Planet A has massa M and radius R. Planet B has half the mass and half the radius of Planet A.If the escape velocities from the Planets A and B are `v_(A) and v_(B),` respectively , then `(v_(A))/(v_(B))=n/4.` The vlaue of n is :

Options

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

Answer

To solve the problem, we need to find the ratio of the escape velocities from two planets, A and B, and determine the value of \( n \) in the equation \( \frac{v_A}{v_B} = \frac{n}{4} \).

### Step-by-step Solution:

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

2. **Identify Parameters for Each Planet**:
- For Planet A:
- Mass \( M_A = M \)
- Radius \( R_A = R \)
- For Planet B:
- Mass \( M_B = \frac{M}{2} \)
- Radius \( R_B = \frac{R}{2} \)

3. **Calculate Escape Velocity for Each Planet**:
- For Planet A:
\[ v_A = \sqrt{\frac{2GM_A}{R_A}} = \sqrt{\frac{2GM}{R}} \]
- For Planet B:
\[ v_B = \sqrt{\frac{2GM_B}{R_B}} = \sqrt{\frac{2G \left(\frac{M}{2}\right)}{\frac{R}{2}}} \]
Simplifying \( v_B \):
\[ v_B = \sqrt{\frac{2G \cdot \frac{M}{2}}{\frac{R}{2}}} = \sqrt{\frac{2G \cdot M}{2} \cdot \frac{2}{R}} = \sqrt{\frac{2GM}{R}} = v_A \]

4. **Find the Ratio of Escape Velocities**:
Now we can find the ratio of the escape velocities:
\[ \frac{v_A}{v_B} = \frac{\sqrt{\frac{2GM}{R}}}{\sqrt{\frac{2GM}{R}}} = 1 \]

5. **Relate the Ratio to Given Equation**:
We know from the problem statement that:
\[ \frac{v_A}{v_B} = \frac{n}{4} \]
Since we found \( \frac{v_A}{v_B} = 1 \), we can set up the equation:
\[ 1 = \frac{n}{4} \]

6. **Solve for \( n \)**:
Multiplying both sides by 4 gives:
\[ n = 4 \]

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