It magnitude of average speed and average velocity over an interval of time are same, then

Options

A. Particle must move with zero acceleration
B. Particle must move with uniform acceleration
C. Particle must be at rest
D. Particle must move in a straight line without turning back

Answer

To solve the question regarding the conditions under which the magnitude of average speed and average velocity over an interval of time are the same, we can follow these steps:

### 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.
- Average velocity is defined as the total displacement (change in position) divided by the total time taken.

2. **Setting Up the Equation**:
- Let \( T \) be the total time for the motion.
- Let \( D \) be the total distance traveled.
- Let \( S \) be the total displacement.
- According to the question, we have:
\[ \text{Average Speed} = \frac{D}{T} \]
\[ \text{Average Velocity} = \frac{S}{T} \]
- Given that the magnitudes of average speed and average velocity are equal, we can write:
\[ \frac{D}{T} = \frac{S}{T} \]

3. **Simplifying the Equation**:
- Since \( T \) is common in both terms, we can cancel it out (assuming \( T \neq 0 \)):
\[ D = S \]

4. **Interpreting the Result**:
- The equality \( D = S \) implies that the total distance traveled is equal to the total displacement.
- This condition holds true only when the object moves in a straight line without changing direction.

5. **Conclusion**:
- Therefore, the condition for the magnitudes of average speed and average velocity to be the same is that the particle must move in a straight line without turning back.

### Final Answer:
The magnitude of average speed and average velocity over an interval of time are the same when the particle moves in a straight line without changing direction.

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