Question

A particle falls from a height h upon a fixed horizontal plane and rebounds. If e is the coefficient of restitution, the total distance travelled before rebounding has stopped is

• A. $h\left(\frac{1+e^{2}}{1-e^{2}}\right)$
• B. $h\left(\frac{1-e^{2}}{1+e^{2}}\right)$
• C. $\frac{h}{2}\left(\frac{1-e^{2}}{1+e^{2}}\right)$
• D. $\frac{h}{2}\left(\frac{1+e^{2}}{1-e^{2}}\right)$

Answer 1

Answer: (A)

Particle falls from height h then formula for height covered by it in nth rebound is given by
$h_{n}=h e^{2 n}$
where $e=$ coefficient of restitution, $n = No$.
of rebound Total distance travelled by particle before rebounding has stopped
$H =h+2 h_{1}+2 h_{2}+2 h_{3}+2 h_{n}+\ldots \ldots$
$=h+2 h e^{2}+2 h e^{4}+2 h e^{6}+2 h e^{8}+\ldots \ldots$
$=h+2 h\left(e^{2}+e^{4}+e^{6}+e^{8}+\ldots \ldots\right)$
$=h+2 h\left[\frac{e^{2}}{1-e^{2}}\right]$
$=h\left[1+\frac{2 e^{2}}{1-e^{2}}\right]$
$=h\left(\frac{1+e^{2}}{1-e^{2}}\right)$