A charged particle X moves directly towards another charged particle Y. For the X plus Y system, the total momentum is p and the total energy is E.

Options

A. (a) p and E are conserved if both X and Y are free to move.
B. (b) (a) is true only if X and Y have similar charges.
C. (c) If Y is fixed, E is conserved but not P.
D. (d) If Y is fixed, neither E nor P is conserved.

Explanation

To solve the problem, we need to analyze the conservation of momentum and energy in the system of two charged particles, X and Y.

### Step-by-Step Solution:

1. **Understanding the System**:
- We have two charged particles, X and Y. We denote their total momentum as \( P \) and their total energy as \( E \).
- We need to determine whether momentum and energy are conserved in different scenarios.

2. **Conservation of Momentum**:
- The momentum of a system is conserved if there are no external forces acting on it.
- In our case, if both particles X and Y are free to move, there are no external forces acting on them. Therefore, the total momentum \( P \) of the system (X + Y) is conserved.

3. **Conservation of Energy**:
- The total energy of a system is conserved if only conservative forces are acting on it.
- In the case of charged particles, the forces between them are electrostatic (which are conservative forces). Therefore, the total mechanical energy (kinetic + potential) of the system is also conserved.

4. **Analyzing the Options**:
- **Option A**: "P and E are conserved if both X and Y are free to move."
- This is correct because both momentum and energy are conserved under the given conditions.
- **Option B**: "A is true only if X and Y have their similar charges."
- This is incorrect. The conservation of momentum and energy does not depend on whether the charges are similar or different.
- **Option C**: "If Y is fixed, E is conserved but P is not."
- This is correct. If Y is fixed, an external force acts on it, which means momentum is not conserved, but energy can still be conserved due to the conservative nature of the forces involved.
- **Option D**: "If Y is fixed, neither A nor B is conserved."
- This is incorrect because energy can still be conserved even if momentum is not.

5. **Conclusion**:
- The correct answers are **Option A** and **Option C**.