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)

After every collision, the velocity of the ball becomes $e$ times of the initial velocity, therefore, the height will become $e^2$ times of initial height.
where $e=$ the coefficient of restitution,
$n=$ The number of rebounds.
Total distance travelled by particle before rebounding has stopped
$H=h+2h_1+2h_2+2h_3+2h_4+\dots \dots$
$H=h+2h{e}^2+2h{e}^4+2h{e}^6+2h{e}^8+\dots \dots \dots$
$H=h+2h\left(e^2+e^4+e^6+e^8+\dots \dots \right)$
$H=h+2h\left[\frac{e^2}{1-e^2}\right]=h\left[1+\frac{2e^2}{1-e^2}\right]$
$H=h\left(\frac{1+e^2}{1-e^2}\right)$ .