A smooth hemisphere of mass m and radius R is at rest. A smooth solid sphere of mass 2m and radius R moving with velocity $V_{0}$ between two horizontal smooth surfaces separated by a distance slightly greater than 2R as shown in figure. Solid sphere collides with the hemisphere. If coefficient of restitution is $\frac{1}{2}$ , then

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The speed of hemisphere after collision is $V_{0}$

The speed of solid sphere after collision is $\frac{V_{0}}{2}$

The loss in kinetic energy of the system is $\frac{m {V_{0}}^{2}}{4}$

The final kinetic energy of hemisphere is $\frac{1}{4} t h$ the initial kinetic energy of the sphere

answer is A, B, C.

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Detailed Solution

$2 M ϑ_{0} = 2 M ϑ_{1} + M ϑ_{2}$ and $ϑ_{2} \cos θ - ϑ_{1} \cos θ = e ϑ_{0} \cos θ$ on solving we get $ϑ_{1} = \frac{(2 - e) ϑ_{0}}{3} = \frac{ϑ_{0}}{2}$
And $ϑ_{2} = (1 + e) \frac{2 ϑ_{0}}{3} = ϑ_{0}$ . $K_{i} = \frac{1}{2} \times 2 m ϑ_{0}^{2}$ and $K_{f} = \frac{1}{2} \times 2 m \times \frac{ϑ_{0}^{2}}{4} + \frac{1}{2} m ϑ_{0}^{2} = \frac{3}{4} m ϑ_{0}^{2}$
$∴$ loss of kinetic energy of the system $= \frac{1}{4} m ϑ_{0}^{2}$