A ball is dropped from a height h on a floor. The coefficient of restitution for the collision between the…

Height achieved in ${\mathrm{n}}^{\mathrm{t}\mathrm{h}}$ collision is given by 

$h_{\mathrm{n}}={\mathrm{e}}^{2\mathrm{n}}\mathrm{h}$
Distance covered in first collision = $h$,

Distance covered in $2^{\mathrm{nd}}$ collision = $2e^{2}h$, and so on.

Total distance covered by the ball before it comes to rest:

$s=h+2e^{2}h+2e^{4}h+......$

$\Rightarrow s=h\left[1+\left(2e^2+2e^4+2e^6+......\right)\right]$

from the sum of infinite geometric progression:

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

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