Question

A sphere of mass m falls on a smooth hemisphere of mass M resting with its plane face on the smooth horizontal table so that at the moment of impact line joining the centres makes an angle α with the vertical. The velocity of the sphere just before impact is u and e is the coefficient of restitution. The velocity of hemisphere after the collision is

Answer 1

The velocity of hemisphere after the collision is

$\boxed{\frac{mu\cos\alpha(1+e)\sin\alpha}{(M+m)}}$

Explanation:

If the velocity of the hemisphere, along the centre after collision is V and that of the sphere if v then

From the conservation of momentum along the centre of the sphere

$mu\cos\alpha+0=MV-mv$  ......... (1)

Also since restitution is e

Therefore,

$e=\frac{V+v}{u\cos\alpha}$

$\implies eu\cos\alpha=V+v$

$\implies v=eu\cos\alpha-V$

Thus, from eq (1)

$mu\cos\alpha=MV-m(eu\cos\alpha-V)$

$\implies mu\cos\alpha=MV-meu\cos\alpha+mV$

$\implies mu\cos\alpha+meu\cos\alpha=(M+m)V$

$\implies mu\cos\alpha(1+e)=(M+m)V$

$\implies V=\frac{mu\cos\alpha(1+e)}{(M+m)}$

This will be the velocity of hemisphere in the direction along its centre

The horizontal component of this velocity will be

$V\sin\alpha$

$=\frac{mu\cos\alpha(1+e)\sin\alpha}{(M+m)}$

This is the velocity with which the hemisphere will move after collision

Hope this answer is helpful.

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