphrase The final velocity of both players is 5.4 m/s [324]. Chapter 9 The momentum of an isolated system
of interacting objects is conserved. 493 09-Phys20-Chap09.qxd 7/24/08 2:44 PM Page 494 Example 9.14 deals with a fireworks bundle that is initially stationary. After it explodes, three fragments (A, B, and C) fly in different directions in a plane. To demonstrate an alternative method of solving collision problems, a vector addition diagram is used to determine the momentum of fragment C. This quantity is then used to calculate its final velocity. Example 9.14 A 0.60-kg fireworks bundle is at rest just before it explodes into three fragments. A 0.20-kg fragment (A) flies at 14.6 m/s [W], and a 0.18-kg fragment (B) moves at 19.2 m/s [S]. What is the velocity of the third fragment (C) just after the explosion? Assume that no mass is lost, and that the motion of the fragments lies in a plane. Practice Problems 1. A 0.058-kg firecracker that is at rest explodes into three fragments. A 0.018-kg fragment moves at 2.40 m/s [N] while a 0.021-kg fragment moves at 1.60 m/s [E]. What will be the velocity of the third fragment? Assume that no mass is lost, and that the motion of the fragments lies in a plane. 2. A 65.2-kg student on a 2.50-kg skateboard moves at 0.40 m/s [W]. He jumps off the skateboard with a velocity of 0.38 m/s [30.0 S of W]. What will be the velocity of the skateboard immediately after he jumps? Ignore friction between the skateboard and the ground. Answers 1. 2.9 m/s [52 S of W] 2. 5.4 m/s [66 N of W] Given mT v i 0.60 kg 0 m/s before W N S E vi 0 m/s mA v Af 0.20 kg 14.6 m/s [W] after mB v Bf N 0.18 kg 19.2 m/s [S] vAf 14.6 m/s W vCf? E vBf 19.2 m/s Figure 9.53 S Required final velocity of fragment C (v Cf ) Analysis and Solution Choose fragments A, B, and C as
an isolated system. Since no mass is lost, find the mass of fragment C. mT mB) 0.60 kg (0.20 kg 0.18 kg) 0.22 kg (mA mC The original firework has an initial velocity of zero. So the system has an initial momentum of zero. p sysi 0 pAf The momentum of each fragment is in the same direction as its velocity. Calculate the momentum of fragments A and B. mBvBf (0.18 kg)(19.2 m/s) 3.46 kgm/s 3.46 kgm/s [S] mAvAf (0.20 kg)(14.6 m/s) 2.92 kgm/s 2.92 kgm/s [W] pBf p Af p Bf 494 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/24/08 2:44 PM Page 495 Apply the law of conservation of momentum to the system. p p sysi 0 p sysf p Bf p Cf Af Use a vector addition diagram to determine the momentum of fragment C. N 1.00 kgm/s pAf 2.92 kgm/s pBf 3.46 kgm/s pCf? θ E Figure 9.54 From Figure 9.54, careful measurements give pCf 50 N of E. 4.53 kgm/s and pCf vCf Divide the momentum of fragment C by its mass to find the velocity. mCvCf pC f m C 4.53 kgm s 0.22 kg 21 m/s 21 m/s [50 N of E] v Cf Paraphrase The velocity of the third fragment just after the explosion is 21 m/s [50 N of E]. Elastic and Inelastic Collisions in Two Dimensions As with one-dimensional collisions, collisions in two dimensions may be either elastic or inelastic. The condition for an elastic two-dimensional Ekf. collision is the same as for an elastic one-dimensional collision, Eki To determine if a collision is elastic, the kinetic energy values before and after collision must be compared. The kinetic energy of an object only depends on the magnitude of the velocity vector. So it does not matter if the velocity vector has only an x component, only a y component, or both x and y components. If you can determine the magnitude of the
velocity vector, it is possible to calculate the kinetic energy. An example of an inelastic collision occurs when two objects join together and move as a unit immediately after impact. If two objects bounce apart after impact, the collision may be either elastic or inelastic, depending on the initial and final kinetic energy of the system. Usually, if one or both colliding objects deform upon impact, the collision is inelastic. e SIM and vfy Predict vfx for an object just after a twodimensional collision using momentum and energy concepts. Follow the eSim links at www.pearsoned.ca/school/ physicssource. Chapter 9 The momentum of an isolated system of interacting objects is conserved. 495 09-Phys20-Chap09.qxd 7/24/08 2:44 PM Page 496 e WEB Research some design improvements in running shoes. Use momentum and collision concepts to explain how these features affect athletes. Write a brief report of your findings, including diagrams where appropriate. Begin your search at www.pearsoned.ca/ school/physicssource. In track sports, the material used for the track has a profound effect on the elasticity of the collision between a runner’s foot and the running surface (Figure 9.55). If a track is made of a very hard material such as concrete, it experiences very little deformation when a runner’s foot comes in contact with it. The collision is more elastic than if the track were made of a more compressible material such as cork. So less kinetic energy of the runner is converted to other forms of energy upon impact. However, running on harder tracks results in a decreased interaction time and an increase in the net force acting on each foot, which could result in more injuries to joints, bones, and tendons. But a track that is extremely compressible is not desirable either, because it slows runners down. With all the pressure to achieve faster times in Olympic and world competitions, researchers and engineers continue to search for the optimum balance between resilience and safety in track construction. On the other hand, some collisions in sporting events present a very low risk of injury to contestants. Example 9.15 involves determining if the collision between two curling stones is elastic. Figure 9.55 Canadian runner Diane Cummins (far right) competing in the 2003 World Championships. The material of a running surface affects the interaction time and the net force acting on a runner’s feet. How would the net force
