Two satellites , A and B , have masses m and 2m respectively . A is in a circular orbit of radius R , and B is in a circular orbit of radius 2R around the earth . The ratio of their energies , `K_A/K_B` is :

Options

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

Answer

To find the ratio of the kinetic energies of two satellites A and B, we can follow these steps:

### Step 1: Understand the Kinetic Energy Formula
The kinetic energy (K.E.) of a satellite in a circular orbit can be expressed as:
\[ K.E. = \frac{GMm}{2R} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( m \) is the mass of the satellite,
- \( R \) is the radius of the orbit.

### Step 2: Calculate Kinetic Energy of Satellite A
For satellite A, with mass \( m \) and orbit radius \( R \):
\[ K_A = \frac{GM \cdot m}{2R} \]

### Step 3: Calculate Kinetic Energy of Satellite B
For satellite B, with mass \( 2m \) and orbit radius \( 2R \):
\[ K_B = \frac{GM \cdot (2m)}{2(2R)} = \frac{GM \cdot 2m}{4R} = \frac{GM \cdot m}{2R} \]

### Step 4: Find the Ratio of Kinetic Energies
Now, we can find the ratio of the kinetic energies \( K_A \) and \( K_B \):
\[ \frac{K_A}{K_B} = \frac{\frac{GM \cdot m}{2R}}{\frac{GM \cdot m}{2R}} = 1 \]

### Conclusion
The ratio of the kinetic energies of satellites A and B is:
\[ \frac{K_A}{K_B} = 1 \]