Question

A particle of mass m = 0.1 kg moving vertically downward with a velocity u = 20 m/s collides with a rough hemispherical body of mass M = 0.3 kg with coefficient of restitution e = 0.5 as shown. The hemispherical body is placed on a rough horizontal surface with coefficient of friction $\mu_0$ = 0.2. The coefficient of friction between the particle and the hemispherical body is $\mu$ = 0.5. Find the velocity (in m/s) of the hemispherical body just after collision.

Answer 1

20/13

Answer 2

Explanation of the solution:

1. Define normal and tangential directions at the point of collision.
2. Decompose the initial velocity of the particle into normal and tangential components.
3. Express the velocity of the point of collision on the hemisphere after collision in terms of the hemisphere's velocity.
4. Apply the coefficient of restitution equation in the normal direction.
5. Apply conservation of momentum in the horizontal direction for the system of particle and hemisphere.
6. Determine the direction of tangential friction based on the relative tangential velocity. Since slipping occurs, the magnitude of tangential impulse is proportional to the magnitude of normal impulse.
7. Use the relationship between tangential and normal impulses to relate the components of velocities after collision.
8. Solve the system of equations to find the velocity of the hemispherical body.

The final answer is $\boxed{20/13}$.