The ratio of binding energy of a satellite at rest on earth's surface to the binding energy of a satellite of same mass revolving around of the earth at a height h above the earth's surface is (R = radius of the earth).

Options

A. (a)`(2(R + h))/(R )`
B. (b)`(R + h)/(2R)`
C. (c)`(R + h)/(R )`
D. (d)`R/(R + h)`

Explanation

To find the ratio of the binding energy of a satellite at rest on the Earth's surface to the binding energy of a satellite of the same mass revolving around the Earth at a height \( h \) above the Earth's surface, we can follow these steps:

### Step 1: Understand the Binding Energy Formula
The binding energy \( E \) of a satellite is given by the formula:
\[ E = -\frac{G M m}{r} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( m \) is the mass of the satellite,
- \( r \) is the distance from the center of the Earth to the satellite.

### Step 2: Calculate Binding Energy at Rest on Earth's Surface
For a satellite at rest on the Earth's surface, the distance \( r \) is equal to the radius of the Earth \( R \):
\[ E_{\text{rest}} = -\frac{G M m}{R} \]

### Step 3: Calculate Binding Energy of Satellite in Orbit
For a satellite revolving at a height \( h \) above the Earth's surface, the distance \( r \) from the center of the Earth is \( R + h \):
\[ E_{\text{orbit}} = -\frac{G M m}{R + h} \]

### Step 4: Find the Ratio of Binding Energies
Now, we need to find the ratio of the binding energy at rest to the binding energy in orbit:
\[ \text{Ratio} = \frac{E_{\text{rest}}}{E_{\text{orbit}}} = \frac{-\frac{G M m}{R}}{-\frac{G M m}{R + h}} = \frac{R + h}{R} \]

### Step 5: Simplify the Ratio
The ratio can be simplified to:
\[ \text{Ratio} = \frac{R + h}{R} = 1 + \frac{h}{R} \]

### Conclusion
Thus, the ratio of the binding energy of a satellite at rest on the Earth's surface to that of a satellite revolving at a height \( h \) above the Earth's surface is:
\[ \text{Ratio} = 1 + \frac{h}{R} \]