In a perfectly elastic collision between two bodies

Options

A. Kinetic energy of the system remains constant .
B. Mechanical energy of the system remains constant .
C. Kinetic energy first decreases and then ncreases .
D. Kinetic energy before collision is equal to kinetic energy after collision .

Explanation

### Step-by-Step Solution:

1. **Understanding Perfectly Elastic Collision**:
In a perfectly elastic collision, both momentum and kinetic energy are conserved. This means that the total kinetic energy before the collision is equal to the total kinetic energy after the collision.

2. **Identifying the Bodies and Their Velocities**:
Let's denote the two bodies as Body 1 and Body 2. Let the initial velocities of Body 1 and Body 2 be \( U_1 \) and \( U_2 \) respectively, where \( U_1 > U_2 \).

3. **Collision Process**:
When Body 1 collides with Body 2, there are two main phases to consider:
- **Time of Compression**: During this phase, the bodies deform, and kinetic energy is converted into elastic potential energy. Thus, the kinetic energy of the system decreases at this moment.
- **Time of Restitution**: After the maximum deformation, the bodies start to return to their original shapes, converting the stored elastic potential energy back into kinetic energy.

4. **Kinetic Energy Before and After Collision**:
- Before the collision, the total kinetic energy of the system is given by:
\[ KE_{initial} = \frac{1}{2} m_1 U_1^2 + \frac{1}{2} m_2 U_2^2 \]
- After the collision, the total kinetic energy is:
\[ KE_{final} = \frac{1}{2} m_1 V_1^2 + \frac{1}{2} m_2 V_2^2 \]
- For a perfectly elastic collision, we have:
\[ KE_{initial} = KE_{final} \]

5. **Mechanical Energy Conservation**:
The total mechanical energy (kinetic + potential) is conserved throughout the process. Therefore, the total mechanical energy before the collision is equal to the total mechanical energy after the collision.

6. **Kinetic Energy Changes During Collision**:
- During the collision, the kinetic energy first decreases (as it is converted to potential energy during deformation) and then increases again (as potential energy is converted back to kinetic energy).

7. **Conclusion**:
- The statements regarding the kinetic energy and mechanical energy can be evaluated:
- **Statement A**: Kinetic energy of the system remains constant (Incorrect during collision).
- **Statement B**: Mechanical energy of the system remains constant (Correct).
- **Statement C**: Kinetic energy first decreases then increases (Correct).
- **Statement D**: Kinetic energy before collision is equal to kinetic energy after collision (Correct).

### Final Answer:
The correct options are B, C, and D.

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