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. `(2(R+h))/R`
B. `(R+h)/(2R)`
C. `(R+h)/R`
D. `R/(R+h)`

Answer

To solve the problem of finding 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 at Earth's Surface
The binding energy \( E_1 \) of a satellite at rest on the Earth's surface can be expressed using the formula for gravitational potential energy. The binding energy is given by:

\[ E_1 = -\frac{G M m}{R} \]

where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( m \) is the mass of the satellite,
- \( R \) is the radius of the Earth.

### Step 2: Understand the Binding Energy of a Satellite in Orbit
For a satellite revolving around the Earth at a height \( h \) above the Earth's surface, the distance from the center of the Earth is \( R + h \). The binding energy \( E_2 \) of this satellite can be expressed as:

\[ E_2 = -\frac{G M m}{R + h} \]

### Step 3: Calculate the Ratio of Binding Energies
We need to find the ratio of the binding energy at the Earth's surface to the binding energy of the satellite in orbit:

\[ \text{Ratio} = \frac{E_1}{E_2} = \frac{-\frac{G M m}{R}}{-\frac{G M m}{R + h}} = \frac{R + h}{R} \]

### Step 4: Simplify the Ratio
The ratio simplifies to:

\[ \text{Ratio} = \frac{R + h}{R} = 1 + \frac{h}{R} \]

### Conclusion
Thus, the final expression for 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 \( h \) is:

\[ \text{Ratio} = 1 + \frac{h}{R} \]