Question

A smooth steel ball strikes a fixed smooth steel plate at an angle $\theta$ with the vertical. If the coefficient of restitution is $e$ , the angle at which the re-bounce will take place is

• A. $\theta$
• B. $ta{n}^{-1}\left[\frac{tan\theta }{e}\right]$
• C. $etan\theta$
• D. $ta{n}^{-1}\left[\frac{e}{tan\theta }\right]$

Answer 1

Answer: (B)

Since, no force is present along the surface of plate. So, momentum conservation principle for ball is applicable along the surface of plate.

$m\upsilon sin{\theta }_1=m\upsilon sin{\theta }_2$
Or $\upsilon sin{\theta }_1={\upsilon }_{1}sin{\theta }_2$ …(i)
$e=\frac{{\upsilon }_{1}cos{\theta }_2}{\upsilon cos{\theta }_1}=\frac{{\upsilon }_{1}cos{\theta }_2}{\upsilon cos\theta }$ ( ${\theta }_1=\theta$ )
$\therefore {\upsilon }_{1}cos{\theta }_2=e\upsilon cos\theta$ …(ii)
$\therefore \frac{{\upsilon }_{1}sin{\theta }_2}{{\upsilon }_{1}cos{\theta }_2}=\frac{\upsilon sin\theta }{e\upsilon cos\theta }=\frac{tan\theta }{e}$
$\therefore tan{\theta }_2=\frac{tan\theta }{e}$
$\therefore {\left(\theta \right)}_2={\left(tan\right)}^{-1}\left(\frac{tan\theta }{e}\right)$