The general equation for conservation of linear momentum for a system of particles is:

Where:

In an inelastic collision between particles, the particles do not behave elastically during the collision. This means that, at the point of impact, the particles do not deform elastically; meaning they may permanently deform, resulting in energy loss during impact. This is unlike an elastic collision where, at the point of impact, the particles deform elastically; meaning they behave like perfectly elastic springs, absorbing and releasing the same amount of energy during impact.

Inelastic Collision Between Two Particles

General equations can be developed for the inelastic collision between two particles.

From equation (1) for the conservation of linear momentum we have

This equation can be expressed as its corresponding (scalar) equations along Cartesian

An additional equation needs to be introduced which accounts for the inelastic nature of the collision. Such an equation needs to relate the initial and final velocities of the particles. This is done using what is called the coefficient of restitution,

For instance, for the general case of collision between two particles the coefficient of restitution

where

The coefficient of restitution is given as

Where:

Equations (3) and (4) can be solved simultaneously.

For the special case of a head on inelastic collision in one dimension, the coefficient of restitution is

From equation (1) for the conservation of linear momentum, we have (in one dimension):

Equations (5) and (6) are then solved simultaneously.

The coefficient of restitution is determined experimentally for different classes of problems and types of materials. Its value varies between 0 and 1. For

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