Two satellite of mass m and 2 m are revolving in two circular orbits of radius r and 2r around an imaginary planet , on the surface of with gravitational force is inversely proportional to distance from its centre . The ratio of orbital speed of satellite is .

Options

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

Answer

To solve the problem of finding the ratio of the orbital speeds of two satellites revolving around an imaginary planet, we can follow these steps:

### Step 1: Understand the Problem
We have two satellites:
- Satellite 1 with mass \( m \) in a circular orbit of radius \( r \).
- Satellite 2 with mass \( 2m \) in a circular orbit of radius \( 2r \).

The gravitational force is inversely proportional to the distance from the center of the planet.

### Step 2: Write the Formula for Orbital Speed
The orbital speed \( v \) of a satellite in a circular orbit is given by the formula:
\[ v = \sqrt{\frac{F}{m}} \]
where \( F \) is the gravitational force acting on the satellite and \( m \) is the mass of the satellite.

### Step 3: Determine the Gravitational Force
Since the gravitational force is inversely proportional to the distance from the center of the planet, we can express the gravitational force \( F \) acting on each satellite as:
\[ F \propto \frac{1}{d} \]
For satellite 1 at distance \( r \):
\[ F_1 = k \cdot \frac{1}{r} \]
For satellite 2 at distance \( 2r \):
\[ F_2 = k \cdot \frac{1}{2r} \]
where \( k \) is a constant of proportionality.

### Step 4: Calculate the Orbital Speeds
1. For Satellite 1:
\[ v_1 = \sqrt{\frac{F_1}{m}} = \sqrt{\frac{k \cdot \frac{1}{r}}{m}} = \sqrt{\frac{k}{mr}} \]

2. For Satellite 2:
\[ v_2 = \sqrt{\frac{F_2}{2m}} = \sqrt{\frac{k \cdot \frac{1}{2r}}{2m}} = \sqrt{\frac{k}{4mr}} \]

### Step 5: Find the Ratio of the Orbital Speeds
Now we can find the ratio of the orbital speeds \( v_1 \) and \( v_2 \):
\[ \frac{v_1}{v_2} = \frac{\sqrt{\frac{k}{mr}}}{\sqrt{\frac{k}{4mr}}} \]
This simplifies to:
\[ \frac{v_1}{v_2} = \frac{\sqrt{\frac{k}{mr}}}{\sqrt{\frac{k}{4mr}}} = \frac{\sqrt{4}}{1} = 2 \]

### Step 6: Conclusion
Thus, the ratio of the orbital speeds of the two satellites is:
\[ \frac{v_1}{v_2} = 2:1 \]