Which of the following statements is correct?

Options

A. (a) Kinetic energy of a system can be changed without changing its momentum.
B. (b) Kinetic energy of a system cannot be changed without changing its momentum.
C. (c) Momentum of a system cannot be changed without changing its kinetic energy.
D. (d) A system cannot have energy without having momentum.

Explanation

To solve the question regarding which statements are correct, we will evaluate each statement one by one based on the principles of physics related to kinetic energy and momentum.

### Step-by-Step Solution:

1. **Evaluate Statement A**: "Kinetic energy of a system can be changed without changing its momentum."
- This statement is **true**.
- Explanation: Consider an explosion where a bomb explodes. Initially, the bomb has no kinetic energy and is at rest (momentum = 0). After the explosion, the fragments of the bomb gain kinetic energy, but the total momentum of the system remains conserved (still 0) because the forces involved are internal. Thus, kinetic energy can change while momentum remains constant.

2. **Evaluate Statement B**: "Kinetic energy of a system cannot be changed without changing its momentum."
- This statement is **false**.
- Explanation: This is the negation of Statement A. Since we established that kinetic energy can change without affecting momentum (as seen in explosions), this statement is incorrect.

3. **Evaluate Statement C**: "Momentum of a system cannot be changed without changing its kinetic energy."
- This statement is **false**.
- Explanation: Consider a particle moving in a straight line with a force applied perpendicular to its motion. The work done by this force is zero (since force and displacement are at 90 degrees), meaning the kinetic energy remains constant. However, the direction of the velocity changes, which means the momentum can change without a change in kinetic energy.

4. **Evaluate Statement D**: "A system cannot have energy without having momentum."
- This statement is **false**.
- Explanation: A particle at a height \( h \) has potential energy due to its position (PE = mgh) but can be at rest (velocity = 0), which means it has no kinetic energy and no momentum. Thus, a system can have energy (potential energy) without having momentum.

### Conclusion:
From the evaluations:
- Statement A is **true**.
- Statements B, C, and D are **false**.

Therefore, the correct answer is that only **Statement A** is correct.