What is the main difference between an inelastic and a perfectly inelastic collision?

See below

Assumption: Inelastic collision imply partially inelastic collision.

Let us define a quantity, Coefficient of restitution ##e##

The coefficient of restitution (COR) is a measure of the kinetic remaining in the objects; involved in collision, after rebound from one another as compared to kinetic energy lost as heat, or as done in deforming the colliding objects.

This is defined as the ratio of relative speeds after and before the collision, taken along the line of the impact.

## e -= \frac{\text{Relative speed after collision}}{\text{Relative speed before collision}} ## ##e## is usually a positive, real number between ##0 and 1##

1. ##e = 0##; This is case of a perfectly inelastic collision. The objects stick together and move as a single object after the collision. Usually such collisions result in loss of maximum kinetic energy. The lost kinetic energy is converted to heat or in to work done in deforming the objects.

2. ##0 < e < 1##; These correspond to real-life situations, partially , or simply called inelastic collisions, in which some kinetic energy is lost as friction, sound and heat. The objects do not stick together and move separately. *Momentum is conserved in all types of inelastic collisions (if the surface on which objects move has zero friction) but the kinetic energy can't be tracked through the collision since some of it gets converted to other forms.

3. ##e = 1##; This is a perfectly elastic collision, in which kinetic energy is also conserved, and the objects rebound from one another with the same relative speed with which they approached.