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 using the principles of conservation of momentum and the coefficient of restitution.

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

### Step 2: Resolve the Velocity of the Sphere
Before the impact, we can resolve the velocity \( u \) of the sphere into two components:
- The vertical component: \( u \cos \alpha \)
- The horizontal component: \( u \sin \alpha \)

However, since the hemisphere is smooth and can only move horizontally, we will focus on the horizontal component for momentum conservation.

### Step 3: Apply Conservation of Momentum
According to the conservation of momentum, the momentum before the collision must equal the momentum after the collision. The momentum before the collision is given by the horizontal component of the sphere's velocity:

\[ \text{Initial momentum} = m \cdot u \cos \alpha \]

Let \( V \) be the velocity of the hemisphere after the impact and \( v \) be the velocity of the sphere after the impact. The momentum after the collision is:

\[ \text{Final momentum} = M \cdot V + m \cdot v \]

Setting the initial momentum equal to the final momentum:

\[ m \cdot u \cos \alpha = M \cdot V + m \cdot v \tag{1} \]

### Step 4: Apply the Coefficient of Restitution
The coefficient of restitution \( e \) relates the relative velocities of the two bodies before and after the collision. It is defined as:

\[ e = \frac{\text{Relative velocity after collision}}{\text{Relative velocity before collision}} \]

In our case, the relative velocity before the collision is \( u \cos \alpha \) (the sphere is moving towards the hemisphere). After the collision, the relative velocity is \( V + v \) (the hemisphere moves with velocity \( V \) and the sphere moves with velocity \( v \)):

\[ e = \frac{V + v}{u \cos \alpha} \tag{2} \]

From equation (2), we can express \( v \) in terms of \( V \):

\[ v = e u \cos \alpha - V \tag{3} \]

### Step 5: Substitute Equation (3) into Equation (1)
Now we substitute equation (3) into equation (1):

\[ m \cdot u \cos \alpha = M \cdot V + m \cdot (e u \cos \alpha - V) \]

Expanding this gives:

\[ m \cdot u \cos \alpha = M \cdot V + m e u \cos \alpha - m V \]

Rearranging terms:

\[ m u \cos \alpha - m e u \cos \alpha = (M - m) V \]

Factoring out \( u \cos \alpha \):

\[ u \cos \alpha (m - m e) = (M - m) V \]

### Step 6: Solve for \( V \)
Now we can solve for \( V \):

\[ V = \frac{u \cos \alpha (m(1 - e))}{M + m} \]

### Step 7: Final Expression
Thus, the final expression for the velocity of the hemisphere after the impact is:

\[ V = \frac{m u \cos \alpha (1 + e)}{M + m} \]

### Conclusion
The correct option that matches our derived expression is:

**Option C:** \( V = \frac{m u \cos \alpha (1 + e)}{M + m} \)