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 given information
- Satellite A has mass \( m \) and is in a circular orbit of radius \( R \).
- Satellite B has mass \( 2m \) and is in a circular orbit of radius \( 2R \).

### Step 2: Write the formula for kinetic energy
The kinetic energy \( K \) of an object in orbit can be expressed as:
\[ K = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the object and \( v \) is its orbital velocity.

### Step 3: Determine the orbital velocity for each satellite
For a satellite in orbit, the gravitational force provides the necessary centripetal force. Thus, we can equate:
\[ \frac{mv^2}{r} = \frac{GMm}{r^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( r \) is the radius of the orbit.

From this equation, we can solve for the velocity \( v \):
\[ v^2 = \frac{GM}{r} \]
Thus, the velocity \( v \) for satellite A (radius \( R \)):
\[ v_A^2 = \frac{GM}{R} \]
And for satellite B (radius \( 2R \)):
\[ v_B^2 = \frac{GM}{2R} \]

### Step 4: Calculate the kinetic energies
Now we can find the kinetic energies for both satellites.

For satellite A:
\[ K_A = \frac{1}{2} m v_A^2 = \frac{1}{2} m \left(\frac{GM}{R}\right) = \frac{GMm}{2R} \]

For satellite B:
\[ K_B = \frac{1}{2} (2m) v_B^2 = \frac{1}{2} (2m) \left(\frac{GM}{2R}\right) = \frac{GMm}{2R} \]

### Step 5: Find the ratio of the kinetic energies
Now we can find the ratio \( \frac{K_A}{K_B} \):
\[ \frac{K_A}{K_B} = \frac{\frac{GMm}{2R}}{\frac{GMm}{2R}} = 1 \]

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