In an elastic collision between two particles

Options

A. the total kinetic energy of the system is always conserved.
B. the kinetic energy of the system before collision is equal to the kinetic energy of the system after collision.
C. the linear momentum of the system is conserved.
D. the mechanical energy of the system before collision is equal to the mechanical energy of the system after collision.

Explanation

To solve the question regarding the elastic collision between two particles, we need to analyze the properties of elastic collisions and the conservation laws that apply.

### Step-by-Step Solution:

1. **Understanding Elastic Collision**:
In an elastic collision, two key properties hold true:
- The coefficient of restitution (e) is equal to 1.
- Both momentum and kinetic energy are conserved during the collision.

2. **Kinetic Energy Conservation**:
- For an elastic collision, the total kinetic energy of the system before the collision is equal to the total kinetic energy of the system after the collision.
- Mathematically, if \( KE_{initial} = KE_{final} \), then \( \frac{1}{2} m_1 v_{1i}^2 + \frac{1}{2} m_2 v_{2i}^2 = \frac{1}{2} m_1 v_{1f}^2 + \frac{1}{2} m_2 v_{2f}^2 \).

3. **Momentum Conservation**:
- The total linear momentum of the system before the collision is equal to the total linear momentum after the collision.
- This can be expressed as \( m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f} \).

4. **Mechanical Energy Conservation**:
- Mechanical energy, which includes both kinetic and potential energy, is conserved in an elastic collision.
- If we consider the potential energy to be negligible or constant during the collision, the mechanical energy before and after the collision remains the same.

5. **Conclusion**:
- Therefore, in an elastic collision, the following statements are true:
- The kinetic energy of the system before the collision is equal to the kinetic energy after the collision (Option B).
- The linear momentum of the system is conserved (Option C).
- The mechanical energy of the system before the collision is equal to the mechanical energy after the collision (Option D).
- Thus, the correct options are B, C, and D.