The satellites when launched from the earth are not given the orbital velocity initially, a multistage rocket propeller carries the spacecraft up to its orbit and during each stage rocket has been fired to increase the velocity to acquire the desired velocity for a particular orbit. The last stage of the rocket brings the satellite in circular/elliptical (desired) orbit.
Consider a satellite of mass `150 kg` in a low circular orbit. In this orbit, we cannot neglect the effect of air drag. This air opposes the motion of satellite and hence the total mechanical energy of earth-satellite system decreases. That means the total energy becomes more negative and hence the orbital radius decreases which causes the increase in `KE` When the satellite comes in the low enough orbit, excessive thermal energy generation due to air friction may cause the satellite to burn up. Based on the above information, answer the following questions.
If due to air drag, the orbital radius of the earth decreases from `R` to `R-/_\R, /_\Rlt ltR`, then the expression for increase in the orbital velocity `/_\v` is

Options

A. `(/_\r)/2 sqrt((GM)/R^3)`
B. `-(/_\R)/2 sqrt((GM)/2)`
C. `/_\R sqrt((GM)/(R^(3)))`
D. `-/_\Rsqrt((GM)/(R^(3)))`

Explanation

To solve the problem of finding the expression for the increase in orbital velocity (\( \Delta v \)) when the orbital radius decreases from \( R \) to \( R - \Delta R \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Orbital Velocity**:
The orbital velocity (\( v \)) of a satellite in a circular orbit is given by the formula:
\[ v = \sqrt{\frac{GM}{r}} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( r \) is the distance from the center of the Earth to the satellite.

2. **Initial and Final Radii**:
The initial radius is \( R \) and the final radius after the decrease due to air drag is \( r = R - \Delta R \).

3. **Calculating Initial Orbital Velocity**:
The initial orbital velocity at radius \( R \) is:
\[ v_i = \sqrt{\frac{GM}{R}} \]

4. **Calculating Final Orbital Velocity**:
The final orbital velocity at radius \( r = R - \Delta R \) is:
\[ v_f = \sqrt{\frac{GM}{R - \Delta R}} \]

5. **Finding the Increase in Orbital Velocity**:
The increase in orbital velocity (\( \Delta v \)) is given by:
\[ \Delta v = v_f - v_i = \sqrt{\frac{GM}{R - \Delta R}} - \sqrt{\frac{GM}{R}} \]

6. **Using Taylor Expansion for Small Changes**:
Since \( \Delta R \) is very small compared to \( R \), we can use a Taylor expansion to approximate \( v_f \):
\[ v_f \approx v_i + \frac{d}{dr}\left(\sqrt{\frac{GM}{r}}\right)\bigg|_{r=R} \Delta R \]
The derivative is:
\[ \frac{d}{dr}\left(\sqrt{\frac{GM}{r}}\right) = -\frac{1}{2}\frac{GM}{r^{3/2}} \]
Evaluating at \( r = R \):
\[ \frac{d}{dr}\left(\sqrt{\frac{GM}{r}}\right)\bigg|_{r=R} = -\frac{1}{2}\frac{GM}{R^{3/2}} \]
Thus,
\[ \Delta v \approx -\frac{1}{2}\frac{GM}{R^{3/2}} \Delta R \]

7. **Final Expression**:
Rearranging gives:
\[ \Delta v = \frac{\Delta R}{2} \cdot \frac{GM}{R^{3/2}} \]

### Final Result:
The expression for the increase in the orbital velocity \( \Delta v \) is:
\[ \Delta v = \frac{\Delta R}{2} \cdot \frac{GM}{R^{3/2}} \]