During uniform circular motion of a particle

Options

A. distance-time graph is a straight line
B. distance-time graph is a parabola
C. displacement-time graph is a straight line
D. displacement-time graph is a parabola

Answer

To solve the question regarding uniform circular motion and the corresponding graphs, we will analyze the properties of distance and displacement in this context.

### Step-by-Step Solution:

1. **Understanding Uniform Circular Motion**:
- In uniform circular motion, a particle moves along a circular path with a constant speed. The direction of the particle's velocity is continuously changing, which means the particle is accelerating even though its speed is constant.

2. **Distance Travelled**:
- The distance travelled by the particle in a given time \( t \) can be calculated using the formula:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
- Since the speed \( v \) is constant, the distance travelled is directly proportional to the time taken. This means:
\[ \text{Distance} \propto \text{Time} \]

3. **Distance-Time Graph**:
- Since distance is directly proportional to time, the graph of distance vs. time will be a straight line passing through the origin. This confirms that option 1 (distance-time graph is a straight line) is correct.

4. **Displacement in Circular Motion**:
- Displacement is defined as the shortest distance from the initial position to the final position. In circular motion, as the particle moves, the displacement can vary significantly depending on the position of the particle on the circle.
- For example, if the particle moves from point A to point B, the displacement is the straight line connecting these two points.

5. **Displacement-Time Graph**:
- As the particle continues to move in a circular path, the displacement does not increase uniformly. It can increase and then decrease depending on the position of the particle. Therefore, the displacement-time graph will not be a straight line.
- Instead, it will exhibit a non-linear pattern, which could resemble a parabolic shape but not strictly follow a parabolic equation. Thus, option 3 (displacement-time graph is a straight line) is incorrect, and option 4 (displacement-time graph is a parabola) is also incorrect.

6. **Conclusion**:
- The only correct option is that the distance-time graph is a straight line, while the displacement-time graph does not follow a simple linear or parabolic form.

### Final Answer:
- The correct option is that the distance-time graph is a straight line (Option 1).