Question

A smooth ball of mass $m$ strikes a horizontal surface with a velocity $v$ in a direction making an angle ${30}^o$ with the normal to the surface as shown in figure. If the coefficient of restitution for the collision between the ball and the surface is $e$ and the ball was in contact with the surface for a small time $\Delta t$, the average force acting on the ball during collision is

• A. $mg$
• B. $\frac{mv(1+e)}{2\Delta t}$
• C. $\sqrt{3}\frac{mv(1-e)}{2\Delta t}$
• D. $\sqrt{3}\frac{mv(1+e)}{2\Delta t}$

Answer 1

The correct option is C $\sqrt{3}\frac{mv(1-e)}{2\Delta t}$

Initial Velocity in vertical direction, $U\cos {30}^o$

Coefficient of restitution, $e=-\frac{finalrelativevelocity}{initialrelativevelocity}=-\frac{0-V}{U\cos {30}^o-0}$

$V=eU\cos {30}^o$

Final velocity in vertical direction, $eU\cos {30}^o$

Force on the ball = change in momentum/ change in time

$F=\frac{m(eV\cos {30}^o-V\cos {30}^o)}{\Delta t}=\frac{\sqrt{3}}{2}\frac{m(e-1)}{\Delta t}$

Hence, magnitude of force act on ball is $\sqrt{3}\frac{mV(1-e)}{2\Delta t}$