change if the track were made of a soft material that deforms easily? Example 9.15 Determine if the collision in Example 9.12 on pages 490 and 491 is elastic. If it is not, what percent of the kinetic energy is retained? Given mA v Bi v Bf 19.6 kg 0 m/s 0.2495 m/s [77.1 W of N] mB v Af 19.6 kg 1.17 m/s [12.0 E of N] v Ai 1.20 m/s [N] Required determine if the collision is elastic Analysis and Solution Choose the two curling stones as an isolated system. Calculate the total initial kinetic energy and the total final kinetic energy of the system. 496 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/24/08 2:44 PM Page 497 Eki 1 1 mA(vAi)2 mB(vBi)2 2 2 1 (19.6 kg)(1.20 m/s)2 0 2 14.11 kgm2/s2 14.11 J Since Eki Find the percent of Ek retained. Ekf, the collision is inelastic. E k 100% % Ek retained f E k i 0 3. 4 1 100% 1 J 1 1. 4 99.4% J Paraphrase The collision is inelastic, and 99.4% of the kinetic energy is retained. (Collisions like this, where very little kinetic energy is lost, may be called “near elastic collisions.”) Conservation Laws and the Discovery of Subatomic Particles Based on the results of experiments, scientists have gained great confidence in the laws of conservation of momentum and of conservation of energy, and have predicted that there are no known exceptions. This confidence has enabled scientists to make discoveries about the existence of particles within atoms as well. You will learn more about subatomic particles in Units VII and VIII. Ekf 1 1 mA(vAf)2 mB(vBf)2 2 2 1 1 (19.6 kg)(0.2495 m/s)2 (19.6 kg)(1.17 m/s)2 2 2 14.03 kgm2/s2 14.03 J Practice Problems 1. A 0.168-kg hockey puck flying at 45.0 m/s [252] is trapped in the pads of an 82.0
-kg goalie moving at 0.200 m/s [0]. The velocity of the centre of mass of the goalie, pads, and puck immediately after collision is 0.192 m/s [333]. Was the collision elastic? If not, calculate the percent of total kinetic energy retained. 2. A 19.0-kg curling stone collides with another identical stationary stone. Immediately after collision, the first stone moves at 0.663 m/s. The second stone, which was stationary, moves at 1.31 m/s. If the collision was elastic, what would have been the speed of the first stone just before collision? Answers 1. inelastic, 0.882% 2. 1.47 m/s In 1930, German scientists Walther Bothe and Wilhelm Becker produced a very penetrating ray of unknown particles when they bombarded the element beryllium with alpha particles (Figure 9.56). An alpha particle is two protons and two neutrons bound together to form a stable particle. In 1932, British scientist James Chadwick (1891–1974) directed rays of these unknown particles at a thin paraffin strip and found that protons were emitted from the paraffin. He analyzed the speeds and angles of the emitted protons and, by using the conservation of momentum, he showed that the protons were being hit by particles of approximately the same mass. In other related experiments, Chadwick was able to determine the mass of these unknown particles very accurately using the conservation of momentum. Earlier experiments had shown that the unknown particles were neutral because they were unaffected by electric or magnetic fields. You will learn about electric and magnetic fields in Unit VI. Chadwick had attempted for several years to find evidence of a suggested neutral particle that was believed to be located in the nucleus of an atom. The discovery of these neutral particles, now called neutrons, resulted in Chadwick winning the Nobel Prize for Physics in 1935. alpha particle 2 neutron beryllium Figure 9.56 The experiment of Bothe and Becker using beryllium paved the way for Chadwick who later discovered the existence of neutrons by creating an experiment where he could detect them. e SIM Practise solving problems involving two-dimensional collisions. Follow the eSim links at www.pearsoned.ca/school/ physicssource. Chapter 9 The momentum of an isolated system of interacting objects is conserved. 497 09-Phys20-Chap09.qxd 7/24/08 2:44
PM Page 498 electron neutron pn 0 kgm/s proton pe pp Figure 9.57 If a neutron is initially stationary, p 0. If the neutron becomes transformed into a proton and an electron moving in the same direction, the momentum of the system is no longer zero. sysi Scientists later found that the neutron, when isolated, soon became transformed into a proton and an electron. Sometimes the electron and proton were both ejected in the same direction, which seemed to contradict the law of conservation of momentum (Figure 9.57). Furthermore, other experiments showed that the total energy of the neutron before transformation was greater than the total energy of both the proton and electron. It seemed as if the law of conservation of energy was not valid either. Austrian physicist Wolfgang Pauli (1900–1958) insisted that the conservation laws of momentum and of energy were still valid, and in 1930, he proposed that an extremely tiny neutral particle produced during the transformation must be moving in the opposite direction at an incredibly high speed. This new particle accounted for the missing momentum and missing energy (Figure 9.58). unknown particle pν neutron pn 0 kgm/s pe electron Many other scientists accepted Pauli’s explanation because they were convinced that the conservation laws of momentum and of energy were valid. For 25 years, they held their belief in the existence of this tiny particle, later called a neutrino, with no other evidence. Then in 1956, the existence of neutrinos was finally confirmed experimentally, further strengthening the universal validity of conservation laws. proton pp Figure 9.58 The existence of another particle accounted for the missing momentum and missing energy observed when a neutron transforms itself into a proton and an electron. THEN, NOW, AND FUTURE Neutrino Research in Canada Canada is a world leader in neutrino research. The SNO project (Sudbury Neutrino Observatory) is a special facility that allows scientists to gather data about these extremely tiny particles that are difficult to detect. The observatory is located in INCO’s Creighton Mine near Sudbury, Ontario, 2 km below Earth’s surface. Bedrock above the mine shields the facility from cosmic rays that might interfere with the observation of neutrinos. The experimental apparatus consists of 1000 t of heavy water encased in an acrylic vessel shaped like a 12-m diameter boiling flask (Figure 9.59). The vessel is surrounded by an array of about 1000 photo detectors,
