A particle is moving on a straight line and its average velocity is found to be zero in an interval of time.

Options

A. Average speed of the particle may also be zero for a given interval of time.
B. Velocity of the particle can never be zero in given interval of time.
C. Velocity of the particle must be zero at a particular instant.
D. Acceleration of particle may be zero.

Explanation

To solve the problem of a particle moving on a straight line with an average velocity of zero over a given interval of time, we can break down the solution into several steps:

### Step-by-Step Solution:

1. **Understanding Average Velocity**:
The average velocity (\(v_{avg}\)) of a particle is defined as the total displacement (\(\Delta x\)) divided by the total time taken (\(\Delta t\)):
\[ v_{avg} = \frac{\Delta x}{\Delta t} \]

2. **Given Condition**:
We are given that the average velocity is zero over a certain interval of time. This implies:
\[ v_{avg} = 0 \implies \Delta x = 0 \]
Therefore, the total displacement of the particle during this time interval is zero.

3. **Interpreting Displacement**:
If the displacement is zero, it means that the final position of the particle (\(X_2\)) is equal to its initial position (\(X_1\)):
\[ X_2 - X_1 = 0 \implies X_2 = X_1 \]
This indicates that the particle has returned to its starting point after the time interval.

4. **Velocity at a Particular Instant**:
Since the particle has returned to its original position, it is possible that at some point during its motion, the particle must have come to a stop (velocity = 0) before changing direction to return to the starting point. This is a critical observation.

5. **Acceleration Consideration**:
For the particle to return to its starting position, it must have experienced some form of acceleration, which could be acting in the opposite direction to its motion. This means that the particle's speed would decrease to zero at some instant before it changes direction.

6. **Conclusion**:
Thus, we can conclude that:
- The average velocity is zero because the displacement is zero.
- The velocity of the particle must be zero at some instant during the motion for it to return to its starting position.

### Final Answer:
The correct option is that the velocity of the particle must be zero at a particular instant.

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