A ball is dropped from some height. After rebound from the floor it ascends to the same height. Draw the velocity-time graph for the given motion.

Options

A. .
B.
C. .
D. .

Explanation

To draw the velocity-time graph for a ball dropped from a height and rebounding to the same height, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Motion**:
- The ball is dropped from a height \( h \) with an initial velocity \( u = 0 \).
- The motion consists of two parts: falling down to the ground and then rebounding back up to the same height.

2. **Analyze the Falling Motion**:
- As the ball falls, it accelerates due to gravity. The acceleration \( a = g \) (approximately \( 9.81 \, \text{m/s}^2 \)).
- The velocity of the ball just before it hits the ground can be calculated using the equation:
\[ v = u + at = 0 + (-gt) = -gt \]
- Here, the negative sign indicates the downward direction.

3. **Graph the Falling Motion**:
- At \( t = 0 \), the velocity \( v = 0 \).
- As time increases, the velocity becomes more negative, indicating that the ball is falling faster.
- The graph will be a straight line with a negative slope, starting from the origin (0,0) and moving downwards.

4. **Analyze the Rebound Motion**:
- Upon hitting the ground, the ball rebounds with the same speed it had just before hitting the ground, but in the opposite direction (upward).
- Thus, the velocity at the moment of rebound is \( v = -v_{\text{impact}} \) (which is positive in the upward direction).

5. **Graph the Rebound Motion**:
- After reaching the ground, the velocity will start from the negative value (just before impact) and increase towards zero as it rises.
- The graph will then continue to rise above the time axis, reaching a maximum height where the velocity becomes zero again.

6. **Final Graph**:
- The complete velocity-time graph will consist of:
- A straight line segment sloping downwards from the origin to a negative velocity value (falling).
- A straight line segment sloping upwards from that negative velocity value back to zero (rebounding).

### Velocity-Time Graph:
- The graph starts at (0,0), slopes downwards to a negative value at the time of impact, and then slopes upwards back to (t, 0) at the maximum height after rebound.