all immersed in a 10-storey chamber of purified water. When a neutrino collides with a heavy water molecule, a tiny burst of light is emitted, which the photo detectors pick up. Despite all that equipment, scientists only detect an average of about 10 neutrinos a day. So experiments acrylic vessel with heavy water vessel of purified water photo detectors Figure 9.59 The Sudbury Neutrino Observatory is a collaborative effort among 130 scientists from Canada, the U.S., and the U.K. must run for a long time in order to collect enough useful data. Scientists are interested in neutrinos originating from the Sun and other distant parts of the universe. At first, it seemed that the Sun was not emitting as many neutrinos as expected. Scientists thought they would have to modify their theories about the reactions taking place within the core of the Sun. Tripling the sensitivity of the detection process, by adding 2 t of salt to the heavy water, showed that 2 of the neutrinos from the 3 Sun were being transformed into different types of neutrinos as they travelled. This discovery has important implications about the basic properties of a neutrino, including its mass. It now appears that scientists’ theories about the reactions within the core of the Sun are very accurate. Continued research at this facility will help answer fundamental questions about matter and the universe. Questions 1. Why was Sudbury chosen as the site for this type of observatory? 2. Explain why, at first, it appeared that the Sun was not emitting the expected number of neutrinos. 3. If a neutrino has a very small mass and travels very fast, why doesn’t it run out of energy? 498 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/24/08 2:44 PM Page 499 9.4 Check and Reflect 9.4 Check and Reflect Knowledge 1. How is a two-dimensional collision different from a one-dimensional collision? Explain, using examples. 2. In your own words, state the law of conservation of momentum for twodimensional collisions. Show how the law relates to x and y components by using an example. 3. In your own words, define the centre of mass of an object. 4. Explain why scientists accepted the existence of the neutrino for so long when there was no direct evidence for it. Applications 5. A cue ball travelling at 0.785 m/s [270] strikes
a stationary five-ball, causing it to move at 0.601 m/s [230]. The cue ball and the five-ball each have a mass of 160 g. What will be the velocity of the cue ball immediately after impact? Ignore frictional and rotational effects. 6. A stationary 230-kg bumper car in a carnival is struck off centre from behind by a 255-kg bumper car moving at 0.843 m/s [W]. The more massive car bounces off at 0.627 m/s [42.0 S of W]. What will be the velocity of the other bumper car immediately after collision? 7. A 0.25-kg synthetic rubber ball bounces to a height of 46 cm when dropped from a height of 50 cm. Determine if this collision is elastic. If not, how much kinetic energy is lost? 8. A football halfback carrying the ball, with a combined mass of 95 kg, leaps toward the goal line at 4.8 m/s [S]. In the air at the goal line, he collides with a 115-kg linebacker travelling at 4.1 m/s [N]. If the players move together after impact, will the ball cross the goal line? 9. A 0.160-kg pool ball moving at 0.563 m/s [67.0 S of W] strikes a 0.180-kg snooker ball moving at 0.274 m/s [39.0 S of E]. The pool ball glances off at 0.499 m/s [23.0 S of E]. What will be the velocity of the snooker ball immediately after collision? 10. A 4.00-kg cannon ball is flying at 18.5 m/s [0] when it explodes into two fragments. One 2.37-kg fragment (A) goes off at 19.7 m/s [325]. What will be the velocity of the second fragment (B) immediately after the explosion? Assume that no mass is lost during the explosion, and that the motion of the fragments lies in the xy plane. 11. A 0.952-kg baseball bat moving at 35.2 m/s [0] strikes a 0.145-kg baseball moving at 40.8 m/s [180]. The baseball rebounds at 37.6 m/s [64.2]. What will be the velocity of the centre of mass of the bat immediately after collision if the batter exerts no force
4-m drop, where the impulse changes the magnitude and direction of the momentum while maintaining the integrity of the balloon. The water balloon must begin with a vertical drop equivalent to eight storeys. Then for the equivalent height of six storeys, the balloon must change direction and come to a stop horizontally. Planning Form a design team of three to five members. Plan and assign roles so that each team member has at least one major task. Roles may include researcher, engineer to perform mathematical calculations, creative designer, construction engineer, materials acquisition officer, and writer, among others. One person may need to perform several roles in turn. Ensure that all team members help along the way. Prepare a time schedule for each task, and for group planning and reporting sessions. Materials • small balloon filled with water • plastic zip-closing bag to contain the water balloon • cardboard and/or wooden frame for apparatus • vehicle or container for balloon • cushioning material • braking device • art materials CAUTION: Test your design in an appropriate area. Make sure no one is in the way during the drop. Assessing Results After completing the project, assess its success based on a rubric designed in class* that considers research strategies experiment and construction techniques clarity and thoroughness of the written report effectiveness of the team’s presentation quality and fairness of the teamwork Procedure 1 Research the range of acceptable accelerations that most people can tolerate. 2 Calculate the maximum speed obtained when an object is dropped from an eight-storey building (equivalent to 24.6 m). 3 Calculate the impulse necessary to change the direction of motion of a 75.0-kg person from vertical to horizontal in the remaining height of six storeys (equivalent to 18.4 m). The person must come to a stop at the end. Assume that the motion follows the arc of a circle. 4 Determine the time required so that the change in the direction of motion and stopping the person meets the maximum acceptable acceleration in step 1. 5 Include your calculations in a report that shows your design and method of changing the motion. 6 Build a working model and test it. Make modifications as necessary to keep the water balloon intact for a fall. Present the project to your teacher and the class. Thinking Further 1. Explain why eight storeys was used in the calculation in step 2, instead of 14 storeys. 2. What other amusement rides has your team thought of while working on this project? What would make each of these rides thrilling and appealing? 3.
