The coefficient of restitution e for a perfectly elastic collision is

Options

A. 1
B. `oo`
C. Zero
D. `-1`

Answer

To determine the coefficient of restitution (e) for a perfectly elastic collision, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Coefficient of Restitution (e)**:
The coefficient of restitution is defined as the ratio of the relative speed of separation to the relative speed of approach of two colliding bodies. It is given by the formula:
\[ e = \frac{\text{Relative speed after collision}}{\text{Relative speed before collision}} \]

2. **Analyzing a Perfectly Elastic Collision**:
In a perfectly elastic collision, both momentum and kinetic energy are conserved. This means that the objects bounce off each other without any loss of kinetic energy.

3. **Relative Speeds in Elastic Collision**:
- Before the collision, let the velocities of two colliding bodies be \( u_1 \) and \( u_2 \).
- The relative speed before collision is given by:
\[ \text{Relative speed before collision} = u_1 - u_2 \]
- After the collision, let their velocities be \( v_1 \) and \( v_2 \).
- The relative speed after collision is given by:
\[ \text{Relative speed after collision} = v_2 - v_1 \]

4. **Applying the Definition**:
For a perfectly elastic collision, the relative speed after the collision is equal to the negative of the relative speed before the collision:
\[ v_2 - v_1 = -(u_1 - u_2) \]
This implies:
\[ v_2 - v_1 = u_2 - u_1 \]

5. **Substituting into the Coefficient of Restitution Formula**:
Now substituting the relative speeds into the coefficient of restitution formula:
\[ e = \frac{v_2 - v_1}{u_1 - u_2} = \frac{-(u_1 - u_2)}{u_1 - u_2} = 1 \]

6. **Conclusion**:
Thus, for a perfectly elastic collision, the coefficient of restitution \( e \) is equal to 1.

### Final Answer:
The coefficient of restitution \( e \) for a perfectly elastic collision is **1**.

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