If magnitude of average speed and average velocity over a time interval are same, then

Options

A. The particle must move with zero acceleration
B. The particle must move with non-zero acceleration
C. The particle must be at rest
D. The particle must move in a straight line without turning back

Answer

To solve the question, we need to analyze the relationship between average speed and average velocity.

### Step-by-Step Solution:

1. **Understand Definitions**:
- Average Speed (S_avg) is defined as the total distance traveled divided by the total time taken.
\[ S_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} \]
- Average Velocity (V_avg) is defined as the total displacement divided by the total time taken.
\[ V_{avg} = \frac{\text{Total Displacement}}{\text{Total Time}} \]

2. **Set the Condition**:
- According to the problem, the magnitude of average speed and average velocity are the same:
\[ |S_{avg}| = |V_{avg}| \]

3. **Equate the Formulas**:
- From the definitions, we can write:
\[ \frac{\text{Total Distance}}{\text{Total Time}} = \frac{\text{Total Displacement}}{\text{Total Time}} \]
- Since the total time is the same in both cases, we can simplify this to:
\[ \text{Total Distance} = \text{Total Displacement} \]

4. **Analyze the Condition**:
- The only scenario where the total distance traveled is equal to the total displacement is when the particle moves in a straight line without changing direction. If the particle turns back or moves in a non-linear path, the distance would be greater than the displacement.

5. **Conclusion**:
- Therefore, for the average speed and average velocity to be equal, the particle must move in a straight line without turning back.

### Final Answer:
The correct option is: **Particle must move in a straight line without turning back.**

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