In what ways could your ideas have a practical use, such as getting people off a high oil derrick or out of a high-rise building quickly and safely? 4. What conditions would cause a person to be an unacceptable candidate for your ride? Write out a list of rules or requirements that would need to be posted. *Note: Your instructor will assess the project using a similar assessment rubric. Unit V Momentum and Impulse 501 09-Phys20-Chap09.qxd 7/24/08 2:44 PM Page 502 UNIT V SUMMARY Unit Concepts and Skills: Quick Reference Concepts CHAPTER 9 Momentum Newton’s second law Impulse Effects of varying the net force and time interval for a given impulse Summary Resources and Skill Building The momentum of an isolated system of interacting objects is conserved. 9.1 Momentum Is Mass Times Velocity Momentum is the product of the mass of an object and its velocity. Momentum is a vector quantity measured in kilogram-metres per second (kgm/s). Newton’s second law states that the net force on an object is equal to the rate of change of its momentum. 9.2 Impulse Is Equivalent to a Change in Momentum The impulse provided to an object is defined as the product of the net force (or average net force if F the interaction time. Impulse is equivalent to the change in momentum of the object. constant) acting on the object during an interaction and net The magnitude of the net force during an interaction and the interaction time determine whether or not injuries or damage to an object occurs. Examples 9.1 & 9.2 9-2 QuickLab Figures 9.12 & 9.13 9-3 Design a Lab Example 9.3 Example 9.4 Net force-time graph Impulse can be determined by calculating the area under a net force-time graph. Conservation of momentum in one dimension Elastic collisions Inelastic collisions Conservation of momentum in two dimensions Elastic and inelastic collisions 9.3 Collisions in One Dimension Momentum is conserved when objects in an isolated system interact in one dimension. A system is the group of objects that interact with each other, and it is isolated if no external net force acts on these objects. 9-4 QuickLab 9-5 Inquiry Lab Examples 9.5–9.8, 9.10 Elastic collisions are collisions in which a system of objects has the same initial and final kinetic energy. So both the momentum and kinetic energy of the system
are conserved. Example 9.9 Inelastic collisions are collisions in which a system of objects has different initial and final kinetic energy values. Example 9.11 9.4 Collisions in Two Dimensions Momentum is conserved when objects in an isolated system interact in two dimensions. An isolated system has no external net force acting on it. Elastic collisions in two dimensions satisfy the same conditions as one-dimensional elastic collisions, that is, Eki Ekf. 9-6 Inquiry Lab Examples 9.12–9.14 Example 9.15 502 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/24/08 2:44 PM Page 503 UNIT V REVIEW Vocabulary 1. Using your own words, define these terms, concepts, principles, or laws. momentum impulse one-dimensional collisions conservation of momentum conservation of energy elastic collisions inelastic collisions two-dimensional collisions centre of mass Knowledge CHAPTER 9 2. Compare and contrast momentum and impulse. 3. Explain the relationship between the units in which momentum and impulse are measured. 4. A student calculated the answer to a problem and got 40 kgm/s [W]. Which quantities could the student have calculated? 5. In your own words, restate Newton’s second law in terms of momentum. 6. What difference does it make that momentum is a vector quantity rather than a scalar quantity? 7. Compare and contrast a net force and an average net force acting on an object during an interaction. 8. Statistics show that less massive vehicles tend to have fewer accidents than more massive vehicles. However, the survival rate for accidents in more massive vehicles is much greater than for less massive ones. How could momentum be used to explain these findings? 9. Using the concept of impulse, explain how the shocks on a high-end mountain bike reduce the chance of strain injuries to the rider. 10. State the quantities, including units, you would need to measure to determine the momentum of an object. 11. State the quantities that are conserved in oneand two-dimensional collisions. Give an example of each type of collision. 12. How do internal forces affect the momentum of a system? 13. What instructions would you give a young gymnast so that she avoids injury when landing on a hard surface? 14. Will the magnitude of the momentum of an object always increase if a net force acts on it? Explain, using an example. 15. What quantity do you get when p is divided by mass? 16
. For a given impulse, what is the effect of (a) increasing the time interval? (b) decreasing the net force during interaction? 17. For each situation, explain how you would effectively provide the required impulse. • to catch a water balloon tossed from some distance • to design a hiking boot for back-country hiking on rough ground • to shoot an arrow with maximum velocity using a bow • for an athlete to win the gold medal in the javelin event with the longest throw • for a car to accelerate on an icy road 18. Why does a hunter always press the butt of a shotgun tight against the shoulder before firing? 19. Describe a method to find the components of a momentum vector. 20. Explain why the conservation of momentum and the conservation of energy are universal laws. 21. Why does the law of conservation of momentum require an isolated system? 