Two satellites A and B are rotating in same orbit. The ratio of their escape velocities, if radius and mass of A is twice to B, is

Options

A. `1:1`
B. `1:2`
C. `1:3`
D. `1:4`

Answer

To solve the problem, we need to find the ratio of the escape velocities of two satellites A and B, given that the radius and mass of satellite A are both twice those of satellite B.

### Step-by-Step Solution:

1. **Understanding Escape Velocity**:
The escape velocity (V_escape) from a gravitational field is given by the formula:
\[ V_{escape} = \sqrt{\frac{2GM}{r}} \]
where \( G \) is the universal gravitational constant, \( M \) is the mass of the celestial body (in this case, Earth), and \( r \) is the distance from the center of the mass to the object.

2. **Given Information**:
- Let the radius of satellite B be \( r_B \) and the mass of satellite B be \( M_B \).
- Therefore, the radius of satellite A will be \( r_A = 2r_B \) and the mass of satellite A will be \( M_A = 2M_B \).

3. **Calculating Escape Velocity for Satellite A**:
Using the escape velocity formula for satellite A:
\[ V_{escape,A} = \sqrt{\frac{2G \cdot M_A}{r_A}} = \sqrt{\frac{2G \cdot (2M_B)}{2r_B}} = \sqrt{\frac{2GM_B}{r_B}} = V_{escape,B} \]

4. **Calculating Escape Velocity for Satellite B**:
Using the escape velocity formula for satellite B:
\[ V_{escape,B} = \sqrt{\frac{2GM_B}{r_B}} \]

5. **Finding the Ratio of Escape Velocities**:
Now, we can find the ratio of the escape velocities of satellites A and B:
\[ \frac{V_{escape,A}}{V_{escape,B}} = \frac{\sqrt{\frac{2GM_A}{r_A}}}{\sqrt{\frac{2GM_B}{r_B}}} = \frac{\sqrt{\frac{2G \cdot (2M_B)}{2r_B}}}{\sqrt{\frac{2GM_B}{r_B}}} \]
Simplifying this gives:
\[ \frac{V_{escape,A}}{V_{escape,B}} = \frac{\sqrt{2GM_B/r_B}}{\sqrt{2GM_B/r_B}} = 1 \]

### Final Result:
The ratio of the escape velocities of satellites A and B is:
\[ \frac{V_{escape,A}}{V_{escape,B}} = 1 \]