The escape velocity from a planet is `v_(0)` The escape velocity from a planet having twice the radius but same density will be

Options

A. `0 . 5 v_(0)`
B. `v_(0)`
C. `2 v_(0)`
D. `4 v_(0)`

Explanation

To find the escape velocity from a planet that has twice the radius but the same density as another planet, we can follow these steps:

### Step 1: Understand the formula for escape velocity
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 the radius of the planet.

### Step 2: Calculate the mass of the first planet
For the first planet (let's call it Planet 1), we have:
- Radius = \( R \)
- Density = \( \rho \)

The mass \( M_1 \) of Planet 1 can be calculated using the formula:
\[ M_1 = \text{Density} \times \text{Volume} = \rho \times \left(\frac{4}{3} \pi R^3\right) \]

### Step 3: Substitute mass into the escape velocity formula for Planet 1
Now, substituting \( M_1 \) into the escape velocity formula:
\[ V_0 = \sqrt{\frac{2G \left(\rho \times \frac{4}{3} \pi R^3\right)}{R}} \]
This simplifies to:
\[ V_0 = \sqrt{\frac{8\pi G \rho R^2}{3}} \]

### Step 4: Calculate the mass of the second planet
For the second planet (Planet 2), which has:
- Radius = \( 2R \)
- Density = \( \rho \)

The mass \( M_2 \) of Planet 2 is:
\[ M_2 = \rho \times \left(\frac{4}{3} \pi (2R)^3\right) = \rho \times \left(\frac{4}{3} \pi \times 8R^3\right) = \frac{32}{3} \pi \rho R^3 \]

### Step 5: Substitute mass into the escape velocity formula for Planet 2
Now, substituting \( M_2 \) into the escape velocity formula for Planet 2:
\[ V = \sqrt{\frac{2G M_2}{2R}} = \sqrt{\frac{2G \left(\frac{32}{3} \pi \rho R^3\right)}{2R}} \]
This simplifies to:
\[ V = \sqrt{\frac{32\pi G \rho R^2}{3}} \]

### Step 6: Relate \( V \) to \( V_0 \)
Now, we can relate \( V \) to \( V_0 \):
\[ V = \sqrt{16} \cdot \sqrt{\frac{2\pi G \rho R^2}{3}} = 4 \cdot \sqrt{\frac{2\pi G \rho R^2}{3}} = 4 \cdot V_0 \]

### Conclusion
Thus, the escape velocity from the planet with twice the radius but the same density is:
\[ V = 2V_0 \]