What is the energy required to launch a `m` kg satellite from earth's surface in a circular orbit at an altitude of 7R ? (R = radius of the earth)

Options

A. `(12)/(13)mgR`
B. `mgR`
C. `(15)/(16)mgR`
D. `(1)/(9)mgR`

Explanation

To find the energy required to launch a satellite of mass \( m \) kg from the Earth's surface to a circular orbit at an altitude of \( 7R \) (where \( R \) is the radius of the Earth), we will follow these steps:

### Step 1: Understand the Energy in Orbit
The total mechanical energy \( E \) of a satellite in a circular orbit is given by the formula:
\[ E = -\frac{G M m}{2r} \]
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: Determine the Distance \( r \)
For a satellite at an altitude of \( 7R \), the distance \( r \) from the center of the Earth is:
\[ r = R + 7R = 8R \]

### Step 3: Calculate the Energy in Orbit
Substituting \( r = 8R \) into the energy formula:
\[ E = -\frac{G M m}{2 \times 8R} = -\frac{G M m}{16R} \]

### Step 4: Calculate the Energy at the Earth's Surface
The potential energy \( U \) of the satellite when it is on the surface of the Earth is given by:
\[ U = -\frac{G M m}{R} \]

### Step 5: Calculate the Change in Energy
The energy required to launch the satellite to the orbit is the difference between the energy in orbit and the energy at the Earth's surface:
\[ \Delta E = E_{\text{orbit}} - E_{\text{surface}} = \left(-\frac{G M m}{16R}\right) - \left(-\frac{G M m}{R}\right) \]
\[ \Delta E = -\frac{G M m}{16R} + \frac{G M m}{R} \]
\[ \Delta E = \frac{G M m}{R} - \frac{G M m}{16R} \]
\[ \Delta E = \frac{G M m}{R} \left(1 - \frac{1}{16}\right) = \frac{G M m}{R} \left(\frac{15}{16}\right) \]

### Final Answer
Thus, the energy required to launch the satellite from the Earth's surface to a circular orbit at an altitude of \( 7R \) is:
\[ \Delta E = \frac{15}{16} \frac{G M m}{R} \]