Question

If the coefficient of restitution is $0.5$, what is the percentage loss of energy on each rebounding of a ball dropped from a height-

• A. $12.5\%$
• B. $25\%$
• C. $50\%$
• D. $75\%$

Answer 1

The correct option is D $75\%$

$\text{Step 1: Coefficient of Restitution}$

Let $V$ be the velocity of collision just before collision & $u$ after collision.

Upward direction is positive and downward direction is negative.

$e=\left(\frac{V_{separation}}{V_{approach}}\right)$

$\Rightarrow \frac{(u-0)}{(0-(-V\left)\right)}=e$ [Refer Fig.]

$\Rightarrow u=Ve$ $....\left(1\right)$

$\text{Step 2: Percentage loss in KE calculation}$

Kinetic Energy (Initial and Final)

$(KE{)}_i=\frac{1}{2}m{V}^2$ & $(KE{)}_f=\frac{1}{2}m{u}^2$

Percentage loss in $KE=\left(\frac{(KE{)}_i-(KE{)}_f}{(KE{)}_i}\right)\times 100$

Percentage $(KE{)}_{loss}=\left(\frac{\frac{1}{2}m{V}^2-\frac{1}{2}m{u}^2}{\frac{1}{2}m{V}^2}\right)\times 100=\left(\frac{V^2-u^2}{V^2}\right)\times 100$

Using eqn. $\left(1\right)$ percentage $(KE{)}_{loss}=(1-e^2)\times 100=75\%$

Hence correct option is $D$.