Question

A ball is dropped from height $5m$ . What is the time (in seconds) after which ball stops rebounding if the coefficient of restitution between ball and ground $e=\frac{1}{2}$ ?

Answer 1

Answer: 3

$T=\frac{v}{g}+\frac{2ev}{g}+\frac{2e^{2}v}{g}+\dots \dots \infty$
$T=\frac{v}{g}\left[1+2e+2e^2+\dots \dots \dots \infty \right]$
$T=\frac{v}{g}\left[2+2e+2e^2+\dots \dots \dots \infty -1\right]$
$T=\frac{v}{g}\left[2\left(1+e+(e{)}^2+\dots \dots \dots \infty \right)-1\right]$
$T=\frac{v}{g}\left[2\times \frac{1}{1-e}-1\right]$
$\Rightarrow T=\frac{\sqrt{2gh}}{g}\left[2\times \frac{1}{1-\frac{1}{2}}-1\right]\Rightarrow T=3s$