The ratio of earth's surface area as covered by two communication satellite moving synchronously in equatorial plane with orbital speed of `1/sqrt2` times and `sqrt3/5` times that of required orbital speed near earths surface is

Options

A. `4//5`
B. `5//2`
C. `5//4`
D. `2//5`

Answer

To solve the problem of finding the ratio of Earth's surface area covered by two communication satellites moving synchronously in the equatorial plane with different orbital speeds, we can follow these steps:

### Step 1: Understanding the Orbital Speed
The required orbital speed \( v \) for a satellite in a circular orbit near the Earth's surface is given by the formula:

\[ v = \sqrt{\frac{GM}{R}} \]

where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth.

### Step 2: Determine the Speeds of the Satellites
Given the speeds of the two satellites:
- Satellite 1: \( v_1 = \frac{1}{\sqrt{2}} v \)
- Satellite 2: \( v_2 = \frac{\sqrt{3}}{5} v \)

### Step 3: Calculate the Orbital Radius
The orbital radius \( r \) for a satellite moving at speed \( v \) is given by:

\[ v = \sqrt{\frac{GM}{r}} \]

Rearranging gives:

\[ r = \frac{GM}{v^2} \]

For each satellite, we can find their respective orbital radii:
- For Satellite 1:

\[ r_1 = \frac{GM}{\left(\frac{1}{\sqrt{2}} v\right)^2} = \frac{GM}{\frac{1}{2} v^2} = 2 \frac{GM}{v^2} = 2R \]

- For Satellite 2:

\[ r_2 = \frac{GM}{\left(\frac{\sqrt{3}}{5} v\right)^2} = \frac{GM}{\frac{3}{25} v^2} = \frac{25}{3} \frac{GM}{v^2} = \frac{25}{3} R \]

### Step 4: Calculate the Surface Area Covered by Each Satellite
The surface area \( A \) covered by a satellite in a circular orbit can be calculated using the formula for the surface area of a sphere:

\[ A = 4\pi r^2 \]

- For Satellite 1:

\[ A_1 = 4\pi (r_1)^2 = 4\pi (2R)^2 = 16\pi R^2 \]

- For Satellite 2:

\[ A_2 = 4\pi (r_2)^2 = 4\pi \left(\frac{25}{3} R\right)^2 = \frac{1000\pi}{9} R^2 \]

### Step 5: Calculate the Ratio of Surface Areas
Now, we can find the ratio of the surface areas covered by the two satellites:

\[ \text{Ratio} = \frac{A_1}{A_2} = \frac{16\pi R^2}{\frac{1000\pi}{9} R^2} = \frac{16 \cdot 9}{1000} = \frac{144}{1000} = \frac{36}{250} = \frac{18}{125} \]

### Final Answer
The ratio of Earth's surface area covered by the two communication satellites is:

\[ \frac{18}{125} \]

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