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

Options

A. The velocity of hemisphere after impact will be zero
B. The hemisphere will move with a velocity `v = ( m u sin alpha)/(M cos^2 alpha)`
C. The hemisphere will move with a velocity `v = (m u cos alpha( 1 + e))/((M + m ))`
D. None of these

Explanation

To solve the problem step by step, we will analyze the situation involving the sphere and the hemisphere, applying the principles of conservation of momentum and the coefficient of restitution.

### Step 1: Analyze the Situation
We have a sphere of mass \( m \) falling towards a hemisphere of mass \( M \). The line joining the centers of the sphere and the hemisphere makes an angle \( \alpha \) with the vertical at the moment of impact. The velocity of the sphere just before impact is \( u \).

### Step 2: Resolve the Velocity of the Sphere
The velocity of the sphere can be resolved into two components:
- The component along the line joining the centers (which is the direction of impact):
\[ u \cos \alpha \]
- The component perpendicular to the line joining the centers is not relevant for the collision analysis.

### Step 3: Apply Conservation of Momentum
Let \( v \) be the velocity of the sphere after the impact and \( V \) be the velocity of the hemisphere after the impact. According to the conservation of momentum along the line joining the centers, we have:
\[ m u \cos \alpha = M V + m v \]
This is our first equation.

### Step 4: Apply the Coefficient of Restitution
The coefficient of restitution \( e \) relates the relative velocities of separation and approach. The equation is given by:
\[ e = \frac{\text{Velocity of separation}}{\text{Velocity of approach}} \]
At the moment of impact, the velocity of approach is \( u \cos \alpha \) and the velocity of separation is \( V + v \). Therefore, we can write:
\[ V + v = e u \cos \alpha \]
This is our second equation.

### Step 5: Substitute and Solve for \( v \)
From the second equation, we can express \( v \) in terms of \( V \):
\[ v = e u \cos \alpha - V \]
Now, substitute this expression for \( v \) into the first equation:
\[ m u \cos \alpha = M V + m (e u \cos \alpha - V) \]
Expanding and rearranging gives:
\[ m u \cos \alpha = M V + m e u \cos \alpha - m V \]
\[ m u \cos \alpha - m e u \cos \alpha = (M - m) V \]
Factoring out \( u \cos \alpha \):
\[ u \cos \alpha (m - m e) = (M - m) V \]
Thus,
\[ V = \frac{u \cos \alpha (m(1 - e))}{M - m} \]

### Step 6: Final Expression for \( v \)
Now, substituting \( V \) back into the expression for \( v \):
\[ v = e u \cos \alpha - \frac{u \cos \alpha (m(1 - e))}{M - m} \]
This gives us the final expression for the velocities after the impact.

### Final Result
The velocities after the impact can be expressed as:
\[ V = \frac{m u \cos \alpha (1 + e)}{M + m} \]