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
$h_n=h{e}^{2n}$
where $e=$ coefficient of restitution, $n=No$.
of rebound Total distance travelled by particle before rebounding has stopped
$H=h+2h_1+2h_2+2h_3+2h_n+\dots \dots$
$=h+2h{e}^2+2h{e}^4+2h{e}^6+2h{e}^8+\dots \dots$
$=h+2h\left(e^2+e^4+e^6+e^8+\dots \dots \right)$
$=h+2h\left[\frac{e^2}{1-e^2}\right]$
$=h\left[1+\frac{2e^2}{1-e^2}\right]$
$=h\left(\frac{1+e^2}{1-e^2}\right)$