A particle falls from a height h and rebounds to a height `h _(1) ( h _(1) lt h),` then which of the graph represents the motion correct ?

Options

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B.
C.
D.

Answer

To solve the problem, we need to analyze the motion of a particle that falls from a height \( h \) and rebounds to a height \( h_1 \) (where \( h_1 < h \)). We will describe the motion in terms of velocity and time, and then determine the correct graph that represents this motion.

### Step-by-Step Solution:

1. **Understanding the Motion**:
- The particle falls from a height \( h \) under the influence of gravity. As it falls, its velocity increases due to gravitational acceleration.
- Upon reaching the ground, it collides with the ground and rebounds to a height \( h_1 \), which is less than \( h \).

2. **Velocity During Free Fall**:
- As the particle falls, its velocity increases. If we take the downward direction as negative, the velocity just before hitting the ground can be calculated using the equation:
\[ v = \sqrt{2gh} \]
- At the moment just before impact, the velocity is at its maximum and directed downwards (negative).

3. **Impact and Rebound**:
- Upon hitting the ground, the particle's velocity direction changes from negative to positive. This is an instantaneous change in direction.
- After the impact, the particle moves upwards, and its velocity decreases as it rises until it reaches the maximum height \( h_1 \) where the velocity becomes zero.

4. **Velocity During Ascent**:
- As the particle ascends, it decelerates due to gravity until it reaches the height \( h_1 \). The velocity decreases from a positive value to zero at the peak of the rebound.

5. **Graph Representation**:
- The graph of velocity versus time will show:
- A negative slope as the particle falls (increasing negative velocity).
- A sudden change in direction at the point of impact (from negative to positive).
- A positive slope as the particle rises (decreasing positive velocity until it reaches zero).

### Conclusion:
The correct graph will show a continuous increase in negative velocity as the particle falls, a sudden change to positive velocity at the point of impact, and a gradual decrease in positive velocity as the particle rises until it reaches the height \( h_1 \).