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P &—1 =m: {]@œ)
tr — = hịc
>> V —V ~==—
EP 4+ _—1 E=iNH) 4144)
pm +ằĂHẶ} ẽ {e
Fig. 10-3. Schematic view of action-reaction experiment with equal
masses.
--- Trang 203 ---
VIEW FROM VIEW FROM
CENTER OF MASS MOVING CAR
(CAR VELOCITY = —v)
v => -——v 2v-> 0
BEFORE COLLISION
v=0 V_>
AFTER COLLISION
Fig. 10-4. TWo views of an inelastic collision between equal masses.
Now the next thing we would like to fñgure out is what happens in a less
simple situation. Suppose we have ÿwo equal masses, one moving with velocity 0
and the other standing still, and they collide and stick; what is goïing to happen?
There is a mass 2mm altogether when we are fñnished, drifting with an unknown
velocity. What velocity? 'That is the problem. 'To find the answer, we make the
assumption that if we ride along in a car, physics will look the same as if we
are standing still. We start with the knowledge that two equal masses, moving
in opposite directions with equal speeds 0, will stop dead when they collide.
Now suppose that while this happens, we are riding by in an automobile, at a
velocity —ø. Then what does ít look like? Since we are riding along with one
of the two masses which are coming together, that one appears to us to have
zero velocity. The other mass, however, going the other way with velocity , will
appear to be coming toward us at a velocity 20 (Eig. 10-4). Finally, the combined
masses after collision will seem to be passing by with velocity 0. We therefore
conclude that an object with velocity 2u, hitting an equal one at rest, will end
up with velocity 0, or what is mathematically exactly the same, an object with
velocity œ hitting and sticking to one at rest will produce an object moving with
velocity 0/2. Note that if we multiply the mass and the velocity beforehand and
add them together, mo + 0, we get the same answer as when we multiply the
mass and the velocity of everything afterwards, 2w times 0/2. So that tells us
what happens when a mass of velocity 0 hits one standing still.
In exactly the same manner we can deduce what happens when equal objects
having am two velocities hit each other.
Suppose we have two equal bodies with velocities ø¡ and 0s, respectively,
which collide and stick together. What is their velocity 0 after the collision?
Again we ride by in an automobile, say at velocity 0a, so that one body appears
to be at rest. The other then appears to have a velocity 0 — 0a, and we have
the same case that we had before. When it is all ñnished they will be moving
--- Trang 204 ---
VIEW FROM. “LAB” VIEW FROM CAR
VỊ va VỊ — Vạ= 0
BEFORE COLLISION
v->~ 1/2(vị — v›)->
AFTER COLLISION
Fig. 10-5. Two views of another inelastic collision between equal
masses.
ab 2(0ị — 0s) with respect to the car. What then is the acbual speed on the
ground?
TEis ø = 3(0Ị — 02) + 0a or š(0 + 92) (Fig. 10-5). Again we note that
T0 -Ƒ tmuua = 2m(01 + 0a) /2. (10.6)
'Thus, using this principle, we can analyze any kind of collision in which Ewo
bodies oŸ equal mass hit each other and stick. In fact, although we have worked
only in one dimension, we can fñnd out a great deal about mụuch more complicated
collisions by imagining that we are riding by ín a car in some oblique direction.
'The prineciple is the same, but the details get somewhat complicated.
In order to test experimentally whether an object moving with velocity 0,
colliding with an equal one at rest, forrms an object moving with velocity 0/2,
we may perform the following experiment with our air-trough apparatus. We
place in the trough three equally massive objects, two of which are initially joined
together with our explosive cylinder device, the third being very near to but
slightly separated from these and provided with a sticky bumper so that it will
stick to another object which hits it. Now, a moment after the explosion, we have
two objects of mass w moving with equal and opposite velocities ø. ÀA moment
after that, one of these collides with the third object and makes an objecE of
mass 2n moving, so we believe, with velocity 0/2. How do we test whether it
1s really 0/2? By arranging the initial positions oŸ the masses on the trough so
that the distances to the ends are not equal, but are in the ratio 2: 1. Thus our
first mass, which continues to move with velocity 0, should cover twice as much
distance in a given tỉme as the 6wo which are sbuck together (allowing for the
small distance travelled by the second object before ¡it collided with the third).
'The mass ?n and the mass 2n should reach the ends at the same time, and when
we try it, we find that they do (Fig. 10-6).
--- Trang 205 ---
NZA — =
[+^—2D+A——>Lm | m ][ m ]<—D_—>{Œ]
~= —v v> 0
HD^~—22 Lm Lm ]}~—D->#¬
"=. vi
LHm_] L_2m _W]
Fig. 10-6. An experiment to verify that a mass m. with velocIty œ
striking a mass mm, with zero velocity gives 2w with velocity 0/2.
'The next problem that we want to work out is what happens If we have two
diferent masses. Let us take a mass rm and a mass 2m and apply our explosive
Interaction. What will happen then? Tf, as a result of the explosion, ?nw mmoves
with velocity 0, with what velocity does 2n move? “The experiment we have
just done may be repeated with zero separation between the second and third
masses, and when we try it we get the same result, namely, the reacting masses
m and 2m attain velocities —u and 0/2. Thus the direct reaction between ?m
and 2m gives the same result as the symmetrical reaction between rn and m,
followed by a collision between rn and a third mass ?m in which they stick together.
Purthermore, we find that the masses rnm and 2n returning from the ends of the
trough, with their velocities (nearly) exactly reversed, sbop dead ïf they stick
together.
Now the next question we may ask is this. What will happen iIÝ a mass rn