A satellite of mass m goes round the earth along a circular path of radius r. Let `m_(E)` be the mass of the earth and `R_(E)` its radius then the linear speed of the satellite depends on.

Options

A. `m, m_(E)` and `r`
B. `m, R_(E)` and `r`
C. `m_(E)` only
D. `m_(E)` and `r`

Answer

To determine how the linear speed of a satellite depends on the mass of the Earth and the radius of its orbit, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Satellite**:
The satellite in orbit experiences two main forces: gravitational force and centripetal force. The gravitational force provides the necessary centripetal force for the satellite to maintain its circular path.

2. **Write the Expression for Centripetal Force**:
The centripetal force \( F_c \) required to keep the satellite moving in a circular path is given by:
\[ F_c = \frac{mv^2}{r} \]
where \( m \) is the mass of the satellite, \( v \) is the linear speed of the satellite, and \( r \) is the radius of the orbit.

3. **Write the Expression for Gravitational Force**:
The gravitational force \( F_g \) between the Earth and the satellite is given by Newton's law of gravitation:
\[ F_g = \frac{G m_E m}{r^2} \]
where \( G \) is the gravitational constant, \( m_E \) is the mass of the Earth, and \( r \) is the distance from the center of the Earth to the satellite.

4. **Set the Centripetal Force Equal to Gravitational Force**:
Since the gravitational force provides the centripetal force, we can set these two forces equal to each other:
\[ \frac{mv^2}{r} = \frac{G m_E m}{r^2} \]

5. **Cancel Out the Mass of the Satellite**:
We can cancel the mass \( m \) of the satellite from both sides of the equation (assuming \( m \neq 0 \)):
\[ \frac{v^2}{r} = \frac{G m_E}{r^2} \]

6. **Rearrange the Equation to Solve for \( v^2 \)**:
Rearranging gives us:
\[ v^2 = \frac{G m_E}{r} \]

7. **Take the Square Root to Find the Linear Speed \( v \)**:
Taking the square root of both sides, we find the linear speed \( v \):
\[ v = \sqrt{\frac{G m_E}{r}} \]

### Conclusion:
The linear speed \( v \) of the satellite depends on the mass of the Earth \( m_E \) and the radius of the orbit \( r \). Thus, we can conclude that:
\[ v \propto \sqrt{m_E} \quad \text{and} \quad v \propto \frac{1}{\sqrt{r}} \]