22. Suppose a problem involves a two-dimensional collision between two objects, and the initial momentum of one object is unknown. Explain how to solve this problem using (a) a vector addition diagram drawn to scale (b) vector components 23. Explain, in terms of momentum, why a rocket does not need an atmosphere to push against when it accelerates. 24. If a firecracker explodes into two fragments of unequal mass, which fragment will have the greater speed? Why? Unit V Momentum and Impulse 503 09-Phys20-Chap09.qxd 7/24/08 2:44 PM Page 504 38. Draw a momentum vector diagram to represent a 575-g basketball flying at 12.4 m/s [26.0 S of E]. 39. Calculate the momentum of a 1250-kg car travelling south at 14.8 m/s. 40. A bowling ball has a momentum of 28 kgm/s [E]. If its speed is 4.5 m/s, what is the mass of the ball? 41. A curling stone has a momentum of 32 kgm/s [W]. What would be the momentum if the mass of the stone is decreased to 7 of its original mass 8 and its speed is increased to 4 of its original 3 speed? 42. A soccer ball has a momentum of 2.8 kgm/s [W]. What would be the momentum if its mass decreased to 3 of its original mass and its 4 speed increased to 9 of its original speed? 8 43. The graph below shows the magnitude of the net force as a function of interaction time for a
volleyball being blocked. The velocity of the ball changes from 18 m/s [N] to 11 m/s [S]. (a) Using the graph, calculate the magnitude of the impulse on the volleyball. (b) What is the mass of the ball? Magnitude of Net Force vs. Interaction Time for a Volleyball Block ) 5000 4000 3000 2000 1000 0 0.0 1.0 3.0 2.0 Time t (ms) 4.0 5.0 44. (a) Calculate the impulse on a soccer ball if a player heads the ball with an average net force of 120 N [210] for 0.0252 s. (b) If the mass of the soccer ball is 0.44 kg, calculate the change in velocity of the ball. 45. At a buffalo jump, a 900-kg bison is running at 6.0 m/s toward the drop-off ahead when it senses danger. What horizontal force must the bison exert to stop itself in 2.0 s? 25. When applying the conservation of momentum to a situation, why is it advisable to find the velocities of all objects in the system immediately after collision, instead of several seconds later? 26. Which physics quantities are conserved in a collision? 27. A curling stone hits another stationary stone off centre. Draw possible momentum vectors for each stone immediately before and immediately after collision, showing both the magnitude and direction of each vector. 28. A Superball™ of rubber-like plastic hits a wall perpendicularly and rebounds elastically. Explain how momentum is conserved. 29. What two subatomic particles were discovered using the conservation of momentum and the conservation of energy? 30. Explain how an inelastic collision does not violate the law of conservation of energy. 31. If a system is made up of only one object, show how the law of conservation of momentum can be used to derive Newton’s first law. 32. A Calgary company, Cerpro, is a world leader in ceramic armour plating for military protection. The ceramic structure of the plate transmits the kinetic energy of an armour-piercing bullet throughout the plate, reducing its penetrating power. Explain if this type of collision is elastic or inelastic. 33. A compact car and a heavy van travelling at approximately the same speed perpendicular to each other collide and stick together. Which vehicle will experience the greatest change in its direction of motion just after impact? Why? 34. Is it possible for
the conservation of momentum to be valid if two objects move faster just before, than just after, collision? Explain, using an example. 35. Fighter pilots have reported that immediately after a burst of gunfire from their jet fighter, the speed of their aircraft decreased by 50–65 km/h. Explain the reason for this change in motion. 36. A cannon ball explodes into three fragments. One fragment goes north and another fragment goes east. Draw the approximate direction of the third fragment. What scientific law did you use to arrive at your answer? Applications 37. Calculate the momentum of a 1600-kg car travelling north at 8.5 m/s. 504 Unit V Momentum and Impulse 09-Phys20-Chap09.qxd 7/24/08 2:44 PM Page 505 46. (a) What is the minimum impulse needed to 52. A 0.146-kg baseball pitched at 40 m/s is hit back give a 275-kg motorcycle and rider a velocity of 20.0 m/s [W] if the motorcycle is initially at rest? (b) If the wheels exert an average force of 710 N [E] to the road, what is the minimum time needed to reach a velocity of 20.0 m/s [W]? (c) Explain how the force directed east causes the motorcycle to accelerate westward. (d) Why is it necessary to specify a minimum impulse and a minimum time? 47. A 1.15-kg peregrine falcon flying at 15.4 m/s [W] captures a 0.423-kg pigeon flying at 4.68 m/s [S]. What will be the velocity of their centre of mass immediately after the interaction? 48. A 275-kg snowmobile carrying a 75-kg driver exerts a net backward force of 508 N on the snow for 15.0 s. (a) What impulse will the snow provide to the snowmobile and driver? (b) Calculate the change in velocity of the snowmobile. 49. The graph below shows the magnitude of the net force as a function of time for a 275-g volleyball being spiked. Assume the ball is motionless the instant before it is struck. (a) Using the graph, calculate the magnitude of the impulse on the volleyball. (b) What is the speed of the ball when it leaves the player’s hand Magnitude of Net Force vs. Interaction Time for a Volleyball Spike 2000
