When a body of mass `m_(1)` collides with another body of mass `m_(2)` kept at rest , then it is found that moving body comes to rest and body at rest starts moving . Which of the following options are possible ?

Options

A. `(m_(1))/(m_(2))lt1`
B. `(m_(1))/(m_(2))gt1`
C. `(m_(1))/(m_(2))=1`
D. `e lt 1 `

Explanation

To solve the problem, we need to analyze the collision between two bodies and apply the principles of conservation of momentum and the coefficient of restitution.

### Step-by-Step Solution:

1. **Identify the Initial Conditions**:
- Let the mass of the moving body be \( m_1 \) and its initial velocity be \( u \).
- Let the mass of the body at rest be \( m_2 \) and its initial velocity be \( 0 \).

2. **Apply Conservation of Momentum**:
- According to the law of conservation of momentum, the total momentum before the collision must equal the total momentum after the collision.
- The initial momentum is given by:
\[ \text{Initial Momentum} = m_1 \cdot u + m_2 \cdot 0 = m_1 \cdot u \]
- After the collision, the moving body comes to rest, so its final momentum is \( 0 \), and the body at rest starts moving with velocity \( v \).
- The final momentum is given by:
\[ \text{Final Momentum} = m_1 \cdot 0 + m_2 \cdot v = m_2 \cdot v \]
- Setting the initial momentum equal to the final momentum:
\[ m_1 \cdot u = m_2 \cdot v \]

3. **Rearranging the Equation**:
- From the equation \( m_1 \cdot u = m_2 \cdot v \), we can express the ratio of masses:
\[ \frac{m_1}{m_2} = \frac{v}{u} \]

4. **Consider the Coefficient of Restitution**:
- The coefficient of restitution \( e \) is defined as the ratio of the relative velocity of separation to the relative velocity of approach.
- Since the moving body comes to rest, the relative velocity of separation is \( v - 0 = v \) and the relative velocity of approach is \( u - 0 = u \).
- Thus, we have:
\[ e = \frac{v}{u} \]
- The coefficient of restitution must satisfy the condition \( 0 \leq e \leq 1 \).

5. **Combine the Results**:
- Since \( e = \frac{v}{u} \) and \( \frac{m_1}{m_2} = \frac{v}{u} \), we can conclude:
\[ \frac{m_1}{m_2} \leq 1 \]
- This means that the mass of the moving body \( m_1 \) must be less than or equal to the mass of the body at rest \( m_2 \).

6. **Evaluate the Options**:
- Based on our findings, we can conclude that:
- \( m_1 \leq m_2 \)
- The coefficient of restitution \( e \) is less than or equal to 1.
- Therefore, the possible options that satisfy these conditions are options A, C, and D.

### Final Conclusion:
- The possible options are A, C, and D.