Question

A smooth sphere of mass $m$ is moving on a horizontal plane with velocity $3\hat{i}+\hat{j}$. It collides with a vertical wall which is parallel to vector$\hat{j}$. If the coefficient of restitution is $0.5$, the angle of velocity of the ball with repect to the normal to the wall after impact will be

• A. ${30}^o$
• B. ${\tan }^{-1}(\frac{1}{3})$
• C. ${\cos }^{-1}\left(\frac{2}{\sqrt{13}}\right)$
• D. ${\sin }^{-1}\left(\frac{2}{\sqrt{13}}\right)$

Answer 1

The correct option is D ${\sin }^{-1}\left(\frac{2}{\sqrt{13}}\right)$

Coefficient of restitution
$e=\frac{Relativevelocityaftercollision}{Relativevelocitybeorecollision}=\frac{\Delta u_{final}}{\Delta u_{initial}}=0.5u_{initial}=3\hat{i}+\hat{j}\Delta u_{initial}=3(\because thewallisparallelto\hat{j})\Rightarrow \frac{\Delta u_{final}}{3}=0.5\Rightarrow \Delta u_{final}=\frac{3}{2}\therefore u_{final}=-\frac{3}{2}\hat{i}+\hat{j}$



$\therefore \sin \theta =\frac{1}{\sqrt{{\left(\frac{3}{2}\right)}^2+1^2}}=\frac{2}{\sqrt{13}}$