A particle falls from rest under gravity. Its potential energy with respect to the ground (PE) and its kinetic energy (KE) are plotted against time (t). Choose the correct graph.

Options

A.
B.
C.
D.

Explanation

To solve the problem of a particle falling from rest under gravity and to analyze its potential energy (PE) and kinetic energy (KE) with respect to time (t), we can follow these steps:

### Step 1: Understand the Initial Conditions
The particle starts from rest at a height \( h \). Therefore, at \( t = 0 \):
- Potential Energy (PE) = \( mgh \) (maximum)
- Kinetic Energy (KE) = 0 (minimum)

### Step 2: Analyze the Motion of the Particle
As the particle falls under the influence of gravity:
- The potential energy decreases as the height decreases.
- The kinetic energy increases as the particle gains speed.

### Step 3: Write the Equations for PE and KE
1. **Potential Energy (PE)**:
The potential energy at any height \( h \) is given by:
\[ PE = mgh \]
As the particle falls, the height \( h \) decreases. The height at time \( t \) can be expressed using the equation of motion:
\[ h(t) = h - \frac{1}{2}gt^2 \]
Therefore, the potential energy as a function of time becomes:
\[ PE(t) = mg\left(h - \frac{1}{2}gt^2\right) = mgh - \frac{1}{2}mgt^2 \]

2. **Kinetic Energy (KE)**:
The velocity \( v \) of the particle at time \( t \) is given by:
\[ v(t) = gt \]
The kinetic energy can be expressed as:
\[ KE = \frac{1}{2}mv^2 = \frac{1}{2}m(gt)^2 = \frac{1}{2}mg^2t^2 \]

### Step 4: Determine the Nature of the Graphs
- **Potential Energy (PE)**:
The equation \( PE(t) = mgh - \frac{1}{2}mgt^2 \) is a downward-facing parabola because the coefficient of \( t^2 \) is negative. It starts at a maximum value \( mgh \) when \( t = 0 \) and decreases to 0 as the particle reaches the ground.

- **Kinetic Energy (KE)**:
The equation \( KE(t) = \frac{1}{2}mg^2t^2 \) is an upward-facing parabola because the coefficient of \( t^2 \) is positive. It starts at 0 when \( t = 0 \) and increases as the particle falls.

### Step 5: Choose the Correct Graph
Based on the analysis:
- The graph of potential energy (PE) will be a downward-facing parabola starting from \( mgh \) at \( t = 0 \) and approaching 0 as \( t \) increases.
- The graph of kinetic energy (KE) will be an upward-facing parabola starting from 0 at \( t = 0 \) and increasing indefinitely as time progresses.

Thus, the correct graph will show:
- PE as a downward parabola starting from \( mgh \).
- KE as an upward parabola starting from 0.

### Conclusion
The correct option is the one that shows the potential energy decreasing from \( mgh \) to 0 and the kinetic energy increasing from 0 to a maximum value as time progresses.

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