The ratio of radii of two satellites is p and the ratio of their acceleration due to gravity is q. the ratio if their escape velocities will be :

Options

A. `(q/p)^(½)`
B. `(p/q)^(½)`
C. `pq`
D. `sqrt(pq)`

Answer

To find the ratio of escape velocities of two satellites given the ratio of their radii and the ratio of their acceleration due to gravity, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Escape Velocity**:
The escape velocity \( v_e \) from the surface of a celestial body can be expressed as:
\[ v_e = \sqrt{2gR} \]
where \( g \) is the acceleration due to gravity at the surface and \( R \) is the radius of the body.

2. **Setting Up Ratios**:
Let:
- \( R_1 \) and \( R_2 \) be the radii of the two satellites.
- \( g_1 \) and \( g_2 \) be the acceleration due to gravity for the two satellites.

According to the problem, we have:
\[ \frac{R_1}{R_2} = p \quad \text{and} \quad \frac{g_1}{g_2} = q \]

3. **Expressing Escape Velocities**:
The escape velocities for the two satellites can be written as:
\[ v_{e1} = \sqrt{2g_1R_1} \quad \text{and} \quad v_{e2} = \sqrt{2g_2R_2} \]

4. **Finding the Ratio of Escape Velocities**:
The ratio of the escape velocities is given by:
\[ \frac{v_{e1}}{v_{e2}} = \frac{\sqrt{2g_1R_1}}{\sqrt{2g_2R_2}} = \sqrt{\frac{g_1R_1}{g_2R_2}} \]
The factor of \( \sqrt{2} \) cancels out since it is common in both escape velocities.

5. **Substituting the Ratios**:
Now, substituting the given ratios:
\[ \frac{v_{e1}}{v_{e2}} = \sqrt{\frac{g_1}{g_2} \cdot \frac{R_1}{R_2}} = \sqrt{q \cdot p} \]

6. **Final Result**:
Therefore, the ratio of the escape velocities of the two satellites is:
\[ \frac{v_{e1}}{v_{e2}} = \sqrt{pq} \]

### Conclusion:
The ratio of the escape velocities of the two satellites is \( \sqrt{pq} \).

---