Two artificial satellites A and B are at a distance `r_(A)` and `r_(B)` above the earth's surface. If the radius of earth is R, then the ratio of their speed will be :-

Options

A. `((r_(B)+R)/(r_(A)+R))^(½)`
B. `((r_(B)+R)/(r_(A)+R))^(2)`
C. `(r_(B)/r_(A))^(2)`
D. `(r_(B)/r_(A))^(½)`

Answer

To solve the problem of finding the ratio of the speeds of two artificial satellites A and B at distances \( r_A \) and \( r_B \) above the Earth's surface, we can follow these steps:

### Step 1: Understand the Formula for Orbital Speed
The orbital speed \( v \) of a satellite is given by the formula:
\[ v = \sqrt{\frac{GM}{r}} \]
where:
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the Earth,
- \( r \) is the distance from the center of the Earth to the satellite.

### Step 2: Determine the Distance from the Center of the Earth
Since the distances \( r_A \) and \( r_B \) are measured from the Earth's surface, we need to express the total distance from the center of the Earth for both satellites:
- For satellite A:
\[ r_A' = r_A + R \]
- For satellite B:
\[ r_B' = r_B + R \]
where \( R \) is the radius of the Earth.

### Step 3: Write the Expressions for the Speeds of Satellites A and B
Using the formula for orbital speed:
- Speed of satellite A:
\[ v_A = \sqrt{\frac{GM}{r_A'}} \]
- Speed of satellite B:
\[ v_B = \sqrt{\frac{GM}{r_B'}} \]

### Step 4: Find the Ratio of the Speeds
To find the ratio of the speeds \( \frac{v_A}{v_B} \):
\[ \frac{v_A}{v_B} = \frac{\sqrt{\frac{GM}{r_A'}}}{\sqrt{\frac{GM}{r_B'}}} = \sqrt{\frac{r_B'}{r_A'}} \]
Substituting the expressions for \( r_A' \) and \( r_B' \):
\[ \frac{v_A}{v_B} = \sqrt{\frac{r_B + R}{r_A + R}} \]

### Final Answer
The ratio of the speeds of the satellites A and B is:
\[ \frac{v_A}{v_B} = \sqrt{\frac{r_B + R}{r_A + R}} \]

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