Following statements are given for the motion of a particle along the straight line. Choose correct statement out of these :

Options

A. Magnitude of the average velocity is equal to the average speed.
B. Distance travelled by the particle is always greater than magnitude of the displacement.
C. A particle having negative acceleration may speed up.
D. A particle having its velocity equal to zero at any instant of the motion, then its acceleration is also zero at that instant.

Answer

To solve the problem of identifying the correct statement regarding the motion of a particle along a straight line, we will analyze each statement one by one.

### Step-by-Step Solution:

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

2. **Analyzing the First Statement**:
- The first statement claims that the magnitude of average velocity is equal to average speed.
- This is generally not true because displacement can be less than the distance traveled if the particle changes direction. Hence, this statement is **incorrect**.

3. **Analyzing the Second Statement**:
- The second statement claims that the distance traveled by the particle is always greater than the magnitude of displacement.
- This is also not universally true. For example, if a particle moves from point A to point B in a straight line and stops there, the distance traveled is equal to the displacement. Therefore, this statement is **incorrect**.

4. **Analyzing the Third Statement**:
- The third statement says that a particle having negative acceleration may speed up.
- This can occur if the velocity of the particle is also in the negative direction. In this case, both the acceleration and velocity are negative, leading to an increase in speed (magnitude of velocity). Thus, this statement is **correct**.

5. **Analyzing the Fourth Statement**:
- The fourth statement claims that if a particle has a velocity of 0 at any instant, then its acceleration is also 0 at that instant.
- This is not necessarily true. A particle can have zero velocity while still experiencing acceleration (for example, at the peak of a projectile's motion). Hence, this statement is **incorrect**.

### Conclusion:
After analyzing all the statements, the only correct statement is that a particle having negative acceleration may speed up.

### Final Answer:
The correct statement is: **A particle having negative acceleration may speed up.**

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