Question

A ball strikes a wall with a velocity $\overrightarrow{u}$ at angle $\theta$ with the normal to the wall surface and rebounds from it at an angle $\beta$ with the surface. Then:

• A. $(\theta +\beta )<{90}^{\circ }$, if the wall is smooth
• B. if the wall is rough, coefficient of restitution $e=\tan \beta /\cot \theta$
• C. if the wall is rough, coefficient of restitution $e<\tan \beta /\cot \theta$
• D. $(\theta +\beta )>{90}^{\circ }$, if the wall is smooth

Answer 1

Answer: (A), (C)

if the wall is rough, coefficient of restitution $e<\tan \beta /\cot \theta$



If wall is smooth, $v\cos \alpha =eu\cos \theta$ ...(i)
& $v\sin \alpha =u\sin \theta$ ... (ii)
where $\alpha =90-\beta$
Dividing the above two equations,
$v\cot \alpha =eu\cot \theta$

Squaring and adding (i) and (ii),
$v^2=u^2+u^2(e^2-1){\cos }^2\theta$
$\Rightarrow v<u$ [since $e<1$]
Therefore, from (ii),
$\alpha >\theta$
$\Rightarrow \alpha +\beta >\theta +\beta$
$\Rightarrow {90}^{\circ }>\theta +\beta$

If the wall is rough, then $v\sin \alpha <u\sin \theta$
i.e $v\cos \beta <u\sin \theta$ ...(iii)
& $v\sin \beta =eu\cos \theta$ ... (iv)
From (iii) and (iv),
$\tan \beta >e\cot \theta$
or $e<\tan \beta /\cot \theta$

Hence, options (a) and (c) are correct.