Two satellites are moving at heights of R and 5R above the surface of the earth of radius R. The ratio of their velocitirs `((V_(1))/(V_(2)))` is

Options

A. `sqrt(5):1`
B. `1 : 1`
C. `sqrt(3) : sqrt(2)`
D. `sqrt(3) : 1`

Explanation

To solve the problem of finding the ratio of the velocities of two satellites moving at heights of R and 5R above the surface of the Earth (with Earth's radius being R), we can follow these steps:

### Step 1: Identify the distances from the center of the Earth
- The radius of the Earth (R) is given.
- For the first satellite (height = R above the surface), the distance from the center of the Earth is:
\[ R_1 = R + R = 2R \]
- For the second satellite (height = 5R above the surface), the distance from the center of the Earth is:
\[ R_2 = R + 5R = 6R \]

### Step 2: Write the formula for orbital velocity
The orbital velocity \( V \) of a satellite is given by the formula:
\[ V = \sqrt{\frac{GM}{R}} \]
where \( G \) is the gravitational constant and \( M \) is the mass of the Earth.

### Step 3: Calculate the velocities of the satellites
- For the first satellite (at distance \( R_1 = 2R \)):
\[ V_1 = \sqrt{\frac{GM}{R_1}} = \sqrt{\frac{GM}{2R}} \]

- For the second satellite (at distance \( R_2 = 6R \)):
\[ V_2 = \sqrt{\frac{GM}{R_2}} = \sqrt{\frac{GM}{6R}} \]

### Step 4: Find the ratio of the velocities
To find the ratio \( \frac{V_1}{V_2} \):
\[ \frac{V_1}{V_2} = \frac{\sqrt{\frac{GM}{2R}}}{\sqrt{\frac{GM}{6R}}} \]
This simplifies to:
\[ \frac{V_1}{V_2} = \sqrt{\frac{6R}{2R}} = \sqrt{\frac{6}{2}} = \sqrt{3} \]

### Final Answer
Thus, the ratio of the velocities of the two satellites is:
\[ \frac{V_1}{V_2} = \sqrt{3} \]