A big ball of mass M , moving with velocity u strikes a small ball of mass m , which is at rest. Finally small ball obtains velocity u and big ball v . Then what is the value of v

Options

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

Answer

To solve the problem, we will use the principle of conservation of momentum. The momentum before the collision must equal the momentum after the collision.

### Step-by-Step Solution:

1. **Identify the masses and velocities**:
- Let the mass of the big ball be \( M \) and its initial velocity be \( u \).
- Let the mass of the small ball be \( m \) and its initial velocity be \( 0 \) (since it is at rest).
- After the collision, the small ball moves with velocity \( u \) and the big ball moves with velocity \( v \).

2. **Write the equation for conservation of momentum**:
The total momentum before the collision is equal to the total momentum after the collision. Mathematically, this can be expressed as:
\[ \text{Momentum before} = \text{Momentum after} \]
Before the collision:
\[ \text{Momentum before} = M \cdot u + m \cdot 0 = M \cdot u \]
After the collision:
\[ \text{Momentum after} = M \cdot v + m \cdot u \]

3. **Set up the equation**:
From the conservation of momentum, we have:
\[ M \cdot u = M \cdot v + m \cdot u \]

4. **Rearrange the equation to solve for \( v \)**:
Rearranging the equation gives:
\[ M \cdot u - m \cdot u = M \cdot v \]
Factor out \( u \) from the left side:
\[ (M - m) \cdot u = M \cdot v \]

5. **Solve for \( v \)**:
Dividing both sides by \( M \):
\[ v = \frac{(M - m) \cdot u}{M} \]

### Final Answer:
Thus, the velocity \( v \) of the big ball after the collision is:
\[ v = \frac{(M - m)}{M} \cdot u \]