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

Explanation

To solve the problem, we need to analyze the statements regarding the two satellites \( S_1 \) and \( S_2 \) that are moving in the same orbit, with the mass of \( S_1 \) being four times that of \( S_2 \).

### Step-by-Step Solution:

1. **Understanding Kinetic Energy**:
The kinetic energy \( KE \) of a satellite in orbit is given by the formula:
\[ KE = \frac{1}{2} m v^2 \]
where \( m \) is the mass of the satellite and \( v \) is its orbital speed.

2. **Orbital Speed**:
The orbital speed \( v \) of a satellite is determined by the formula:
\[ v = \sqrt{\frac{GM}{R}} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the orbit. Since both satellites are in the same orbit, they have the same \( R \) and \( v \).

3. **Kinetic Energy Comparison**:
Since \( S_1 \) has a mass \( m_1 = 4m_2 \) (where \( m_2 \) is the mass of \( S_2 \)), we can express their kinetic energies:
- For \( S_1 \):
\[ KE_1 = \frac{1}{2} (4m_2) v^2 = 2 m_2 v^2 \]
- For \( S_2 \):
\[ KE_2 = \frac{1}{2} m_2 v^2 = \frac{1}{2} m_2 v^2 \]

Clearly, \( KE_1 \neq KE_2 \). Thus, the first statement that the kinetic energies of the two satellites are equal is **incorrect**.

4. **Time Period Calculation**:
The time period \( T \) of a satellite in orbit is given by:
\[ T = \frac{2\pi R}{v} \]
Since both satellites have the same orbital radius \( R \) and the same speed \( v \), their time periods are equal:
\[ T_1 = T_2 \]
Therefore, the second statement that the time period of \( S_1 \) is 4 times that of \( S_2 \) is **incorrect**.

5. **Potential Energy Comparison**:
The gravitational potential energy \( U \) of a satellite is given by:
\[ U = -\frac{GMm}{R} \]
For both satellites:
- For \( S_1 \):
\[ U_1 = -\frac{GM(4m_2)}{R} = -\frac{4GMm_2}{R} \]
- For \( S_2 \):
\[ U_2 = -\frac{GMm_2}{R} \]
Since the potential energies depend on the mass of the satellites, \( U_1 \neq U_2 \). Thus, the third statement that the potential energies of the two satellites are equal is **incorrect**.

6. **Speed Comparison**:
Since we established that both satellites have the same orbital speed \( v \), the fourth statement that \( S_1 \) and \( S_2 \) are moving with the same speed is **correct**.

### Conclusion:
The only true statement is that both satellites \( S_1 \) and \( S_2 \) are moving with the same speed.