A satellite is revolving in a circular orbit at a height 'h' from the earth's surface (radius of earth R). The minimum increase in its orbital velocity required, So that the satellite could escape from the earth's gravitational field, is close to :(Neglect the effect of atomsphere.)

Options

A. `sqrt(2gR)`
B. `sqrt(gR)`
C. `sqrt(gR//2)`
D. `sqrt(gR)(sqrt(2)-1)`

Answer

To solve the problem of determining the minimum increase in orbital velocity required for a satellite to escape from the Earth's gravitational field, we can follow these steps:

### Step 1: Understand the parameters
- Let the radius of the Earth be \( R \).
- Let the height of the satellite above the Earth's surface be \( h \).
- The distance from the center of the Earth to the satellite is \( R + h \).

### Step 2: Calculate the orbital velocity (\( V_0 \))
The gravitational force provides the necessary centripetal force for the satellite in orbit. Therefore, we can set up the following equation:

\[ \frac{GMm}{(R + h)^2} = \frac{mv_0^2}{(R + h)} \]

Where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( m \) is the mass of the satellite,
- \( v_0 \) is the orbital velocity.

By simplifying this equation, we can find the expression for the orbital velocity:

\[ v_0^2 = \frac{GM}{R + h} \]

Thus, the orbital velocity is:

\[ v_0 = \sqrt{\frac{GM}{R + h}} \]

### Step 3: Calculate the escape velocity (\( V_e \))
The escape velocity from a distance \( R + h \) from the center of the Earth is given by:

\[ V_e = \sqrt{\frac{2GM}{R + h}} \]

### Step 4: Calculate the increase in velocity (\( \Delta V \))
The increase in velocity required for the satellite to escape is the difference between the escape velocity and the orbital velocity:

\[ \Delta V = V_e - V_0 \]

Substituting the expressions for \( V_e \) and \( V_0 \):

\[ \Delta V = \sqrt{\frac{2GM}{R + h}} - \sqrt{\frac{GM}{R + h}} \]

### Step 5: Factor out the common term
Factoring out \( \sqrt{\frac{GM}{R + h}} \):

\[ \Delta V = \sqrt{\frac{GM}{R + h}} \left( \sqrt{2} - 1 \right) \]

### Step 6: Express in terms of \( g \)
We know that \( g = \frac{GM}{R^2} \). Therefore, we can express \( GM \) in terms of \( g \):

\[ \Delta V = \sqrt{gR} \left( \sqrt{2} - 1 \right) \]

### Final Expression
Thus, the minimum increase in the orbital velocity required for the satellite to escape from the Earth's gravitational field is:

\[ \Delta V = \sqrt{gR} \left( \sqrt{2} - 1 \right) \]

### Conclusion
The correct answer is option D: \( \sqrt{gR} \cdot (\sqrt{2} - 1) \).