1500 1000 500 0 0.0 1.0 2.0 3.0 Time t (ms) 4.0 5.0 6.0 50. A Centaur rocket engine expels 520 kg of exhaust gas at 5.0 104 m/s in 0.40 s. What is the magnitude of the net force on the rocket that will be generated? 51. An elevator with passengers has a total mass of 1700 kg. What is the net force on the cable needed to give the elevator a velocity of 4.5 m/s [up] in 8.8 s if it is starting from rest? toward the pitcher at a speed of 45 m/s. (a) What is the impulse provided to the ball? (b) The bat is in contact with the ball for 8.0 ms. What is the average net force that the bat exerts on the ball? 53. An ice dancer and her 80-kg partner are both gliding at 2.5 m/s [225]. They push apart, giving the 45-kg dancer a velocity of 3.2 m/s [225]. What will be the velocity of her partner immediately after the interaction? 54. Two students at a barbecue party put on inflatable Sumo-wrestling outfits and take a run at each other. The 87.0-kg student (A) runs at 1.21 m/s [N] and the 73.9-kg student (B) runs at 1.51 m/s [S]. The students are knocked off their feet by the collision. Immediately after impact, student B rebounds at 1.03 m/s [N]. (a) Assuming the collision is completely elastic, calculate the speed of student A immediately after impact using energy considerations. (b) How different would your answer be if only conservation of momentum were used? Calculate to check. (c) How valid is your assumption in part (a)? 55. A cannon mounted on wheels has a mass of 1380 kg. It shoots a 5.45-kg projectile at 190 m/s [forward]. What will be the velocity of the cannon immediately after firing the projectile? 56. A 3650-kg space probe travelling at 1272 m/s [0.0] has a directional thruster rocket exerting a force of 1.80 104 N [90.0] for 15.6 s. What will be the newly adjusted velocity of the probe? 57. In a movie stunt, a 1.
60-kg pistol is struck by a 15-g bullet travelling at 280 m/s [50.0]. If the bullet moves at 130 m/s [280] after the interaction, what will be the velocity of the pistol? Assume that no external force acts on the pistol. 58. A 52.5-kg snowboarder, travelling at 1.24 m/s [N] at the end of her run, jumps and kicks off her 4.06-kg snowboard. The snowboard leaves her at 2.63 m/s [62.5 W of N]. What is her velocity just after she kicks off the snowboard? 59. A 1.26-kg brown bocce ball travelling at 1.8 m/s [N] collides with a stationary 0.145-kg white ball, driving it off at 0.485 m/s [84.0 W of N]. (a) What will be the velocity of the brown ball immediately after impact? Ignore friction and rotational effects. (b) Determine if the collision is elastic. Unit V Momentum and Impulse 505 09-Phys20-Chap09.qxd 7/24/08 2:44 PM Page 506 60. Two people with a combined mass of 128 kg are sliding downhill on a 2.0-kg toboggan at 1.9 m/s. A third person of mass 60 kg inadvertently stands in front and upon impact is swept along with the toboggan. If all three people remain on the toboggan after impact, what will be its velocity after impact? 61. An aerosol paint can is accidentally put in a fire pit. After the fire is lit, the can is heated and explodes into two fragments. A 0.0958-kg fragment (A) flies off at 8.46 m/s [E]. The other fragment (B) has a mass of 0.0627 kg. The 0.0562-kg of gas inside bursts out at 9.76 m/s [N]. What will be the velocity of fragment B immediately after the explosion? Assume that no mass is lost during the explosion, and that the motion of the fragments lies in a plane. 62. A 0.185-kg golf club head travelling horizontally at 28.5 m/s hits a 0.046-kg golf ball, driving it straight off at 45.7 m/s. (a) Suppose the golfer does not exert
an external force on the golf club after initial contact with the ball. If the collision between the golf club and the ball is elastic, what will be the speed of the club head immediately after impact? (b) Show that the law of conservation of momentum is valid in this interaction. 63. A student on a skateboard is travelling at 4.84 m/s [0], carrying a 0.600-kg basketball. The combined mass of the student and skateboard is 50.2 kg. He throws the basketball to a friend at a velocity of 14.2 m/s [270]. What is the resulting velocity of the centre of mass of the student-skateboard combination immediately after the throw? Ignore frictional effects. 64. An oxygen molecule of mass 5.31 10–26 kg with a velocity of 4.30 m/s [0.0°] collides headon with a 7.31 10–26-kg carbon dioxide molecule which has a velocity of 3.64 m/s [180.0°]. After collision the oxygen molecule has a velocity of 4.898 m/s [180.0°]. (a) Calculate the velocity of the carbon dioxide molecule immediately after collision. (b) Determine by calculation whether or not the collision is elastic. 65. An isolated stationary neutron is transformed into a 9.11 1031-kg electron travelling at 4.35 105 m/s [E] and a 1.67 1027-kg proton travelling at 14.8 m/s [E]. What is the momentum of the neutrino that is released? 506 Unit V Momentum and Impulse 66. An 8.95-kg bowling ball moving at 3.62 m/s [N] hits a 0.856-kg bowling pin, sending it off at 3.50 m/s [58.6 E of N]. (a) What will be the velocity of the bowling ball immediately after collision? (b) Determine if the collision is elastic. 67. A wooden crate sitting in the back of a pickup truck travelling at 50.4 km/h [S] has a momentum of magnitude 560 kgm/s. (a) What is the mass of the crate? (b) What impulse would the driver have to apply with the brakes to stop the vehicle in 5.25 s at an amber traffic light? Use mT for the total mass of the truck. (c) If the coefficient of friction
