Two satellite of the same mass are orbiting round the earth at height f R and 4R above the earth's surface, R being the radius of the earth. Then their kinetic energies are in the ratio of

Options

A. `4:1`
B. `3:2`
C. `4:3`
D. `5:2`

Explanation

To find the ratio of the kinetic energies of two satellites orbiting the Earth at different heights, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the heights of the satellites:**
- Let the radius of the Earth be \( R \).
- The height of the first satellite above the Earth's surface is \( h_1 = R \).
- The height of the second satellite above the Earth's surface is \( h_2 = 4R \).

2. **Calculate the orbital radius for each satellite:**
- The orbital radius \( R_1 \) for the first satellite is:
\[ R_1 = R + h_1 = R + R = 2R \]
- The orbital radius \( R_2 \) for the second satellite is:
\[ R_2 = R + h_2 = R + 4R = 5R \]

3. **Use the formula for kinetic energy of a satellite:**
- The kinetic energy \( K \) of a satellite in orbit is given by the formula:
\[ K = \frac{GMm}{2R} \]
- Where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( m \) is the mass of the satellite,
- \( R \) is the orbital radius.

4. **Calculate the kinetic energy for the first satellite \( K_{e1} \):**
- For the first satellite:
\[ K_{e1} = \frac{GMm}{2R_1} = \frac{GMm}{2(2R)} = \frac{GMm}{4R} \]

5. **Calculate the kinetic energy for the second satellite \( K_{e2} \):**
- For the second satellite:
\[ K_{e2} = \frac{GMm}{2R_2} = \frac{GMm}{2(5R)} = \frac{GMm}{10R} \]

6. **Find the ratio of the kinetic energies:**
- The ratio of the kinetic energies \( \frac{K_{e1}}{K_{e2}} \) is:
\[ \frac{K_{e1}}{K_{e2}} = \frac{\frac{GMm}{4R}}{\frac{GMm}{10R}} = \frac{10R}{4R} = \frac{10}{4} = \frac{5}{2} \]

7. **Final Result:**
- The ratio of the kinetic energies of the two satellites is:
\[ \text{Ratio} = 5 : 2 \]