Question

A ball is dropped on to a horizontal plate from a height $h=9m$ above it. If the coefficient of restitution is $e=\frac{1}{2}$, the total distance travelled before the ball comes to rest is:

• A. $10m$
• B. $15m$
• C. $20m$
• D. $25m$

Answer 1

Answer: (B)

$15m$

$H_1=9m{V}_1=\sqrt{2\times 10\times 9}=\sqrt{180}$

$V_2=e{V}_1{H}_2=\frac{V_2^2}{2g}=\frac{e^2{V}_1^2}{2g}$

$V_3=e^2{V}_1{H}_3=\frac{V_3^2}{2g}=\frac{e^4{V}_1^2}{2g}$

Total distance travelled

$=H_1+2(H_2+....)$

$=H_1+2(\frac{e^2{V}_1^2}{2g}+\frac{e^4{V}_1^2}{2g}+\frac{e^6{V}_1^2}{2g}....)$

$=H_1+\frac{e^2{V}_1^2}{g}(1+e^2+e^4+....)$

$=H_1+\frac{e^2{V}_1^2}{g}(1+\frac{1}{4}+\frac{1}{16}+....)$

$=H_1+\frac{e^2{V}_1^2}{gI}\left(\frac{1}{1-\frac{1}{4}}\right)$

$=H_1+\frac{e^2{V}_1^2}{g}\left(\frac{4}{3}\right)$

$=9+\frac{\left(\frac{1}{4}\right)\left(180\right)}{10}\left(\frac{4}{3}\right)$

$=15m$.