Datasets:
vnpthack
/
vnpt_data

Dataset card Data Studio Files
xet
 Community
1
text
stringlengths
0
6.73k
s=ur+t0+1p+ ¬- `...
có dt dị. dị dt
s=c hiền 0
s=tut te LG œdu bdu c du
có dc C\u dc dc. +” đt
8-4 Distance as an integral
Now we have to discuss the inverse problem. Suppose that instead of a table of
distances, we have a table of speeds at diferent times, starting from zero. Eor the
falling ball, such speeds and tỉmes are shown in Table 8-4. A similar table could
be constructed for the velocity of the car, by recording the speedometer reading
every minute or half-minute. If we know how fast the car is goiïng at any tỉme,
can we determine how far it goes? This problem is just the inverse of the one
solved above; we are given the velocity and asked to ñnd the distance. How can
we find the distance if we know the speed? If the speed of the car is not constant,
and the lady goes sixty miles an hour for a moment, then slows down, speeds up,
Table 8-4
Velocity of a Falling Ball
£ (sec) | ® (ñ/sec)
--- Trang 171 ---
and so on, how can we determine how far she has gone? 'Phat is easy. We use
the same idea, and express the distance in terms of infnitesimals. Let us say, “In
the fñrst second her speed was such and such, and from the formula As = Af
we can calculate how far the car went the first second at that speed.” Now ín the
next second her speed is nearly the same, but slightly diferent; we can calculate
how far she went in the next second by taking the new speed times the time.
W©e proceed similarly for each second, to the end of the run. We now have a
number of little distances, and the total distance will be the sum of all these little
pieces. Thhat is, the distance will be the sum of the velocities times the times,
or s= 0 Af, where the Greek letter ồ” (sigma) is used to denote addition. To
be more precise, it is the sum of the velocity at a certain time, let us say the
¡-bh time, multipled by A¿.
s= » u(;) At. (8.6)
The rule for the times is that f;¿‡¡ = f¿ + A. However, the distance we obtain by
this method will not be correct, because the velocity changes during the time
interval A¿. TH we take the times short enough, the sum is precise, so we take
them smaller and smaller until we obtain the desired accuracy. 'Phe true s is
s= Am. » u(;) At. (8.7)
The mathematicians have invented a symbol for this limit, analogous to the
symbol for the diferemial. The A turns into a đ to remind us that the time is as
small as it can be; the velocity is then called 0 at the time f, and the addition is
written as a sum with a great “s,” ƒ (from the Latin sưmzna), which has become
distorted and is now unfortunately just called an integral sign. Thus we write
s= Tao đt. (8.8)
This process of adding all these terms together is called integration, and it 1s
the opposite process to diferentiation. "The derivative of this integral 1s 0, SO
one operabor (đ) undoes the other (ƒ). One can get formulas for integrals by
taking the formulas for derivatives and running them backwards, because they are
related to each other inversely. 'Thus one can work out his own table of integrals
--- Trang 172 ---
by diferentiating all sorts of functions. For every formula with a diferential, we
get an integral formula if we turn it around.
lvery function can be diferentiated analytically, i.e., the process can be
carried out algebraically, and leads to a defñnite function. But it is not possible
in a simple manner to write an analytical value for any integral at will. You can
calculate it, for instance, by doing the above sum, and then doïng it again with a
fñner interval A‡ and again with a fñiner interval until you have it nearly right. In
general, given some particular function, ït is not possible to find, analytically, what
the integral is. One may always try to ñnd a function which, when diferentiated,
gives some desired function; but one may not fnd ït, and it may not exist, In
the sense oŸ being expressible in terms oŸ functions that have already been given
names.
8-5 Acceleration
The next step in developing the equations of motion is to introduce another
idea which goes beyond the concept of velocity to that oŸ change of velocity,
and we now ask, “How does the velocity change?” In previous chapters we have
discussed cases in which forces produce changes in velocity. You may have heard
with great excitement about some car that can get from rest to 60 miles an hour
in ten seconds at. From such a performance we can see how fast the speed
changes, but only on the average. What we shall now discuss is the next level of
complexity, which is how fast the velocity is changing. In other words, by how
many feet per second does the velocity change in a second, that is, how many
feet per second, per second? We previously derived the formula for the velocity
of a falling body as = 32f, which is charted in Table 8-4, and now we want to
fnd out how much the velocity changes per second; this quantity ¡is called the
acceleration.
Acceleration is defined as the time rate of change of velocity. From the
preceding discussion we know enough already to write the acceleration as the
derivative đu /đứ, in the same way that the velocity is the derivative of the distance.
Tf we now diferentiate the formula = 32 we obtain, for a falling body,
a= TE 32. (8.9)
[To diferentiate the term 32 we can utilize the result obtained in a previous
problem, where we found that the derivative of Đứ is simply Ö (a constant). So
--- Trang 173 ---
by letting = 32, we have at once that the derivative of 32 is 32.] This means
that the velocity of a falling body is changing by 32 feet per second, per second
always. We also see from Table 8-4 that the velocity increases by 32 ft/sec in each
second. 'Phis is a very simple case, for accelerations are usually not constant. The
reason the acceleration is constant here is that the force on the falling body is
constant, and Newton”s law says that the acceleration is proportional to the force.
As a further example, let us find the acceleration in the problem we have
already solved for the velocity. Starting with
s= Af + Bt+
we obtained, for ø = ds/dt,
0 =3Ai/2 + B.
Since acceleration is the derivative of the velocity with respect to the time, we
need to diferentiate the last expression above. Recall the rule that the derivative
of the two terms on the right equals the sum of the derivatives of the individual
terms. To diferentiate the first of these terms, instead of going through the
fundamental process again we note that we have already difÑferentiated a quadratic
term when we differentiated 16/2, and the efect was to double the numerical