Compute the additional velocity required by a satellite orbiting around earth with radius 2R to become free from earth's gravitational field. Mass of earth is M.

Options

A. `sqrt((2GM)/(R))(sqrt(2)-1)`
B. `sqrt((GM)/(2R))(sqrt(2)-1)`
C. `sqrt((GM)/(R))(sqrt(3)-1)`
D. `sqrt((GM)/(R))(sqrt(2)+1)`

Explanation

To solve the problem of computing the additional velocity required by a satellite orbiting around Earth at a radius of \(2R\) to escape Earth's gravitational field, we can follow these steps:

### Step 1: Understand the Escape Velocity
The escape velocity (\(v_e\)) from a distance \(r\) from the center of the Earth is given by the formula:
\[ v_e = \sqrt{\frac{2GM}{r}} \]
where \(G\) is the gravitational constant and \(M\) is the mass of the Earth.

### Step 2: Calculate the Escape Velocity at Radius \(2R\)
For a satellite at a distance of \(2R\) from the center of the Earth, we can substitute \(r = 2R\) into the escape velocity formula:
\[ v_e = \sqrt{\frac{2GM}{2R}} = \sqrt{\frac{GM}{R}} \]

### Step 3: Calculate the Orbital Velocity at Radius \(2R\)
The orbital velocity (\(v_0\)) of a satellite in circular orbit at a distance \(r\) is given by:
\[ v_0 = \sqrt{\frac{GM}{r}} \]
Substituting \(r = 2R\):
\[ v_0 = \sqrt{\frac{GM}{2R}} = \frac{1}{\sqrt{2}} \sqrt{\frac{GM}{R}} \]

### Step 4: Find the Additional Velocity Required
The additional velocity (\(\Delta v\)) required for the satellite to escape Earth's gravitational field is the difference between the escape velocity and the orbital velocity:
\[ \Delta v = v_e - v_0 \]
Substituting the values we calculated:
\[ \Delta v = \sqrt{\frac{GM}{R}} - \frac{1}{\sqrt{2}} \sqrt{\frac{GM}{R}} \]
Factoring out \(\sqrt{\frac{GM}{R}}\):
\[ \Delta v = \sqrt{\frac{GM}{R}} \left(1 - \frac{1}{\sqrt{2}}\right) \]

### Step 5: Simplify the Expression
To simplify \(1 - \frac{1}{\sqrt{2}}\):
\[ 1 - \frac{1}{\sqrt{2}} = \frac{\sqrt{2} - 1}{\sqrt{2}} \]
Thus, we have:
\[ \Delta v = \sqrt{\frac{GM}{R}} \cdot \frac{\sqrt{2} - 1}{\sqrt{2}} \]

### Final Answer
The additional velocity required by the satellite to escape Earth's gravitational field is:
\[ \Delta v = \sqrt{\frac{GM}{R}} \cdot \frac{\sqrt{2} - 1}{\sqrt{2}} \]