A particle covers equal distance around a circular path, in equal intervals of time. Which of the following quantities connected with the motion of the particle remains constant with time?

Options

A. displacement
B. velocity
C. speed
D. acceleration

Explanation

To solve the problem, we need to analyze the motion of a particle moving in a circular path while covering equal distances in equal intervals of time. We will evaluate the quantities of displacement, velocity, speed, and acceleration to determine which one remains constant.

### Step-by-Step Solution:

1. **Understanding the Motion**:
- The particle is moving in a circular path and covers equal distances in equal time intervals. This implies that the particle is moving with uniform motion along the circular path.

2. **Analyzing Speed**:
- Speed is defined as the total distance traveled divided by the time taken. Since the particle covers equal distances in equal intervals of time, the speed of the particle remains constant.

3. **Analyzing Velocity**:
- Velocity is a vector quantity that depends on both the magnitude (speed) and the direction of motion. In circular motion, even if the speed is constant, the direction of the velocity vector changes continuously as the particle moves around the circle. Therefore, the velocity is not constant.

4. **Analyzing Displacement**:
- Displacement is the shortest distance from the initial to the final position of the particle. In circular motion, as the particle moves around the circle, the displacement changes with each position. Hence, displacement is not constant.

5. **Analyzing Acceleration**:
- Acceleration is also a vector quantity and is defined as the rate of change of velocity. In uniform circular motion, the particle experiences centripetal acceleration directed towards the center of the circle. Although the magnitude of this acceleration remains constant, its direction changes continuously as the particle moves. Thus, acceleration is not constant.

6. **Conclusion**:
- Among the quantities analyzed, the only quantity that remains constant with time is **speed**.

### Final Answer:
**The quantity that remains constant with time is speed.**