A Tennis ball is released from a height h and after freely falling on a wooden floor it rebounds and reaches height `h/2`. The velocity versus height of the ball during its motion may be represented graphically by :
(graph are drawn schematically and on not to scale )

Options

A.
B.
C.
D.

Answer

To solve the problem, we need to analyze the motion of the tennis ball as it falls and rebounds. We will derive the relationship between velocity and height and determine the correct graphical representation.

### Step-by-Step Solution:

1. **Understanding the Motion**:
- The tennis ball is released from a height \( h \).
- It falls freely under the influence of gravity until it hits the ground.
- After hitting the ground, it rebounds to a height of \( \frac{h}{2} \).

2. **Finding the Velocity Just Before Impact**:
- Using the equation of motion:
\[ v^2 = u^2 + 2gh \]
where \( u = 0 \) (initial velocity), \( g \) is the acceleration due to gravity, and \( h \) is the height.
- Thus, the velocity just before impact (when height is 0) is:
\[ v = \sqrt{2gh} \]

3. **Finding the Velocity Just After Impact**:
- After rebounding to a height of \( \frac{h}{2} \), we can use the same equation to find the velocity just after the impact when it reaches the height \( \frac{h}{2} \):
\[ v' = \sqrt{2g\left(\frac{h}{2}\right)} = \sqrt{gh} \]

4. **Analyzing the Velocity vs. Height Graph**:
- As the ball falls from height \( h \) to height \( 0 \), its velocity increases from \( 0 \) to \( \sqrt{2gh} \).
- After the impact, as the ball rises from height \( 0 \) to height \( \frac{h}{2} \), its velocity decreases from \( \sqrt{2gh} \) to \( 0 \).

5. **Determining the Nature of the Graph**:
- The relationship between velocity and height is not linear. As the ball falls, the velocity increases, and as it rises, the velocity decreases.
- The slope of the velocity vs. height graph will change, indicating a non-linear relationship.

6. **Conclusion**:
- The correct representation of the velocity vs. height graph will show an increase in velocity as height decreases (falling) and a decrease in velocity as height increases (rising), confirming that the graph is not linear.