A body of mass `m_(1)` moving with a velocity `3 m//s` collides with another body at rest of `m_(2)` . After collision the velocities of the two bodies are `2 m//s` and `5 m//s` , respectively , along the direction of motion of `m_(1)`. The ratio `m_(1)//m_(2)` is

Options

A. `5/12`
B. `5`
C. `1/5`
D. `12/5`

Explanation

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 since no external forces are acting on the system.

### Step-by-Step Solution:

1. **Identify the given values:**
- Mass of the first body: \( m_1 \)
- Initial velocity of \( m_1 \): \( u_1 = 3 \, \text{m/s} \)
- Mass of the second body: \( m_2 \)
- Initial velocity of \( m_2 \): \( u_2 = 0 \, \text{m/s} \) (at rest)
- Final velocity of \( m_1 \): \( v_1 = 2 \, \text{m/s} \)
- Final velocity of \( m_2 \): \( v_2 = 5 \, \text{m/s} \)

2. **Write the equation for conservation of momentum:**
\[ \text{Initial Momentum} = \text{Final Momentum} \]
\[ m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \]

3. **Substituting the known values into the equation:**
Since \( u_2 = 0 \):
\[ m_1 \cdot 3 + m_2 \cdot 0 = m_1 \cdot 2 + m_2 \cdot 5 \]
This simplifies to:
\[ 3m_1 = 2m_1 + 5m_2 \]

4. **Rearranging the equation:**
\[ 3m_1 - 2m_1 = 5m_2 \]
\[ m_1 = 5m_2 \]

5. **Finding the ratio \( \frac{m_1}{m_2} \):**
\[ \frac{m_1}{m_2} = 5 \]

Thus, the ratio \( \frac{m_1}{m_2} \) is \( 5 \).

### Final Answer:
The ratio \( \frac{m_1}{m_2} \) is \( 5 \).