The speed -time graph of the ball in the above situation is.

Options

A.
B.
C.
D.

Answer

To determine the correct speed-time graph for a ball dropped from a height and bouncing on a horizontal floor with a coefficient of restitution of 1/2, we can analyze the motion step by step.

### Step-by-Step Solution:

1. **Initial Drop**:
- The ball is dropped from a height \( h \).
- Initially, the speed \( v_0 = 0 \) m/s (the ball is at rest before being dropped).
- As it falls, it accelerates downwards due to gravity \( g \) (approximately \( 9.8 \, \text{m/s}^2 \)).
- The speed of the ball just before it hits the ground can be calculated using the equation:
\[ v_1 = gt \]
- The graph will start at the origin (0,0) and rise as the speed increases.

2. **Impact with the Ground**:
- Upon hitting the ground, the ball will have a speed \( v_1 \).
- Due to the coefficient of restitution \( e = \frac{1}{2} \), the speed after the bounce will be:
\[ v_2 = e \cdot v_1 = \frac{1}{2} v_1 \]
- This means that the speed decreases after the bounce.

3. **Upward Motion**:
- After bouncing, the ball will move upwards, decelerating due to gravity until it reaches its maximum height where its speed becomes 0.
- The speed-time graph will show a decrease from \( v_2 \) to 0 as it moves upward.

4. **Second Impact**:
- When the ball falls back down, it will again accelerate downwards and hit the ground with a speed \( v_3 \), which will be:
\[ v_3 = e \cdot v_2 = e \cdot \left(\frac{1}{2} v_1\right) = \frac{1}{4} v_1 \]
- The speed will again increase as it falls back down.

5. **Continued Bouncing**:
- This process will continue, with the speed before each impact decreasing due to the coefficient of restitution.
- The speeds will follow the pattern: \( v_1, v_2, v_3, v_4 \) where \( v_1 > v_2 > v_3 > v_4 \) and so on.

6. **Graph Characteristics**:
- The graph will show a series of peaks (the speeds just before each impact) that get progressively lower.
- The segments of the graph will alternate between increasing and decreasing slopes, reflecting the acceleration downwards and the deceleration upwards.

### Conclusion:
Based on the analysis, the correct speed-time graph will show:
- An initial increase in speed as the ball falls.
- A drop in speed after each bounce, with the peaks getting lower each time.
- The graph will not cross into negative speed values, as speed cannot be negative in this context.

Thus, the correct option for the speed-time graph is **B**.