Two planets A and B have the same material density. If the radius of A is twice that of B, then the ratio of the escape velocity `V_(A)//V_(B)` is

Options

A. 2
B. `sqrt(2)`
C. `1//sqrt(2)`
D. `1//2`

Answer

To find the ratio of the escape velocities of two planets A and B, given that they have the same material density and that the radius of planet A is twice that of planet B, we can follow these steps:

### Step 1: Define the Variables
Let:
- \( R_A \) = radius of planet A
- \( R_B \) = radius of planet B
- Given that \( R_A = 2R_B \)
- Let the density of both planets be \( \rho \)

### Step 2: Calculate the Mass of Each Planet
The mass \( M \) of a planet can be calculated using the formula:
\[ M = \text{Volume} \times \text{Density} \]
The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi R^3 \]

For planet A:
\[ M_A = \frac{4}{3} \pi R_A^3 \rho \]
For planet B:
\[ M_B = \frac{4}{3} \pi R_B^3 \rho \]

### Step 3: Substitute the Radius of Planet A
Since \( R_A = 2R_B \), we substitute this into the equation for \( M_A \):
\[ M_A = \frac{4}{3} \pi (2R_B)^3 \rho = \frac{4}{3} \pi (8R_B^3) \rho = \frac{32}{3} \pi R_B^3 \rho \]

### Step 4: Write the Escape Velocity Formula
The escape velocity \( V \) from a planet is given by:
\[ V = \sqrt{\frac{2GM}{R}} \]
where \( G \) is the gravitational constant.

### Step 5: Calculate the Escape Velocity for Both Planets
For planet A:
\[ V_A = \sqrt{\frac{2GM_A}{R_A}} = \sqrt{\frac{2G \left(\frac{32}{3} \pi R_B^3 \rho\right)}{2R_B}} = \sqrt{\frac{32G \pi R_B^3 \rho}{3R_B}} = \sqrt{\frac{32G \pi R_B^2 \rho}{3}} \]

For planet B:
\[ V_B = \sqrt{\frac{2GM_B}{R_B}} = \sqrt{\frac{2G \left(\frac{4}{3} \pi R_B^3 \rho\right)}{R_B}} = \sqrt{\frac{8G \pi R_B^3 \rho}{3R_B}} = \sqrt{\frac{8G \pi R_B^2 \rho}{3}} \]

### Step 6: Find the Ratio of Escape Velocities
Now, we can find the ratio of escape velocities \( \frac{V_A}{V_B} \):
\[ \frac{V_A}{V_B} = \frac{\sqrt{\frac{32G \pi R_B^2 \rho}{3}}}{\sqrt{\frac{8G \pi R_B^2 \rho}{3}}} \]
The \( G \), \( \pi \), \( R_B^2 \), and \( \rho \) terms cancel out:
\[ \frac{V_A}{V_B} = \sqrt{\frac{32}{8}} = \sqrt{4} = 2 \]

### Final Answer
Thus, the ratio of the escape velocities is:
\[ \frac{V_A}{V_B} = 2 \]