Which of the following graph is correct between kinetic energy `E`, potential energy `(U)` and height `(h)` from the ground of the partical

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining the correct relationship between kinetic energy (E), potential energy (U), and height (h) from the ground for a particle, we can follow these steps:

### Step 1: Understand the Concepts
- **Potential Energy (U)**: The potential energy of an object at height \( h \) is given by the formula:
\[ U = mgh \]
where \( m \) is the mass of the object, \( g \) is the acceleration due to gravity, and \( h \) is the height above the ground.

- **Kinetic Energy (E)**: The kinetic energy of an object is given by:
\[ E = \frac{1}{2}mv^2 \]
where \( v \) is the velocity of the object.

### Step 2: Analyze Energy Conservation
- According to the law of conservation of mechanical energy, the total mechanical energy (sum of kinetic and potential energy) of the particle remains constant if only conservative forces (like gravity) are acting on it. Thus, we can write:
\[ E + U = \text{constant} \]
This implies that if potential energy increases, kinetic energy must decrease, and vice versa.

### Step 3: Relationship Between Height, Kinetic Energy, and Potential Energy
- As height \( h \) increases:
- Potential energy \( U \) increases because \( U = mgh \).
- Kinetic energy \( E \) decreases due to the conservation of energy.

### Step 4: Graphical Representation
- The graph of potential energy \( U \) versus height \( h \) will be a straight line that increases linearly.
- The graph of kinetic energy \( E \) versus height \( h \) will decrease as height increases, leading to a downward slope.

### Conclusion
- The correct graph will show:
- An increasing linear graph for potential energy \( U \) as height \( h \) increases.
- A decreasing graph for kinetic energy \( E \) as height \( h \) increases.

### Final Answer
The correct option is the one that shows potential energy increasing linearly with height and kinetic energy decreasing as height increases.