A body of mass `m_(1)` collides elastically with a stationary body of mass `m_(2)` and returns along the same line with one fourth of its initial speed, then `m_(1)//m_(2)=`

Options

A. 1
B. 2
C. `0.66`
D. `0.33`

Answer

To solve the problem, we need to analyze the elastic collision between two bodies, where one body is initially stationary. We will use the principles of conservation of momentum and the properties of elastic collisions.

### Step-by-Step Solution:

1. **Define Initial Conditions**:
- Let the mass of the first body be \( m_1 \) and its initial velocity be \( v \).
- The mass of the second body is \( m_2 \) and its initial velocity is \( 0 \) (since it is stationary).

2. **Post-Collision Velocities**:
- After the collision, the first body (mass \( m_1 \)) moves in the opposite direction with a velocity of \( -\frac{v}{4} \).
- Let the velocity of the second body (mass \( m_2 \)) after the collision be \( v_2 \).

3. **Apply Conservation of Momentum**:
- The law of conservation of momentum states that the total momentum before the collision equals the total momentum after the collision.
- Initial momentum:
\[ p_{\text{initial}} = m_1 \cdot v + m_2 \cdot 0 = m_1 v \]
- Final momentum:
\[ p_{\text{final}} = m_1 \left(-\frac{v}{4}\right) + m_2 v_2 \]
- Setting initial momentum equal to final momentum:
\[ m_1 v = -\frac{m_1 v}{4} + m_2 v_2 \]

4. **Rearranging the Equation**:
- Rearranging gives:
\[ m_1 v + \frac{m_1 v}{4} = m_2 v_2 \]
- Simplifying:
\[ \frac{5}{4} m_1 v = m_2 v_2 \]

5. **Use Elastic Collision Property**:
- For elastic collisions, the relative velocity of separation is equal to the relative velocity of approach:
\[ v_2 - \left(-\frac{v}{4}\right) = v - 0 \]
- This simplifies to:
\[ v_2 + \frac{v}{4} = v \]
- Rearranging gives:
\[ v_2 = v - \frac{v}{4} = \frac{3v}{4} \]

6. **Substituting \( v_2 \) Back**:
- Substitute \( v_2 \) back into the momentum equation:
\[ \frac{5}{4} m_1 v = m_2 \left(\frac{3v}{4}\right) \]
- Cancel \( v \) from both sides (assuming \( v \neq 0 \)):
\[ \frac{5}{4} m_1 = \frac{3}{4} m_2 \]

7. **Finding the Ratio \( \frac{m_1}{m_2} \)**:
- Rearranging gives:
\[ \frac{m_1}{m_2} = \frac{3}{5} \]

### Final Answer:
\[ \frac{m_1}{m_2} = \frac{3}{5} \]