**The principal of conservation of momentum states that the total momentum of a system of objects remains constant provided no resultant external force acts on the system.**

Head-on collisions are one-dimensional collisions, which occur, in the same straight line without any deviations to this original straight path.

## Elastic Collision

Elastic collision in which both total momentum and total kinetic energy are conserved.

**In an elastic collision**, the kinetic energy lost by one body during an elastic collision is transferred to the other body so that the total kinetic energy of the colliding bodies is conserved.

For an elastic collision, it can be shown that the relative speed of approach of the bodies before collision is equal to the relative speed of separation of the bodies after collisions.

## Inelastic Collision

Inelastic collision in which total momentum is conserved but total kinetic energy is not conserved.

**In inelastic collisions**, some of the kinetic energy is converted to heat, sound or some other forms of energy when the objects are deformed after the collision. Therefore, in an inelastic collision, the total kinetic energy is not conserved.

### Completely Inelastic Collision

**Completely inelastic collision **in which total momentum is conserved and the particles stick together after collision so that their final velocities are the same. Total kinetic energy is not conserved.

## Momentum & Kinetic Energy in Collisions

The total momentum of the colliding bodies is always conserved in all types of collisions, as long as there is no resultant force acting on the system. However, the momentum of each individual colliding body is not conserved.

When finding the total kinetic energy of the colliding bodies, the direction of travel of bodies is not taken into account since kinetic energy is scalar, unlike the case when we are finding the total momentum (vector).

Note:

- When using the equation of relative velocities for elastic collision, direction of travel of the particle is important.
- If the particles approach towards each other, the relative speed of approach would be the sum of the approaching speeds; if they separate from each other in opposite directions, the relative speed of separation would be the sum of the separating speeds.

Special cases:

For elastic collision of two particles of equal mass, the particles simply exchange their velocities after the collision.

If $u_{2}$ = 0 (particle M originally at rest:

- If m = M (identical particle):
- The two particles exchange velocities

- If m << M:
- m rebounds with almost same speed while M remains stationary.

- If m >>M (a heavy particle hitting a very light particle)
- m continues with same speed, M goes off with twice speed of m.