The K.E. of a satellite in an orbit close to the surface of the earth is E. Its max K.E. so as to escape from the gravitational field of the earth is

Options

A. 2E
B. 4E
C. `2 sqrt(2) E`
D. `sqrt(2) E`

Explanation

To solve the problem, we need to find the maximum kinetic energy (K.E.) required for a satellite to escape from the gravitational field of the Earth, given that its kinetic energy in a low orbit is \( E \).

### Step-by-step Solution:

1. **Understanding Kinetic Energy in Orbit**:
The kinetic energy (K.E.) of a satellite in a circular orbit close to the Earth's surface is given by the formula:
\[ K.E. = \frac{G M m}{2R} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( m \) is the mass of the satellite,
- \( R \) is the radius of the Earth.

According to the problem, this kinetic energy is given as \( E \):
\[ E = \frac{G M m}{2R} \]

2. **Escape Velocity**:
The escape velocity \( v_e \) from the Earth's surface is given by the formula:
\[ v_e = \sqrt{\frac{2GM}{R}} \]

3. **Calculating Maximum Kinetic Energy for Escape**:
The maximum kinetic energy required to escape the gravitational field of the Earth is given by:
\[ K.E. = \frac{1}{2} m v_e^2 \]
Substituting the expression for escape velocity:
\[ K.E. = \frac{1}{2} m \left(\sqrt{\frac{2GM}{R}}\right)^2 \]
Simplifying this gives:
\[ K.E. = \frac{1}{2} m \cdot \frac{2GM}{R} = \frac{GMm}{R} \]

4. **Relating to Given Kinetic Energy**:
From the earlier step, we know:
\[ E = \frac{G M m}{2R} \]
Therefore, we can express \( \frac{G M m}{R} \) in terms of \( E \):
\[ \frac{G M m}{R} = 2E \]

5. **Final Result**:
Thus, the maximum kinetic energy required to escape from the gravitational field of the Earth is:
\[ K.E. = 2E \]

### Conclusion:
The maximum kinetic energy required for the satellite to escape from the gravitational field of the Earth is \( 2E \).

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