A ball is allowed to fall freely from a height of h metres on a fixed plate. If the successive rebound heights are h₁, h₂, h₃,.., the distance covered by the ball before coming to rest if e = 0.5 will be

Velocity of the ball when it strikes the first plate
${\mathrm{v}}_1=\sqrt{\left(2\mathrm{gh}\right)}$
Velocity after first rebound $={\mathrm{ev}}_1=\mathrm{e}\sqrt{\left(2\mathrm{gh}\right)}$
Maximum height to which it will rebounds after first impact
h₁ =e² h
Similarly, heights after successive rebounds are
${\mathrm{e}}^4\mathrm{h}, {\mathrm{e}}^6\mathrm{h}, {\mathrm{e}}^8\mathrm{h},\dots$
Total distance 
$\begin{array}{l}=\mathrm{h}+2{\mathrm{h}}_1+2{\mathrm{h}}_2+2{\mathrm{h}}_3+\dots \\ =\mathrm{h}+2{\mathrm{e}}^2\mathrm{h}+2{\mathrm{e}}^4\mathrm{h}+2{\mathrm{e}}^6\mathrm{h}+\dots \\ =\mathrm{h}\left[1+2{\mathrm{e}}^2\left(1+{\mathrm{e}}^2+{\mathrm{e}}^4+\dots \right)\right]\\ =\mathrm{h}\left[1+\left(\frac{2{\mathrm{e}}^2}{1-{\mathrm{e}}^2}\right]\right]\\ =\mathrm{h}\left[\frac{1+{\mathrm{e}}^2}{1-{\mathrm{e}}^2}\right]=\mathrm{h}\left[\frac{1+0\cdot 25}{1-0\cdot 25}\right]\\ =\frac{5\mathrm{h}}{3}\end{array}$