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with velocity 0, say, hits and sticks to another mass 2 at rest? 'This is very
easy to answer using our prineiple of Galilean relativity, for we simply watch
the collision which we have just described from a car moving with velocity —0/2
(Fig. 10-7). Erom the car, the velocities are
U =0— 0(car) =0u+0/2=30/2
0 = —0/2~ 0(car) = —0/2+/2=0.
After the collision, the mass 3n appears to us to be moving with velocity 0/2.
Thus we have the answer, I.e., the ratio of velocitles before and after collision
1s 3 to 1: if an object of mass mm collides with a stationary object of mass 2m,
then the whole thing moves of, stuck together, with a velocity 1/3 as mụuch. The
general rule again is that the sum of the produects of the masses and the velocities
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VIEW FROM VIEW FROM
CM SYSTEM CAR
vV —Vv/2 3v/2 0
— —— —
BEFORE COLLISION
0 v/2->
AFTER COLLISION
Fig. 10-7. TWo views of an inelastic collision between m and 2m.
stays the same: ?zø + 0 equals 3mm tỉmes 0/3, so we are gradually building up
the theorem of the conservation of momentum, piece by piece.
Now we have one against two. Ủsing the same arguments, we can predict the
result oŸ one against three, two against three, etc. The case of two against three,
starting from rest, is shown in Fig. 10-8.
0 Ø0 vw¿ 0 0 0
0 -~——v v+> 0 0
-——v/2 v/2> 0
~——v/2 v/3->
Fig. 10-8. Action and reaction between 2m and 3m.
In every case we find that the mass of the first obJect times its velocity, plus
the mass of the second object times its velocity, is equal to the total mass of the
fnal obJect times its velocity. Thhese are all examples, then, of the conservation
of momentum. Starting from simple, symmetrical cases, we have demonstrated
the law for more complex cases. We could, in fact, do 1 for any rational mass
ratio, and since every ratio is exceedingly close to a rational ratio, we can handle
every ratio as precisely as we wish.
10-4 Momentum and energy
All the foregoing examples are simple cases where the bodies collide and stick
together, or were initially stuck together and later separated by an explosion.
--- Trang 207 ---
However, there are situations in which the bodies do no cohere, as, for example,
two bodies of equal mass which collide with equal speeds and then rebound. Eor
a brief moment they are in contact and both are compressed. At the instant of
mmaximum compression they both have zero velocity and energy is stored in the
elastic bodies, as in a compressed spring. This energy is derived from the kinetic
energy the bodies had before the collision, which becomes zero at the instant
their velocity is zero. The loss of kinetic energy is only momentary, however. The
compressed condition is analogous to the cap that releases energy in an explosion.
The bodies are immediately decompressed in a kind of explosion, and fy apart
again; but we already know that case—the bodies ñy apart with equal speeds.
However, this speed of rebound is less, in general, than the initial speed, because
not all the energy is available for the explosion, depending on the material. If
the material is putty no kinetic energy is recovered, but IŸ it is something more
rigid, some kinetic energy is usually regained. In the collision the rest of the
kinetic energy is transformed into heat and vibrational energy——the bodies are
hot and vibrating. 'Phe vibrational energy also is soon transformed into heat. lt
is possible to make the colliding bodies rom highly elastic materials, such as
sieel, with carefully designed spring bumpers, so that the collision generates very
little heat and vibration. In these circumstances the velocities oŸ rebound are
practically equal to the initial velocities; such a collision is called elastic.
That the speeds 0efore and a/fter an elastic collision are equal is not a matter oŸ
conservation oŸ momentum, but a matter of conservation of kinefic energu. That
the veloeities of the bodies rebounding after a symmetrical collision are equal to
and opposite each other, however, is a matter of conservation of momentum.
We might similarly analyze collisions between bodies of diferent masses,
diferent initial velocities, and various degrees of elasticity, and determine the
ñnal velocities and the loss of kinetic energy, but we shall not go into the details
of these processes.
Bilastic collisions are especially interesting for systems that have no internal
“gears, wheels, or parts.” Then when there is a collision there is nowhere for the
energy to be impounded, because the objects that move apart are in the same
condition as when they collided. 'Therefore, bebween very elementary obJects,
the collisions are always elastic or very nearly elastic. For instance, the collisions
between atoms or molecules in a gas are said to be perfectly elastic. Although
this is an excellent approximation, even such collisions are not perƒectlu elastic;
otherwise one could not understand how energy in the form of light or heat
radiation could come out of a gas. Once in a while, in a gas collision, a low-energy
--- Trang 208 ---
infrared ray is emitted, but this occurrence is very rare and the energy emitted is
very small. So, for most purposes, collisions of molecules in gases are considered
to be perfectly elastic.
As an interesting example, let us consider an eÏasfic collision between two
objects of eguøl rmass. If they come together with the same speed, they would
come apart at that same speed, by symmetry. But now look at this in another
circumstanece, in which one oŸ them is moving with velocity ø and the other one
1s at rest. What happens? We have been through this before. We watch the
symmetrical collision from a car moving along with one of the objects, and we
ñnd that if a stationary body is struck elastically by another body of exactly the
same mass, the moving body stops, and the one that was standing still now moves
away with the same speed that the other one had; the bodies simply exchange
velocities. 'This behavior can easily be demonstrated with a suitable impaect
apparatus. More generally, If both bodies are moving, with diferent velocities,
they simply exchange velocity at impact.
Another example of an almost elastic interaction is magnetism. ÏIÝ we arrange
a païr of U-shaped magnets in our glide blocks, so that they repel each other,
when one drifts quietly up to the other, it pushes it away and stands perfectly
still, and now the other goes along, frictionlessly.
The principle of conservation of momentum is very useful, because it enables
us to solve many problems without knowing the details. We did not know the