IF e is the coefficient of restitution then which one of the following gives the condition for perfectly elastic bodies ?

Options

A. e = 0
B. e = 0.5
C. e = 0.8
D. e = 1.0

Answer

To determine the condition for perfectly elastic bodies in terms of the coefficient of restitution (e), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definition of Perfectly Elastic Bodies**:
- A perfectly elastic body is one where the total kinetic energy before and after a collision remains the same. This means there is no loss of kinetic energy during the collision.

2. **Kinetic Energy Conservation**:
- For two colliding bodies with masses \( m_1 \) and \( m_2 \), and initial velocities \( v_1 \) and \( v_2 \), the initial kinetic energy (KE_initial) is given by:
\[ KE_{\text{initial}} = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 \]
- After the collision, if the velocities change to \( v_1' \) and \( v_2' \), the final kinetic energy (KE_final) is:
\[ KE_{\text{final}} = \frac{1}{2} m_1 (v_1')^2 + \frac{1}{2} m_2 (v_2')^2 \]

3. **Condition for Perfectly Elastic Collision**:
- For a perfectly elastic collision, we require:
\[ KE_{\text{initial}} = KE_{\text{final}} \]
- This implies that the kinetic energy is conserved.

4. **Coefficient of Restitution (e)**:
- The coefficient of restitution (e) is defined as the ratio of the relative velocity of separation to the relative velocity of approach. Mathematically, it is given by:
\[ e = \frac{\text{Relative velocity after collision}}{\text{Relative velocity before collision}} = \frac{v_2' - v_1'}{v_1 - v_2} \]

5. **Condition for Perfectly Elastic Bodies**:
- For perfectly elastic bodies, since there is no loss of kinetic energy, the value of the coefficient of restitution (e) must be equal to 1:
\[ e = 1 \]

6. **Conclusion**:
- Therefore, the condition for perfectly elastic bodies is that the coefficient of restitution \( e \) is equal to 1.

### Final Answer:
The condition for perfectly elastic bodies is given by:
\[ e = 1 \]