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.

Answer

To solve the problem of determining the correct graph for the potential energy (PE) and kinetic energy (KE) of a particle falling from rest under gravity, we need to analyze how both energies change over time.

### Step-by-Step Solution:

1. **Understanding the Motion**:
- A particle falls from rest under the influence of gravity. Initially, it has zero kinetic energy and maximum potential energy.

2. **Potential Energy (PE)**:
- The potential energy of the particle with respect to the ground can be expressed as:
\[ PE = mgh \]
where \( h \) is the height of the particle above the ground. As the particle falls, \( h \) decreases, leading to a decrease in potential energy.

3. **Kinetic Energy (KE)**:
- The kinetic energy of the particle can be expressed as:
\[ KE = \frac{1}{2} mv^2 \]
where \( v \) is the velocity of the particle. As the particle falls, its velocity increases, leading to an increase in kinetic energy.

4. **Relationship Between PE and Time**:
- Since the particle falls under gravity, the height \( h \) can be expressed as:
\[ h = \frac{1}{2} gt^2 \]
- Substituting this into the potential energy formula gives:
\[ PE = mg\left(\frac{1}{2}gt^2\right) = \frac{1}{2} mg t^2 \]
- This shows that potential energy is proportional to \( t^2 \) and decreases as the particle falls.

5. **Relationship Between KE and Time**:
- The velocity \( v \) of the particle can be expressed as:
\[ v = gt \]
- Substituting this into the kinetic energy formula gives:
\[ KE = \frac{1}{2} m(gt)^2 = \frac{1}{2} mg^2 t^2 \]
- This shows that kinetic energy is also proportional to \( t^2 \) and increases as the particle falls.

6. **Graphical Representation**:
- As time progresses:
- Potential energy decreases quadratically (downward curve).
- Kinetic energy increases quadratically (upward curve).
- The total mechanical energy (PE + KE) remains constant if we ignore air resistance.

### Conclusion:
The correct graph will show:
- A downward-opening parabola for potential energy (PE) as it decreases over time.
- An upward-opening parabola for kinetic energy (KE) as it increases over time.