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.
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Answer

To solve the problem of determining which graph represents the motion of a particle falling from height \( h \) and rebounding to height \( h_1 \) (where \( h_1 < h \)), we will analyze the motion step by step.

### Step 1: Understand the Motion
The particle falls freely from height \( h \) under the influence of gravity. Initially, it has zero velocity and accelerates downwards due to gravity.

### Step 2: Analyze the Falling Motion
When the particle is falling, its velocity \( v \) can be expressed as:
\[ v = -gt \]
where \( g \) is the acceleration due to gravity and \( t \) is the time of fall. The negative sign indicates that the velocity is directed downwards.

### Step 3: Identify the Velocity During Fall
As the particle falls, its velocity increases in the negative direction (downwards). Therefore, during this phase, the graph of velocity vs. time should show a line with a negative slope starting from zero and moving downwards.

### Step 4: Analyze the Rebound Motion
Upon reaching the ground, the particle rebounds to a height \( h_1 \). The rebound is not elastic, meaning it does not reach the original height \( h \). The velocity just before the rebound is negative (downward), and upon hitting the ground, it instantaneously changes direction to positive.

### Step 5: Express the Velocity During Rebound
For the upward motion after the rebound, the velocity can be expressed as:
\[ v = u - gt \]
where \( u \) is the initial velocity at the moment of rebound (which is now positive). The particle will decelerate until it reaches the peak height \( h_1 \) where the velocity will be zero.

### Step 6: Identify the Velocity During Rebound
During the rebound phase, the velocity starts from a positive value and decreases to zero as the particle reaches the maximum height \( h_1 \). Thus, the graph should show a line starting from a positive value and decreasing to zero.

### Step 7: Combine Both Phases
The complete motion consists of two parts:
1. A downward motion with negative velocity increasing in magnitude.
2. An upward motion starting from a positive velocity and decreasing to zero.

### Step 8: Determine the Correct Graph
From the analysis, the correct graph should:
- Start from zero, go downwards (negative velocity) as time increases during the fall.
- Then, instantaneously change direction at the ground (velocity becomes positive).
- Finally, decrease to zero as it reaches the height \( h_1 \).

### Conclusion
The graph that represents this motion correctly is the one that shows negative velocity during the fall and a positive velocity that decreases to zero during the rebound.