Among the four graphs shown in the figure there is only one graph for which average velocity over the time interval (0,T) can vanish for a suitably chosen `T`. Which one is it ?

Options

A.
B.
C.
D.

Explanation

To determine which graph allows the average velocity over the time interval (0, T) to vanish for a suitably chosen T, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Average Velocity**:
- Average velocity (v_avg) is defined as the total displacement (Δx) divided by the total time (Δt). Mathematically, it is given by:
\[ v_{avg} = \frac{\Delta x}{\Delta t} \]

2. **Identify Displacement**:
- Displacement is the difference between the final position and the initial position. For the average velocity to be zero, the total displacement must be zero.

3. **Analyze Each Graph**:
- We need to examine the four graphs provided to see which one can yield a total displacement of zero over some time interval (0, T).

4. **Graph A**:
- If the graph shows a continuous increase in displacement, the average velocity cannot be zero since the displacement is always positive.

5. **Graph B**:
- If the graph has segments where the object moves forward and then returns to the starting point, it can have a total displacement of zero. This means there are two times (t1 and t2) where the object has the same displacement, allowing for a zero average velocity.

6. **Graph C**:
- If this graph shows a continuous increase or decrease without returning to the starting point, the average velocity cannot be zero.

7. **Graph D**:
- Similar to Graph A, if there is no point where the displacement returns to zero, the average velocity will not vanish.

8. **Conclusion**:
- After analyzing the graphs, we conclude that **Graph B** is the only graph where the average velocity can vanish for a suitably chosen T, as it allows for a scenario where the total displacement is zero.

### Final Answer:
The correct graph is **Graph B**.