Radius of orbit of satellite of earth is `R`. Its kinetic energy is proportional to

Options

A. `(1)/(sqrt(R ))`
B. `(1)/(R )`
C. `R`
D. `(1)/(R^(3//2))`

Answer

To determine the relationship between the kinetic energy of a satellite in orbit and the radius of its orbit, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Forces**:
A satellite in orbit experiences a gravitational force that acts as the centripetal force required to keep it in circular motion. The gravitational force \( F_G \) between the Earth and the satellite is given by Newton's law of gravitation:
\[ F_G = \frac{G M m}{r^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, \( m \) is the mass of the satellite, and \( r \) is the distance from the center of the Earth to the satellite.

2. **Centripetal Force Requirement**:
For the satellite to maintain its circular orbit, the gravitational force must equal the centripetal force required for circular motion:
\[ F_C = \frac{m v^2}{r} \]
where \( v \) is the orbital speed of the satellite.

3. **Setting the Forces Equal**:
Setting the gravitational force equal to the centripetal force gives:
\[ \frac{G M m}{r^2} = \frac{m v^2}{r} \]

4. **Canceling Mass and Rearranging**:
We can cancel the mass of the satellite \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{G M}{r^2} = \frac{v^2}{r} \]
Multiplying both sides by \( r \) gives:
\[ v^2 = \frac{G M}{r} \]

5. **Finding Kinetic Energy**:
The kinetic energy \( K \) of the satellite is given by:
\[ K = \frac{1}{2} m v^2 \]
Substituting the expression for \( v^2 \):
\[ K = \frac{1}{2} m \left( \frac{G M}{r} \right) \]
Simplifying this, we find:
\[ K = \frac{G M m}{2r} \]

6. **Determining Proportionality**:
From the equation \( K = \frac{G M m}{2r} \), we can see that the kinetic energy \( K \) is inversely proportional to the radius \( r \):
\[ K \propto \frac{1}{r} \]

### Conclusion:
Thus, the kinetic energy of a satellite in orbit is proportional to \( \frac{1}{R} \), where \( R \) is the radius of the orbit.