Two satellites of earth `S_(1)` and `S_(2)` are moving in the same orbit. The mass of `S_(1)` is four times the mass of `S_(2)`. Which one of the following statements is true?

Options

A. The time period of `S_(1)` is four times that of `S_(2)`
B. The potential energies of the earth and satellite in the two cases are equal
C. `S_(1)` and `S_(2)` are moving with the same speed
D. The kinetic energies of the two satellites are equal

Answer

To solve the problem, we need to analyze the characteristics of two satellites \( S_1 \) and \( S_2 \) that are moving in the same orbit around the Earth, where the mass of \( S_1 \) is four times the mass of \( S_2 \).

### Step-by-Step Solution:

1. **Define the Masses of the Satellites**:
- Let the mass of satellite \( S_2 \) be \( m \).
- Therefore, the mass of satellite \( S_1 \) will be \( 4m \).

2. **Understand the Orbit**:
- Both satellites are in the same orbit, which means they are at the same distance \( r \) from the center of the Earth.

3. **Time Period of Satellites**:
- The time period \( T \) of a satellite in a circular orbit is given by the formula:
\[ T = 2\pi \sqrt{\frac{r^3}{GM}} \]
where \( G \) is the gravitational constant and \( M \) is the mass of the Earth.
- Since both satellites are at the same distance \( r \) from the center of the Earth, their time periods \( T_1 \) and \( T_2 \) will be the same:
\[ T_1 = T_2 \]

4. **Speed of the Satellites**:
- The speed \( v \) of a satellite in orbit is given by:
\[ v = \frac{2\pi r}{T} \]
- Since \( T_1 = T_2 \) and both satellites are at the same distance \( r \), it follows that:
\[ v_1 = v_2 \]
- Thus, both satellites are moving with the same speed.

5. **Kinetic Energy of the Satellites**:
- The kinetic energy \( KE \) of a satellite is given by:
\[ KE = \frac{1}{2} mv^2 \]
- For satellite \( S_1 \):
\[ KE_1 = \frac{1}{2} (4m) v_1^2 = 2m v_1^2 \]
- For satellite \( S_2 \):
\[ KE_2 = \frac{1}{2} m v_2^2 = \frac{1}{2} m v_1^2 \]
- Since \( v_1 = v_2 \), we can see that:
\[ KE_1 \neq KE_2 \]
- Therefore, the kinetic energies of the two satellites are not equal.

6. **Potential Energy of the Satellites**:
- The gravitational potential energy \( U \) between the Earth and a satellite is given by:
\[ U = -\frac{GMm}{r} \]
- For satellite \( S_1 \):
\[ U_1 = -\frac{GM(4m)}{r} \]
- For satellite \( S_2 \):
\[ U_2 = -\frac{GMm}{r} \]
- Clearly, \( U_1 \neq U_2 \).

### Conclusion:
From the analysis, we find that:
- The time periods of both satellites are the same.
- The speeds of both satellites are the same.
- The kinetic energies of both satellites are not equal.
- The potential energies of both satellites are not equal.

The correct statement is that **both satellites are moving with the same speed**.