Two identical satellites A and B revolve round the earth in circular orbits at distance R and 3R from the surface of the earth. The ratio of the linear momenta of A and B is (R = radius of the earth)

Options

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

Answer

To solve the problem of finding the ratio of the linear momenta of two identical satellites A and B revolving around the Earth at different distances, we can follow these steps:

### Step 1: Understand the distances of the satellites from the center of the Earth
- The radius of the Earth is denoted as \( R \).
- Satellite A is at a distance \( R \) from the surface of the Earth, so its distance from the center of the Earth is:
\[ r_1 = R + R = 2R \]
- Satellite B is at a distance \( 3R \) from the surface of the Earth, so its distance from the center of the Earth is:
\[ r_2 = R + 3R = 4R \]

### Step 2: Determine the orbital speeds of the satellites
The orbital speed \( v \) of a satellite in a circular orbit is given by the formula:
\[ v = \sqrt{\frac{GM}{r}} \]
where \( G \) is the gravitational constant and \( M \) is the mass of the Earth.

- For satellite A (at distance \( r_1 = 2R \)):
\[ v_1 = \sqrt{\frac{GM}{2R}} \]

- For satellite B (at distance \( r_2 = 4R \)):
\[ v_2 = \sqrt{\frac{GM}{4R}} \]

### Step 3: Find the ratio of the speeds
To find the ratio of the speeds \( \frac{v_1}{v_2} \):
\[ \frac{v_1}{v_2} = \frac{\sqrt{\frac{GM}{2R}}}{\sqrt{\frac{GM}{4R}}} = \frac{\sqrt{4R}}{\sqrt{2R}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]

### Step 4: Calculate the linear momenta of the satellites
The linear momentum \( p \) of an object is given by:
\[ p = mv \]
where \( m \) is the mass of the object and \( v \) is its velocity.

Since both satellites are identical, they have the same mass \( m \). Therefore, the linear momenta of the satellites can be expressed as:
- For satellite A:
\[ p_1 = mv_1 \]
- For satellite B:
\[ p_2 = mv_2 \]

### Step 5: Find the ratio of the linear momenta
The ratio of the linear momenta \( \frac{p_1}{p_2} \) is:
\[ \frac{p_1}{p_2} = \frac{mv_1}{mv_2} = \frac{v_1}{v_2} = \sqrt{2} \]

### Final Result
Thus, the ratio of the linear momenta of satellites A and B is:
\[ \frac{p_1}{p_2} = \sqrt{2} : 1 \]