Inelastic Collisions and the Coefficient of Restitution

In any real collision, momentum is always conserved (so long as no significant external force acts during the very short time of contact), but kinetic energy is conserved only in the idealised, perfectly elastic case treated in §5.10. In every other collision -- which is to say, in every collision actually observed in practice, to at least some degree -- some kinetic energy is lost, converted irreversibly into heat generated by internal friction and deformation at the point of contact, into sound, and into the permanent (plastic) deformation of the colliding bodies themselves. Such collisions are called inelastic.

Perfectly inelastic collisions. At the opposite extreme from a perfectly elastic collision lies the perfectly inelastic collision, in which the two colliding bodies stick together and move off afterward with one single, common velocity $v'$. Momentum conservation alone determines this common velocity:

Because the two bodies are constrained to share exactly one final velocity (rather than each being free to take on its own final velocity, as in the more general elastic case), a perfectly inelastic collision represents the situation of maximum possible kinetic-energy loss consistent with the given masses, initial velocities, and momentum conservation -- no other outcome, consistent with momentum being conserved, loses more kinetic energy than sticking together completely does.

The coefficient of restitution. To measure, with a single number, exactly how elastic or inelastic any particular collision is, the coefficient of restitution $e$ is defined as the ratio of the relative velocity of separation (after the collision) to the relative velocity of approach (before the collision), both measured along the line of impact:

$e = \frac{v_2' - v_1'}{u_1 - u_2}$ …