For a moving particle, which of the following options may be correct? Here, `V_(av)` is average velocity and `v_(av)` the average speed.

Options

A. `|V_(av)|ltv_(av)`
B. `|V_(av)|gtv_(av)`
C. `V_(av)=0` but `v_(av) !=0`
D. `V_(av)!=0 but v_(av) =0`

Explanation

To solve the question regarding the relationship between average velocity (\(V_{av}\)) and average speed (\(v_{av}\)) for a moving particle, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Definitions**:
- Average speed (\(v_{av}\)) is defined as the total distance traveled divided by the total time taken:
\[ v_{av} = \frac{\text{Total Distance}}{\text{Total Time}} \]
- Average velocity (\(V_{av}\)) is defined as the total displacement divided by the total time taken:
\[ V_{av} = \frac{\text{Total Displacement}}{\text{Total Time}} \]

2. **Relationship Between Distance and Displacement**:
- It is important to note that the total distance traveled is always greater than or equal to the magnitude of the total displacement. This is because displacement is a vector quantity that considers direction, while distance is a scalar quantity that does not:
\[ \text{Total Distance} \geq |\text{Total Displacement}| \]

3. **Implications for Average Speed and Average Velocity**:
- From the relationship established in the previous step, we can conclude that:
\[ v_{av} \geq |V_{av}| \]
- This means that the average speed is always greater than or equal to the magnitude of the average velocity.

4. **Consider Special Cases**:
- If the average velocity (\(V_{av}\)) is zero, it implies that the total displacement is zero. However, the average speed (\(v_{av}\)) can still be greater than zero if the particle has traveled a distance but returned to its starting point.
- Conversely, if the average speed (\(v_{av}\)) is zero, it means the particle has not moved at all, which implies that the average velocity (\(V_{av}\)) must also be zero.

5. **Conclusion**:
- Therefore, the correct statements regarding average velocity and average speed are:
- Average speed is always greater than or equal to the magnitude of average velocity.
- Average velocity can be zero while average speed is not zero, but not the other way around.

### Final Answer:
- Option 1: \(v_{av} \geq |V_{av}|\) is correct.
- Option 3: \(V_{av} = 0\) does not imply \(v_{av} = 0\) is also correct.