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342 CHAPTER 11 Collisions

SOLUTION: The initial momentum of the head is

p mv 5.0 kg 15 m s 75 kg m s
x x
When the head stops, the final momentum is zero. Hence the average force is

p p x p x
x
F
x
¢t ¢t
75 kg m s
3
5.0 10 N
0.015 s
The average acceleration is
3
F x 5.0 10 N
3
a 1.0 10 m s 2
x m 5.0 kg
which is about 100 standard g’s!

Often it is not possible to calculate the motion of the colliding bodies by direct
solution of Newton’s equation of motion because the impulsive forces that act during
the collision are not known in sufficient detail.We must then glean whatever information
we can from the general laws of conservation of momentum and energy, which do not
depend on the details of these forces. In some simple instances, these general laws
permit the deduction of the motion after the collision from what is known about the
motion before the collision.
In all collisions between two or more particles, the total momentum of the system is con-
served. Whether or not the mechanical energy is conserved depends on the character
of the forces that act between the particles. A collision in which the total kinetic energy
elastic collision before and after the collision is the same is called elastic. (This usage of the word elastic is
consistent with the usage we encountered previously when discussing the restoring
force of a deformable body in Section 6.2. For example, if the colliding bodies exert a
force on each other by means of a massless elastic spring placed between them, then
the kinetic energy before and after the collision will indeed be the same—that is, the
collision will be elastic.) Collisions between macroscopic bodies are usually not elas-
tic—during the collision some of the kinetic energy is transformed into heat by the
internal friction forces and some is used up in doing work to change the internal con-
figuration of the bodies. For example, the automobile collision shown in Fig. 11.1 is
highly inelastic; almost the entire initial kinetic energy is used up in doing work on
the automobile parts, changing their shape. On the other hand, the collision of a “Super
Ball” and a hard wall or the collision of two billiard balls comes pretty close to being
elastic—that is, the kinetic energies before and after the collision are almost the same.
Collisions between “elementary” particles—such as electrons, protons, and
neutrons—are often elastic.These particles have no internal friction forces which could
dissipate kinetic energy. A collision between such particles can be inelastic only if it
involves the creation of new particles; such new particles may arise either by conver-
sion of some of the available kinetic energy into mass or else by transmutation of the
old particles by means of a change of their internal structure.

A Super Ball, made of a rubberlike plastic, is thrown against a
EXAMPLE 3
hard, smooth wall.The ball strikes the wall from a perpendicular
direction with speed v. Assuming that the collision is elastic, find the speed of the
ball after the collision.