Two planets of masses `M` and `M/2` have radii `R` and `R/2` respectively. If ratio of escape velocities from their surfaces `(v_1)/(v_2)` is `n/4` , then find n :

Options

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

Answer

To solve the problem, we need to find the ratio of escape velocities from the surfaces of two planets with given masses and radii. Let's denote the escape velocities from the surfaces of the two planets as \( v_1 \) and \( v_2 \).

### Step-by-Step Solution:

1. **Understand the formula for escape velocity**:
The escape velocity \( v \) from the surface of a planet is given by the formula:
\[ v = \sqrt{\frac{2GM}{R}} \]
where \( G \) is the universal gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet.

2. **Identify the masses and radii of the planets**:
- For Planet 1: Mass \( M_1 = M \) and Radius \( R_1 = R \)
- For Planet 2: Mass \( M_2 = \frac{M}{2} \) and Radius \( R_2 = \frac{R}{2} \)

3. **Calculate the escape velocities for both planets**:
- For Planet 1:
\[ v_1 = \sqrt{\frac{2GM_1}{R_1}} = \sqrt{\frac{2GM}{R}} \]
- For Planet 2:
\[ v_2 = \sqrt{\frac{2GM_2}{R_2}} = \sqrt{\frac{2G \cdot \frac{M}{2}}{\frac{R}{2}}} = \sqrt{\frac{2G \cdot \frac{M}{2} \cdot 2}{R}} = \sqrt{\frac{2GM}{R}} = v_1 \]

4. **Find the ratio of escape velocities**:
Now, we can find the ratio of the escape velocities:
\[ \frac{v_1}{v_2} = \frac{\sqrt{\frac{2GM}{R}}}{\sqrt{\frac{2GM}{R}}} = 1 \]

5. **Relate the ratio to the given condition**:
According to the problem, we have:
\[ \frac{v_1}{v_2} = \frac{n}{4} \]
Setting this equal to our calculated ratio:
\[ 1 = \frac{n}{4} \]

6. **Solve for \( n \)**:
To find \( n \), we multiply both sides by 4:
\[ n = 4 \]

### Conclusion:
The value of \( n \) is \( 4 \).