Question

A ball is allowed to fall freely from a height of 3 metres on to a fixed plate. The successive rebound heights are $h_1,h_2,h_3,\dots$ lf the distance covered by the ball before coming to rest is y metres, find y. (Given that the coefficient of restitution is 0.5)

Answer 1

Let $v_1$ be the velocity with which the ball strikes the plate $v_1^{\prime }$ be the velocity of rebound.
Let H be the height from which the ball is dropped.

$v_1=\sqrt{2gH}$

$v_1^{\prime }=\sqrt{2g{h}_1}$

$\Rightarrow e=\frac{v_{1^{\prime }}}{v_1}=\sqrt{\frac{h_1}{H}}$

$e^2=\frac{h_1}{H}\Rightarrow h_1=e^{2}H$

Similarly,
$h_2=e^2{h}_1=e^{4}H$ and so on.

Total distance described before it comes to stop

$y=H+2h_1+2h_2+2h_3+....$

$=H+2e^{2}H+2e^{4}H+2e^{6}H+..$

$=H(1+2e^2+2e^4+2e^6+....)$

$=H[1+2e^2(1+e^2+e^4+...\left)\right]$

$=H[1+\frac{2e^2.1}{1-e^2}]$

$=H\left[\frac{1+e^2}{1-e^2}\right]$

$=3\left[\frac{1+{0.5}^2}{1-{0.5}^2}\right]=5$metres