To solve the problem, we need to analyze the motion of a particle moving in a circular path while covering equal distances in equal time intervals. This type of motion is known as uniform circular motion. Let's break down the solution step by step.
### Step-by-Step Solution:
1. **Understanding the Motion**:
- The particle moves in a circular path.
- It covers equal distances in equal intervals of time, which means it has a constant speed.
**Hint**: Remember that uniform circular motion implies constant speed, but not constant velocity.
2. **Identifying Key Quantities**:
- The main quantities we need to consider are speed, velocity, displacement, and acceleration.
- Speed is a scalar quantity, while velocity and acceleration are vector quantities.
3. **Analyzing Speed**:
- Since the particle covers equal distances in equal time intervals, the speed remains constant throughout the motion.
- Let's denote this constant speed as \( V \).
**Hint**: Speed is constant if the distance covered in equal time intervals is the same.
4. **Analyzing Velocity**:
- Velocity is a vector quantity that depends on both speed and direction.
- Although the speed is constant, the direction of the particle's motion changes continuously as it moves along the circular path.
- Therefore, the velocity is not constant.
**Hint**: Remember that a change in direction means a change in velocity, even if speed is constant.
5. **Analyzing Displacement**:
- Displacement is the shortest distance from the initial position to the final position.
- As the particle moves around the circular path, the displacement changes with time, depending on the position of the particle on the circle.
- Thus, displacement is not constant.
**Hint**: Displacement can vary even if the distance traveled is constant.
6. **Analyzing Acceleration**:
- In uniform circular motion, the acceleration is centripetal, directed towards the center of the circular path.
- While the magnitude of centripetal acceleration remains constant (given by \( a = \frac{V^2}{R} \)), its direction changes continuously as the particle moves around the circle.
- Therefore, acceleration is not constant.
**Hint**: Acceleration can have a constant magnitude but still change direction.
7. **Conclusion**:
- The only quantity that remains constant in this scenario is the speed of the particle.
- Thus, the correct answer is that the speed remains constant with time.
### Final Answer:
The quantity connected with the motion of the particle that remains constant with time is **speed**.
