The correct graph representing the variation of total energy `(E_(t))`, kinetic energy `(E_(k))` and potential energy `(U)` of a satellite with its distance form the centre of earth is

Options

A.
B.
C.
D.

Explanation

To analyze the variation of total energy (E_t), kinetic energy (E_k), and potential energy (U) of a satellite with its distance from the center of the Earth, we can follow these steps:

### Step 1: Understand the Energy Formulas
1. **Potential Energy (U)**: The potential energy of a satellite in orbit is given by the formula:
\[ U = -\frac{G M m}{r} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, \( m \) is the mass of the satellite, and \( r \) is the distance from the center of the Earth.

2. **Kinetic Energy (E_k)**: The kinetic energy of the satellite is given by:
\[ E_k = \frac{G M m}{2r} \]

3. **Total Energy (E_t)**: The total energy of the satellite is the sum of kinetic and potential energy:
\[ E_t = E_k + U = \frac{G M m}{2r} - \frac{G M m}{r} = -\frac{G M m}{2r} \]

### Step 2: Analyze the Variation of Energies with Distance
- As the distance \( r \) increases (moving away from the Earth):
- The **Potential Energy (U)** becomes less negative (increases) because it approaches zero.
- The **Kinetic Energy (E_k)** decreases because it is inversely proportional to \( r \).
- The **Total Energy (E_t)** also becomes less negative, following the same trend as potential energy.

### Step 3: Sketch the Graphs
1. **Kinetic Energy (E_k)**:
- Starts positive and decreases as \( r \) increases.
- The graph will be a decreasing curve starting from a positive value.

2. **Potential Energy (U)**:
- Starts from a very negative value and approaches zero as \( r \) increases.
- The graph will be a decreasing curve starting from a negative value and approaching zero.

3. **Total Energy (E_t)**:
- Starts from a value slightly less than the potential energy and remains negative, decreasing as \( r \) increases.
- The graph will be a decreasing curve below the potential energy graph.

### Step 4: Combine the Graphs
- Plot all three graphs on the same axes with energy on the y-axis and distance from the center of the Earth on the x-axis.
- The graph of kinetic energy will be above the potential energy graph, and the total energy graph will be below both.

### Conclusion
Based on the analysis, the correct graph representing the variation of total energy, kinetic energy, and potential energy of a satellite with its distance from the center of the Earth corresponds to option C.

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