Two satellites orbiting around the earth have their critical speeds in the ratio 4 : 5. What is the ratio of their orbital radii ?

Options

A. `(5)/(4)`
B. `(4)/(5)`
C. `(25)/(16)`
D. `(12)/(5)`

Explanation

To solve the problem, we need to find the ratio of the orbital radii of two satellites given that their critical speeds are in the ratio of 4:5.

### Step-by-Step Solution:

1. **Understanding Critical Speed**:
The critical speed (orbital speed) \( V_c \) of a satellite in orbit around Earth is given by the formula:
\[ V_c = \sqrt{\frac{GM}{R}} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the distance from the center of the Earth to the satellite.

2. **Setting Up the Ratios**:
Let \( V_{c1} \) be the critical speed of satellite 1 and \( V_{c2} \) be the critical speed of satellite 2. According to the problem:
\[ \frac{V_{c1}}{V_{c2}} = \frac{4}{5} \]

3. **Expressing Critical Speeds in Terms of Radii**:
Using the formula for critical speed, we can express the critical speeds of the two satellites as:
\[ V_{c1} = \sqrt{\frac{GM}{R_1}} \quad \text{and} \quad V_{c2} = \sqrt{\frac{GM}{R_2}} \]

4. **Substituting into the Ratio**:
Substitute \( V_{c1} \) and \( V_{c2} \) into the ratio:
\[ \frac{\sqrt{\frac{GM}{R_1}}}{\sqrt{\frac{GM}{R_2}}} = \frac{4}{5} \]
The \( GM \) terms cancel out:
\[ \frac{\sqrt{R_2}}{\sqrt{R_1}} = \frac{4}{5} \]

5. **Squaring Both Sides**:
To eliminate the square roots, square both sides of the equation:
\[ \frac{R_2}{R_1} = \left(\frac{4}{5}\right)^2 = \frac{16}{25} \]

6. **Finding the Ratio of Radii**:
Rearranging gives us:
\[ \frac{R_1}{R_2} = \frac{25}{16} \]

### Final Answer:
The ratio of the orbital radii \( R_1 : R_2 \) is \( 25 : 16 \).