between the crate and the truck bed is 0.30, will the crate slide forward as the truck stops? Justify your answer with calculations. 68. A firecracker bursts into three fragments. An 8.5-g fragment (A) flies away at 25 m/s [S]. A 5.6-g fragment (B) goes east at 12 m/s. Calculate the velocity of the 6.7-g fragment (C). Assume that no mass is lost during the explosion, and that the motion of the fragments lies in a plane. 69. A spherical molecule with carbon atoms arranged like a geodesic dome is called a buckyball. A 60-atom buckyball (A) of mass 1.2 10–24 kg travelling at 0.92 m/s [E] collides with a 70-atom buckyball (B) of mass 1.4 10–24 kg with a velocity of 0.85 m/s [N] in a laboratory container. Buckyball (A) bounces away at a velocity of 1.24 m/s [65° N of E]. (a) Calculate the speed of buckyball (B) after the collision assuming that this is an elastic collision. (b) Use the conservation of momentum to find the direction of buckyball (B) after the collision. 70. A moose carcass on a sled is being pulled by a tow rope behind a hunter’s snowmobile on a horizontal snowy surface. The sled and moose have a combined mass of 650 kg and a momentum of 3.87 103 kgm/s [E]. (a) Calculate the velocity of the moose and sled. (b) The magnitude of the force of friction between the sled and the snow is 1400 N. As the hunter uniformly slows the snowmobile, what minimum length of time is needed for him to stop and keep the sled from running into the snowmobile (i.e., keep the same distance between the sled and the snowmobile)? 09-Phys20-Chap09.qxd 7/24/08 2:44 PM Page 507 71. A 940-kg car is travelling at 15 m/s [W] when it is struck by a 1680-kg van moving at 20 m/s [50.0 N of E]. If both vehicles join together after impact, what will be the velocity of their centre of mass immediately after impact? 78. Research
[36.0]. The cue ball and three-ball each have a mass of 0.160 kg. Calculate the velocity of the cue ball immediately after collision. Ignore friction and rotational effects. 74. A hunter claims to have shot a charging bear through the heart and “dropped him in his tracks.” To immediately stop the bear, the momentum of the bullet would have to be as great as the momentum of the charging bear. Suppose the hunter was shooting one of the largest hunting rifles ever sold, a 0.50 caliber Sharps rifle, which delivers a 2.27 102 kg bullet at 376 m/s. Evaluate the hunter’s claim by calculating the velocity of a 250-kg bear after impact if he was initially moving directly toward the hunter at a slow 0.675 m/s [S]. 75. An object explodes into three fragments (A, B, and C) of equal mass. What will be the approximate direction of fragment C if (a) both fragments A and B move north? (b) fragment A moves east and fragment B moves south? (c) fragment A moves [15.0] and fragment B moves [121]? Extensions 76. Research the physics principles behind the design of a Pelton wheel. Explain why it is more efficient than a standard water wheel. Begin your search at www.pearsoned.ca/school/physicssource. 77. A fireworks bundle is moving upward at 2.80 m/s when it bursts into three fragments. A 0.210-kg fragment (A) moves at 4.52 m/s [E]. A 0.195-kg fragment (B) flies at 4.63 m/s [N]. What will be the velocity of the third fragment (C) immediately after the explosion if its mass is 0.205 kg? Assume that no mass is lost during the explosion. 79. Two billiard balls collide off centre and move at right angles to each other after collision. In what directions did the impulsive forces involved in the collision act? Include a diagram in your answer. 80. A 2200-kg car travelling west is struck by a 2500-kg truck travelling north. The vehicles stick together upon impact and skid for 20 m [48.0 N of W]. The coefficient of friction for the tires on the road surface is 0.78. Both drivers claim to have been travelling at 90 km/h before the crash. Determine the truth of their
statements. 81. Research the developments in running shoes that help prevent injuries. Interview running consultants, and consult sales literature and the Internet. How does overpronation or underpronation affect your body’s ability to soften the road shock on your knees and other joints? Write a brief report of your findings. Begin your search at www.pearsoned.ca/school/physicssource. 82. A 3.5-kg block of wood is at rest on a 1.75-m high fencepost. When a 12-g bullet is fired horizontally into the block, the block topples off the post and lands 1.25 m away. What was the speed of the bullet immediately before collision? Consolidate Your Understanding 83. Write a paragraph describing the differences between momentum and impulse. Include an example for each concept. 84. Write a paragraph describing how momentum and energy concepts can be used to analyze the motion of colliding objects. Include two examples: One is a one-dimensional collision and the other is a two-dimensional collision. Include appropriate diagrams. Think About It Review your answers to the Think About It questions on page 447. How would you answer each question now? e TEST To check your understanding of momentum and impulse, follow the eTest links at www.pearsoned.ca/school/physicssource. Unit V Momentum and Impulse 507 10-PearsonPhys30-Chap10 7/24/08 2:51 PM Page 508 U N I T VI Forces Forces and Fields and Fields On huge metal domes, giant electrostatic charge generators can create voltages of 5 000 000 V, compared with 110 V in most of your household circuits. How are electrostatic charges produced? What is voltage? What happens when electric charges interact? e WEB The person in this photo is standing inside a Faraday cage. To find out how the Faraday cage protects her from the huge electrical discharges, follow the links at www.pearsoned.ca/school/physicssource. 508 Unit VI 10-PearsonPhys30-Chap10 7/24/08 2:51 PM Page 509 Unit at a Glance C H A P T E R 1 0 Physics laws can explain the behaviour of electric charges. 10.1 Electrical Interactions 10.2 Coulomb’s Law C H A P T E R 1 1 Electric field theory describes electrical phenomena. 11.1 Forces and Fields 11.2 Electric Field
