Head-on Elastic Collisions, Target at Rest

Conservation of momentum
Conservation of kinetic energy

Dividing these relationships gives

which may be substituted into equation (2) above to obtain:

Note that these equations apply only to the case where the target is at rest.

Velocities After Collision

may be used along with conservation of momentum equation

to obtain expressions for the individual velocities after the collision.

These relationships may be used for any head-on collision by transforming to the frame of the target particle before using them, and then transforming back after the calculation. However, one may calculate the case for head-on collisions where both particles are moving with the relationships: