Question

In the figure shown a particle $P$ strikes the inclined smooth plane horizontally and rebounds vertically. If the angle $\theta$ is ${60}^{\circ }$, then the coefficient of restitution is :

• A. $\frac{1}{3}$
• B. $\frac{1}{\sqrt{3}}$
• C. $\frac{1}{2}$
• D. $1$

Answer 1

The correct option is A $\frac{1}{3}$

Let $u$ be velocity of particle before collision
$v$ be velocity of particle after collision.
Resolution of velocities along and perpendicular to plane are shown in the figure below,


As the plane is smooth there is no force exerted by inclined plane on the particle along the inclined plane ,so the velocity of particle doesnot change along the plane
$v\cos {30}^{\circ }=u\cos {60}^{\circ }$
$u=\sqrt{3}v\dots \dots \left(1\right)$

Coefficient of restitution along common normal
$e=\frac{v_2-v_1}{u_1-u_2}$
$e=\frac{v\sin 30-0}{0-(-u\sin 60)}$
$e=\frac{\frac{v}{2}}{\left(\sqrt{3}v\right)\frac{\sqrt{3}}{2}}\Rightarrow \frac{1}{3}$