A particle may have

Options

A. variable velocity without variable speed.
B. variable speed without variable velocity.
C. zero acceleration with variable velocity
D. nonzero acceleration with constant speed.

Answer

To solve the question regarding the possible scenarios for a particle's motion, we will analyze each option based on the definitions of speed, velocity, and acceleration.

### Step-by-Step Solution:

1. **Understanding Key Concepts**:
- **Speed**: A scalar quantity that represents how fast an object is moving, regardless of its direction. It is the magnitude of velocity.
- **Velocity**: A vector quantity that includes both speed and direction. It can change if either the speed or the direction changes.
- **Acceleration**: A vector quantity that represents the rate of change of velocity over time. It can be non-zero even if speed is constant, as long as the direction changes.

2. **Analyzing Option A**:
- **Variable velocity without variable speed**:
- This is possible if the speed (magnitude of velocity) remains constant while the direction changes. For example, an object moving in a circular path at a constant speed has variable velocity due to the change in direction.
- **Conclusion**: This option is **correct**.

3. **Analyzing Option B**:
- **Variable speed without variable velocity**:
- This scenario is not possible because if the speed varies, the magnitude of velocity must also change. Thus, it cannot remain constant.
- **Conclusion**: This option is **incorrect**.

4. **Analyzing Option C**:
- **Zero acceleration with variable velocity**:
- Acceleration is defined as the rate of change of velocity. If velocity is variable (changing), then acceleration cannot be zero.
- **Conclusion**: This option is **incorrect**.

5. **Analyzing Option D**:
- **Non-zero acceleration with constant speed**:
- This is possible if the object is changing direction while maintaining a constant speed. For example, a car turning around a curve at a steady speed experiences non-zero acceleration due to the change in direction.
- **Conclusion**: This option is **correct**.

### Final Answer:
The correct options are A and D.