Inelastic Collisions:

A collision or impact is the sudden, forceful coming together in direct contact of two bodies, such as, for example, two billiard balls, a golf club and a ball, a hammer and a nail head, two railroad cars when being coupled together, or a falling object and a floor. Apart from the properties of the materials of the two objects, two factors affect the result of impact: the force and the time during which the objects are in contact. It is a matter of common experience that a hard steel ball dropped on a steel plate will rebound to almost the position from which it was dropped, whereas with a ball of putty or lead there is no rebound. The impact between the steel ball and plate is said to be elastic, and that between the putty or lead balls and plate is inelastic, or plastic; between these extremes there are varying degrees of elasticity and corresponding responses to impact. In a perfectly elastic impact (attained only at the atomic level), none of the kinetic energy of the coacting bodies is lost; in a perfectly plastic impact, the loss of kinetic energy is at a maximum.

In all of the examples of colliding bodies here referred to, the time of contact is extremely short and the force of contact extremely large. It can be shown that, in the limiting case of an "infinite" force acting for an "infinitesimal" time, there is an instantaneous change in the velocity of a body but no change in its position during the period of contact. Forces of this nature are known as impulsive forces and, being difficult to measure or estimate, their effects are measured by the change in the momentum (mass times velocity) of the body. The ballistic pendulum is a device based on this principle.

When two bodies collide, the equal and opposite impulsive forces are internal forces in what is called a two-body system and consequently have no effect on the total momentum of the system. This phenomenon means that the sum of the momenta of the bodies before impact is equal to the sum of the momenta after impact. The relation between the kinetic energies before and after impact depends, as previously noted, on the elasticity of the bodies. Knowing the initial velocities, the final velocities can be obtained by the simultaneous solution of the momentum and energy equations.

Excerpt from the Encyclopedia Britannica without permission.