Two identical satellites are orbiting are orbiting at distances R and 7R from the surface of the earth, R being the radius of the earth. The ratio of their

Options

A. kinetic energies is 4
B. potential energies is 4
C. total energies is 4
D. All of these

Explanation

To solve the problem, we need to find the ratios of the kinetic energies, potential energies, and total energies of two identical satellites orbiting at different distances from the Earth's surface.

### Given:
- Radius of Earth, \( R \)
- Distance of the first satellite from the surface of the Earth, \( R_1 = R \)
- Distance of the second satellite from the surface of the Earth, \( R_2 = 7R \)

### Step 1: Calculate the distances from the center of the Earth
- The distance of the first satellite from the center of the Earth:
\[ r_1 = R + R = 2R \]
- The distance of the second satellite from the center of the Earth:
\[ r_2 = R + 7R = 8R \]

### Step 2: Kinetic Energy Calculation
The formula for the kinetic energy \( K \) of a satellite in orbit is given by:
\[ K = \frac{GMm}{2r} \]
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

For the first satellite:
\[ K_1 = \frac{GMm}{2(2R)} = \frac{GMm}{4R} \]

For the second satellite:
\[ K_2 = \frac{GMm}{2(8R)} = \frac{GMm}{16R} \]

### Step 3: Ratio of Kinetic Energies
Now, we can find the ratio of the kinetic energies:
\[ \frac{K_1}{K_2} = \frac{\frac{GMm}{4R}}{\frac{GMm}{16R}} = \frac{16R}{4R} = 4 \]

### Step 4: Potential Energy Calculation
The formula for potential energy \( U \) of a satellite in orbit is given by:
\[ U = -\frac{GMm}{r} \]

For the first satellite:
\[ U_1 = -\frac{GMm}{2R} \]

For the second satellite:
\[ U_2 = -\frac{GMm}{8R} \]

### Step 5: Ratio of Potential Energies
Now, we can find the ratio of the potential energies:
\[ \frac{U_1}{U_2} = \frac{-\frac{GMm}{2R}}{-\frac{GMm}{8R}} = \frac{8R}{2R} = 4 \]

### Step 6: Total Energy Calculation
The total energy \( E \) of a satellite is given by:
\[ E = K + U \]

For the first satellite:
\[ E_1 = K_1 + U_1 = \frac{GMm}{4R} - \frac{GMm}{2R} = \frac{GMm}{4R} - \frac{2GMm}{4R} = -\frac{GMm}{4R} \]

For the second satellite:
\[ E_2 = K_2 + U_2 = \frac{GMm}{16R} - \frac{GMm}{8R} = \frac{GMm}{16R} - \frac{2GMm}{16R} = -\frac{GMm}{16R} \]

### Step 7: Ratio of Total Energies
Now, we can find the ratio of the total energies:
\[ \frac{E_1}{E_2} = \frac{-\frac{GMm}{4R}}{-\frac{GMm}{16R}} = \frac{16R}{4R} = 4 \]

### Conclusion
The ratios of kinetic energies, potential energies, and total energies of the two satellites are all equal to 4. Therefore, the correct option is:
**All of these.**