If the escape velocity on the earth is `11.2km-s^-1`, its value for a planet having double the radius and 8 times the mass of earth is

Options

A. `10.2 kms^(-1)`
B. `22.4 kms^(-1)`
C. `10 kms^(-1)`
D. 0

Answer

To find the escape velocity for a planet with double the radius and eight times the mass of Earth, we can use the formula for escape velocity:

### Step 1: Write down the formula for escape velocity
The escape velocity \( v_e \) is given by the formula:
\[ v_e = \sqrt{\frac{2GM}{R}} \]
where \( G \) is the universal gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet.

### Step 2: Identify the escape velocity for Earth
The escape velocity for Earth is given as \( 11.2 \, \text{km/s} \).

### Step 3: Set up the parameters for the new planet
For the new planet:
- Mass \( M' = 8M \) (8 times the mass of Earth)
- Radius \( R' = 2R \) (double the radius of Earth)

### Step 4: Substitute the new parameters into the escape velocity formula
Now, substituting these values into the escape velocity formula for the new planet:
\[ v_{e}' = \sqrt{\frac{2G(8M)}{2R}} \]

### Step 5: Simplify the expression
This simplifies to:
\[ v_{e}' = \sqrt{\frac{8 \cdot 2GM}{2R}} = \sqrt{4 \cdot \frac{2GM}{R}} = \sqrt{4} \cdot \sqrt{\frac{2GM}{R}} = 2 \cdot v_e \]

### Step 6: Calculate the new escape velocity
Since \( v_e \) (escape velocity for Earth) is \( 11.2 \, \text{km/s} \):
\[ v_{e}' = 2 \cdot 11.2 \, \text{km/s} = 22.4 \, \text{km/s} \]

### Final Answer
The escape velocity for the new planet is \( 22.4 \, \text{km/s} \).

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