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 escape velocities for both planets A and B and then determine the ratio of these velocities.

### Step 1: Write 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.

### Step 2: Calculate escape velocity for Planet A
For Planet A, with mass \( M \) and radius \( R \):
\[ v_A = \sqrt{\frac{2GM}{R}} \]

### Step 3: Calculate escape velocity for Planet B
Planet B has half the mass and half the radius of Planet A. Therefore, the mass \( M_B \) and radius \( R_B \) of Planet B are:
\[ M_B = \frac{M}{2}, \quad R_B = \frac{R}{2} \]
Now, substituting these values into the escape velocity formula for Planet B:
\[ v_B = \sqrt{\frac{2G \left(\frac{M}{2}\right)}{\frac{R}{2}}} \]
This simplifies to:
\[ v_B = \sqrt{\frac{2G \cdot \frac{M}{2}}{\frac{R}{2}}} = \sqrt{\frac{GM}{R}} = \sqrt{\frac{2GM}{R}} \cdot \sqrt{\frac{1}{2}} = \frac{v_A}{\sqrt{2}} \]

### Step 4: Find the ratio of escape velocities
Now we can find the ratio of the escape velocities:
\[ \frac{v_A}{v_B} = \frac{v_A}{\frac{v_A}{\sqrt{2}}} = \sqrt{2} \]

### Step 5: Relate the ratio to the given condition
According to the problem, we have:
\[ \frac{v_A}{v_B} = \frac{n}{4} \]
Setting the two expressions equal gives:
\[ \sqrt{2} = \frac{n}{4} \]

### Step 6: Solve for \( n \)
To find \( n \), we multiply both sides by 4:
\[ n = 4\sqrt{2} \]

### Final Answer
Thus, the value of \( n \) is:
\[ n = 4\sqrt{2} \]