# Perfectly Elastic Head-on Collision

Let two bodies of masses

*m*_{1}and*m*_{2}moving with initial velocities*u*_{1}and*u*_{2}in the same direction and they collide such that after collision their final velocities are*v*_{1}and*v*_{2}, respectively (Fig. 2).According to the law of conservation of momentum,

According to the law of conservation of kinetic energy

Dividing (4) by (2),

Relative velocity of approach = Relative velocity of separation

*Notes:*- The ratio of relative velocity of separation and relative velocity of approach is defined as coefficient of restitution.
- For perfectly elastic collision,
*e*= 1. - For perfectly inelastic collision,
*e*= 0. - For inelastic collision, 0 <
*e*< 1.*e*is the degree of elasticity of collision and it is dimension-less quantity.

Further from (

*5*) we get*v*_{2}=*v*_{1}+*u*_{1}â€“*u*_{2}.Substituting this value of

*v*_{2}in (1) and rearranging, we getSimilarly, we get