When two objects collide head-on, the ratio of their relative velocities after collision and their relative velocities before collision is called the coefficient of restitution e.

Thus                                                                                   ... (i)

Or             (along the line of impact)                   ... (ii)

Where             v₁ = The speed of first body after collision.

                        v₂ = The speed of second body after collision.

                        u₁ = the speed of first body before collision.

                        u₂ = the speed of second body before collision.

The ratio e is known as the coefficient of restitution and is constant for two particular objects.

In general,  0≤e≤1         

e = 0, for completely inelastic collision, as both the bodies stick together. So, their relative speed is zero or e = 0.

e = 1, for an elastic collision, as we can show that from equation (i)

                        v₂ - v₁ = u₁ - u₂                                                            ... (iii)

Or                    separation speed = approach speed

Or                    e = 1

Let us now search the velocities of two particles after collision if they collide directly and the coefficient of restitution between them is given as e.

Applying conservation of linear momentum,                         (iv)

Further, separation speed = e (approach speed)       or         u₁ - u₂ = e (v₂ - v₁)                  (v)

Solving equations (iv) and (v) we get ,                       (vi)

and                                                                        (vii)

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