A stone is projected vertically upwards `y=0` second. The net displacement of stone is zero in time interval between `t=0` second to `t=T` seconds. Pick up the correct statement

Options

A. From time `t=(T)/(4)` seconds to `t=(3T)/(4)` second, the average velocity is zero.
B. the change in velocity from time `t=0` to `t=(T)/(4)` second is same as change in velocity from `t=(T)/(8)` second to `t=(3T)/(8)` second
C. The distance travelled from `t=0` to `t=(T)/(4)` second is larger than distance travelled from `t=(T)/(4)` second to `t=(3T)/(4)` second
D. The distance travelled from `t=(T)/(2) second to t=(3T)/(4) second` is half the distance travelled from `t=(T)/(2)` second to `t=T` second.

Explanation

To solve the problem, we need to analyze the motion of the stone projected vertically upwards and determine the correct statement based on the given conditions.

### Step-by-Step Solution:

1. **Understanding the Motion**:
The stone is projected vertically upwards at time \( t = 0 \) seconds. As it rises, it will eventually reach a maximum height and then fall back down due to gravity.

2. **Net Displacement**:
The problem states that the net displacement of the stone is zero in the time interval from \( t = 0 \) seconds to \( t = T \) seconds. This means that the stone returns to its original position after time \( T \).

3. **Time of Flight**:
For a stone projected upwards, the total time of flight \( T \) can be divided into two equal halves: the time taken to reach the maximum height \( T/2 \) and the time taken to fall back down \( T/2 \).

4. **Average Velocity Calculation**:
The average velocity \( v_{avg} \) over the time interval \( t = 0 \) to \( t = T \) is given by:
\[ v_{avg} = \frac{\text{Total Displacement}}{\text{Total Time}} = \frac{0}{T} = 0 \]
Since the net displacement is zero, the average velocity is also zero.

5. **Analyzing the Options**:
- **Option A**: The average velocity from \( t = T/4 \) to \( t = 3T/4 \) is zero. This is correct because the stone moves up and then down, returning to the same position.
- **Option B**: The change in velocity from \( t = 0 \) to \( t = T/4 \) is equal to the change in velocity from \( t = T/8 \) to \( t = 3T/8 \). This is also correct since the stone experiences uniform acceleration due to gravity.
- **Option C**: The distance traveled from \( t = 0 \) to \( t = T/4 \) is larger than the distance traveled from \( t = T/4 \) to \( t = 3T/4 \). This is incorrect because the stone travels upwards for the first quarter and then downwards for the next quarter, covering the same distance.
- **Option D**: The distance traveled from \( t = T/2 \) to \( t = 3T/4 \) is not equal to the distance traveled from \( t = 3T/4 \) to \( t = T \). This is also incorrect.

6. **Conclusion**:
The correct statements are Option A and Option B.

### Final Answer:
- **Correct Statements**: Option A and Option B are correct.