Coefficient of restitution

Real collisions -- unlike the idealised perfectly elastic case -- generally lose some kinetic energy, and the amount lost is captured by a single dimensionless number, the coefficient of restitution (COR), usually written $e$. Dropping a rubber ball and a plastic ball onto the same floor makes the difference vivid: the rubber ball bounces back much higher, because it loses far less kinetic energy in the bounce.

The coefficient of restitution is defined as the ratio of the relative velocity of separation after the collision to the relative velocity of approach before it:

For a perfectly elastic collision, the relative velocity of separation exactly equals the relative velocity of approach (in magnitude), which was shown directly in the elastic-collision derivation -- so $e=1$: physically, no kinetic energy at all is lost, and the bodies "bounce back" with their full relative speed intact.

For a perfectly plastic (perfectly inelastic) collision, the bodies do not separate at all after colliding -- their relative velocity of separation is zero -- so $e=0$.

Every real collision falls somewhere between these two extremes:

$0 \le e \le 1$ …