A batsman hits a sixes and the ball touches the ground outside the cricket ground. Which of the following graph describes the variation of the cricket ball's vertical velocity v with time between the time `t_1` as it hits the bat and time `t_2` when it touches the ground?

Options

A.
B.
C.
D.

Explanation

To solve the problem of the variation of the cricket ball's vertical velocity \( v \) with time between the time \( t_1 \) (when the batsman hits the ball) and time \( t_2 \) (when it touches the ground), we can analyze the motion of the ball step by step.

### Step-by-Step Solution:

1. **Understanding the Motion**:
- When the batsman hits the ball, it is propelled upwards. Initially, at time \( t_1 \), the ball has a maximum vertical velocity directed upwards.

2. **Velocity Decrease During Ascent**:
- As the ball rises, it moves against the force of gravity. The gravitational force acts downwards, causing the ball's upward velocity to decrease until it reaches its maximum height. At this point, the vertical velocity becomes zero.

3. **Maximum Height**:
- The maximum height is reached when the vertical velocity is zero. This occurs at a certain time \( t_{max} \) between \( t_1 \) and \( t_2 \).

4. **Velocity Increase During Descent**:
- After reaching the maximum height, the ball begins to fall back down. The gravitational force accelerates the ball downwards, causing the vertical velocity to increase in the negative direction (downwards) until it touches the ground at time \( t_2 \).

5. **Graphical Representation**:
- The graph of vertical velocity \( v \) versus time \( t \) will start at a maximum positive value at \( t_1 \), decrease to zero at the maximum height, and then increase negatively until it reaches a maximum negative value just before it touches the ground at \( t_2 \).

6. **Choosing the Correct Graph**:
- The correct graph will show:
- A maximum positive velocity at \( t_1 \),
- A decrease to zero at the maximum height,
- An increase in negative velocity as the ball falls back down to the ground.

### Conclusion:
The graph that best describes this motion will show a curve that starts from a positive maximum, decreases to zero, and then continues into the negative region, reflecting the increasing downward velocity as the ball descends.