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. **Understanding Average Speed and Average Velocity**:
- Average speed is defined as the total distance traveled divided by the total time taken.
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]
- Average velocity is defined as the total displacement divided by the total time taken.
\[ \text{Average Velocity} = \frac{\text{Total Displacement}}{\text{Total Time}} \]

2. **Setting Up the Condition**:
- According to the problem, the magnitude of average speed and average velocity are the same:
\[ \left| \text{Average Speed} \right| = \left| \text{Average Velocity} \right| \]

3. **Equating the Two Definitions**:
- 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. **Interpreting the Result**:
- The equality of total distance and total displacement implies that the object has not changed direction during its motion.
- This means that the object must have traveled in a straight line without turning back.

5. **Conclusion**:
- Therefore, if the magnitude of average speed and average velocity over a time interval are the same, it indicates that the particle must move in a straight line without changing direction.

### Final Answer:
The correct conclusion is that the particle must move in a straight line without turning back.

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