Coefficient of restitution is defined as the ratio of relative speeds before and after collision.
This can also be expressed as the square-root of the ratio of kinetic energies before and after collision.
e \equiv \frac{v_{post}}{v_{pre}} = \sqrt{\frac{K_{post}}{K_{pre}}}.
Suppose if K_0 is the kinetic energy before the first collision and K_1, K_2 and K_3 are kinetic energies after the first, second and third collisions respectively,
e=\sqrt{\frac{K_1}{K_0}} = \sqrt{\frac{K_2}{K_1}} = \sqrt{\frac{K_3}{K_2}}
K_1 = e^2K_0; \qquad K_2 = e^2K_1=e^4K_0; \qquad K_3 = e^2K_2=e^6K_0
The height h that a ball will ascend when thrown with a kinetic energy K is found by applying the mechanical energy conservation condition,
mgh = K; \qquad \rightarrow \qquad h = K/(mg);
The ball is initially dropped from the height h_0 and acquires a kinetic energy K_0 when it hits the ground, before the first collision.
mgh_0 = K_0;
If h_1, h_2 and h_3 are the heights after the first, second and third collisions,
h_1 = K_1/(mg) = e^2 K_0/(mg) = e^2 (cancel{mg}h_0)/(cancel{mg}) = e^2h_0;
h_2 = K_2/(mg) = e^4K_0/(mg) = e^4 (cancel{mg}h_0)/(cancel{mg}) = e^4h_0;
h_3 = K_3/(mg) = e^6K_0/(mg) = e^6 (cancel{mg}h_0)/(cancel{mg}) = e^6h_0;
In general, the height gained by the ball after the n^{th} collision is:
h_n = e^{2n}h_0
