The escape velocity for the earth is 11.2 km / sec . The mass of another planet is 100 times that of the earth and its radius is 4 times that of the earth. The escape velocity for this planet will be

Options

A. 56.0 km/s
B. 280 km/s
C. 112 km/s
D. 11.2 km/s

Answer

To find the escape velocity for the new planet, we can use the formula for escape velocity:

\[ V_e = \sqrt{\frac{2GM}{R}} \]

Where:
- \( V_e \) is the escape velocity,
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the planet,
- \( R \) is the radius of the planet.

### Step 1: Identify the parameters for Earth
Given:
- Escape velocity for Earth, \( V_e = 11.2 \, \text{km/s} \)
- Mass of Earth, \( M \)
- Radius of Earth, \( R \)

### Step 2: Identify the parameters for the new planet
The mass of the new planet is given as:
\[ M' = 100M \]
The radius of the new planet is given as:
\[ R' = 4R \]

### Step 3: Substitute the new planet's parameters into the escape velocity formula
Using the escape velocity formula for the new planet:
\[ V_e' = \sqrt{\frac{2G \cdot M'}{R'}} \]
Substituting \( M' \) and \( R' \):
\[ V_e' = \sqrt{\frac{2G \cdot (100M)}{4R}} \]

### Step 4: Simplify the expression
We can simplify the expression:
\[ V_e' = \sqrt{\frac{100 \cdot 2GM}{4R}} = \sqrt{\frac{100}{4}} \cdot \sqrt{\frac{2GM}{R}} = \sqrt{25} \cdot \sqrt{\frac{2GM}{R}} = 5 \cdot V_e \]

### Step 5: Calculate the escape velocity for the new planet
Now, substituting \( V_e = 11.2 \, \text{km/s} \):
\[ V_e' = 5 \cdot 11.2 \, \text{km/s} = 56 \, \text{km/s} \]

### Final Answer
The escape velocity for the new planet is \( 56 \, \text{km/s} \).

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