A smooth sphere of mass M moving with velocity u directly collides elastically with another sphere of mass m at rest. After collision their final velocities are V and v respectively. The value of v is

Options

A. `(2uM)/M`
B. `(2uM/M)`
C. `2u/(1+m/M)`
D. `(2u/(1+M/m)`

Explanation

To solve the problem of finding the final velocity \( v \) of the sphere of mass \( m \) after an elastic collision with a sphere of mass \( M \) moving with an initial velocity \( u \), we will use the principles of conservation of momentum and conservation of kinetic energy.

### Step-by-Step Solution:

1. **Identify Initial Conditions**:
- Mass of the first sphere (moving): \( M \)
- Initial velocity of the first sphere: \( u \)
- Mass of the second sphere (at rest): \( m \)
- Initial velocity of the second sphere: \( 0 \)

2. **Apply Conservation of Momentum**:
The total momentum before the collision must equal the total momentum after the collision.
\[ \text{Initial Momentum} = \text{Final Momentum} \]
\[ Mu + 0 = MV + mv \]
This simplifies to:
\[ Mu = MV + mv \quad \text{(1)} \]

3. **Apply Conservation of Kinetic Energy**:
The total kinetic energy before the collision must equal the total kinetic energy after the collision.
\[ \text{Initial Kinetic Energy} = \text{Final Kinetic Energy} \]
\[ \frac{1}{2}Mu^2 + 0 = \frac{1}{2}MV^2 + \frac{1}{2}mv^2 \]
This simplifies to:
\[ Mu^2 = MV^2 + mv^2 \quad \text{(2)} \]

4. **Solve for \( V \) from Equation (1)**:
Rearranging equation (1) gives:
\[ MV = Mu - mv \]
Therefore,
\[ V = \frac{Mu - mv}{M} \quad \text{(3)} \]

5. **Substitute \( V \) into Equation (2)**:
Substitute equation (3) into equation (2):
\[ Mu^2 = M\left(\frac{Mu - mv}{M}\right)^2 + mv^2 \]
Expanding the left side:
\[ Mu^2 = \frac{M(Mu - mv)^2}{M^2} + mv^2 \]
Simplifying gives:
\[ Mu^2 = \frac{(Mu - mv)^2}{M} + mv^2 \]

6. **Expand and Simplify**:
Expanding \( (Mu - mv)^2 \):
\[ (Mu - mv)^2 = M^2u^2 - 2Mumv + m^2v^2 \]
Substituting back:
\[ Mu^2 = \frac{M^2u^2 - 2Mumv + m^2v^2}{M} + mv^2 \]
Multiplying through by \( M \):
\[ M^2u^2 = M^2u^2 - 2mumv + m^2v^2 + Mv^2 \]

7. **Rearranging Terms**:
\[ 0 = -2mumv + (m^2 + M)v^2 \]
Factoring out \( v \):
\[ v( (m^2 + M)v - 2mu) = 0 \]
This gives two solutions:
\[ v = 0 \quad \text{or} \quad v = \frac{2mu}{m + M} \]

8. **Final Result**:
Since \( v = 0 \) is not a physically meaningful solution in this context, we have:
\[ v = \frac{2Mu}{m + M} \]

### Final Answer:
The value of \( v \) is:
\[ v = \frac{2Mu}{m + M} \]