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

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B.
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D.

Answer

To solve the problem of the tennis ball's motion and to determine the appropriate velocity versus height graph, we can follow these steps:

### Step 1: Understand the Motion of the Ball
The tennis ball is released from a height \( h \) and falls freely under the influence of gravity until it hits the wooden floor. After bouncing, it reaches a height of \( \frac{h}{2} \).

### Step 2: Analyze the Falling Motion
When the ball is released from height \( h \):
- The initial velocity \( u = 0 \) (since it is released).
- As it falls, its velocity increases due to gravitational acceleration \( g \).
- The velocity just before it hits the floor can be calculated using the equation of motion:
\[ v^2 = u^2 + 2gh \implies v^2 = 0 + 2gh \implies v = \sqrt{2gh} \]
- This velocity is directed downwards, so we consider it negative in our graph.

### Step 3: Analyze the Bouncing Motion
Upon hitting the floor:
- The ball rebounds to a height of \( \frac{h}{2} \).
- At the peak of its rebound, the velocity becomes \( 0 \) again.
- The velocity just after leaving the floor can be calculated using the same principle:
\[ v^2 = u^2 - 2g\left(\frac{h}{2}\right) \implies v^2 = 0 - gh \implies v = -\sqrt{gh} \]
- After the bounce, the ball moves upwards, and its velocity becomes positive as it rises to \( \frac{h}{2} \).

### Step 4: Graphical Representation
- The graph will start from height \( h \) with a negative velocity (falling down).
- As the ball falls, the velocity increases in the negative direction until it reaches the ground (height \( 0 \)).
- After bouncing, the ball will have a positive velocity as it rises to \( \frac{h}{2} \), and the velocity will decrease to \( 0 \) at the peak.

### Step 5: Determine the Shape of the Graph
- The velocity versus height graph will not be linear due to the nature of the equations of motion (quadratic relationships).
- The downward motion will show an increasing negative velocity (steeper slope) as it approaches the ground.
- The upward motion will show a decreasing positive velocity (less steep slope) as it approaches the height \( \frac{h}{2} \).

### Conclusion
Based on the analysis:
- The graph starts with a negative slope (falling) that becomes steeper as the height decreases, and then it transitions to a positive slope (rising) that becomes less steep as the height increases to \( \frac{h}{2} \).