Lines and Electric Potential 11.3 Electrical Interactions and the Law of Conservation of Energy C H A P T E R 1 2 Properties of electric and magnetic fields apply in nature and technology. 12.1 Magnetic Forces and Fields 12.2 Moving Charges and Magnetic Fields 12.3 Current-carrying Conductors and Magnetic Fields 12.4 Magnetic Fields, Moving Charges, and New and Old Technologies Unit Themes and Emphases • Energy and Matter • Nature of Science • Scientific Inquiry Focussing Questions While studying this unit, you will investigate how the science of electricity, magnetism, and electromagnetism evolved and its corresponding effect on technology. As you work through this unit, consider these questions. • How is the value of the elementary charge determined? • What is the relationship between electricity and magnetism? • How does magnetism assist in the understanding of fundamental particles? Unit Project Building a Model of a Direct Current Generator • By the time you complete this unit, you will have the knowledge and skills to build a model of a direct current generator. For this task, you will research wind power and design and build a model of an electric generator that uses wind energy. Unit VI Forces and Fields 509 10-PearsonPhys30-Chap10 7/24/08 2:51 PM Page 510 Physics laws can explain the behaviour of electric charges. Figure 10.1 A thunderbird on a totem pole in Vancouver Abolt of lightning flashing across dark cloudy skies, followed a few moments later by the deafening sound of thunder, is still one of the most awe-inspiring physical events unleashed by nature. What is the cause of lightning? Why is it so dangerous? So powerful is this display that many early civilizations reasoned these events must be the actions of gods. To the Romans, lightning was the sign that Jove, the king of the gods, was angry at his enemies. In some First Nations traditions, lightning flashed from the eyes of the enormous thunderbird, while thunder boomed from the flapping of its huge wings (Figure 10.1). In this chapter, you will learn how relating lightning to simpler phenomena, such as the sparking observed as you stroke a cat, initially revealed the electrical nature of matter. Further studies of the nature of electric charges and the electrical interactions between them will enable you to understand laws that describe their behaviour. Finally, you will investigate the force acting on electric charges by studying the variables that determine this force and the law that describes how to calculate such forces. C H A P
T E R 10 Key Concepts In this chapter, you will learn about: electric charge conservation of charge Coulomb’s law Learning Outcomes When you have completed this chapter, you will be able to: Knowledge explain electrical interactions using the law of conservation of charge explain electrical interactions in terms of the repulsion and attraction of charges compare conduction and induction explain the distribution of charge on the surfaces of conductors and insulators use Coulomb’s law to calculate the electric force on a point charge due to a second point charge explain the principles of Coulomb’s torsion balance experiment determine the magnitude and direction of the electric force on a point charge due to one or more stationary point charges in a plane compare, qualitatively and quantitatively, the inverse square relationship as it is expressed by Coulomb’s law and by Newton’s universal law of gravitation Science, Technology, and Society explain that concepts, models, and theories are often used in predicting, interpreting, and explaining observations explain that scientific knowledge may lead to new technologies and new technologies may lead to scientific discoveries 510 Unit VI 10-PearsonPhys30-Chap10 7/24/08 2:51 PM Page 511 10-1 QuickLab 10-1 QuickLab Charging Objects Using a Van de Graaff Generator Problem What can demonstrations on the Van de Graaff generator reveal about the behaviour and interactions of electric charges? Materials and Equipment Van de Graaff generator and grounding rod small piece of animal fur (approximately 15 cm x 15 cm) 5 aluminium pie plates small foam-plastic cup with confetti soap bubble dispenser and soap CAUTION! Follow your teacher’s instructions to avoid getting an electric shock. Procedure 1 Copy Table 10.1 into your notebook. Make the table the full width of your page so you have room to write in your observations and explanations. Table 10.1 Observations and Explanations from Using a Van de Graaff Generator 2 Watch or perform each of the demonstrations in steps 3 to 9. 3 Place a piece of animal fur, with the fur side up, on the top of the charging sphere of the Van de Graaff generator. 4 Turn on the generator and let it run. 5 Record your observations in Table 10.1, making sure that your description is precise. 6 Ground the sphere with the grounding rod, and turn off the generator. 7 Repeat steps 3 to 6, replacing the animal fur with the aluminium pie plates (stacked upside down), and then the foam-plastic

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