The escape velocity at the surface of Earth is approximately 8 km/s. What is the escape velocity for a planet whose radius is 4 times and whose mass is 100 times that of Earth?

Options

A. 1.6 km/s
B. 8 km/s
C. 40 km/s
D. 200 km/s

Explanation

To find the escape velocity for a planet whose radius is 4 times that of Earth and whose mass is 100 times that of Earth, we can use the formula for escape velocity:

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

where:
- \( v_e \) = escape velocity
- \( G \) = gravitational constant
- \( M \) = mass of the planet
- \( R \) = radius of the planet

### Step 1: Write the escape velocity for Earth
The escape velocity at the surface of Earth is given as approximately 8 km/s. We can express this as:

\[ v_{e, \text{Earth}} = \sqrt{\frac{2GM_{\text{Earth}}}{R_{\text{Earth}}}} = 8 \text{ km/s} \]

### Step 2: Write the escape velocity for the new planet
For the new planet:
- Mass \( M_{\text{planet}} = 100 M_{\text{Earth}} \)
- Radius \( R_{\text{planet}} = 4 R_{\text{Earth}} \)

Substituting these values into the escape velocity formula, we have:

\[ v_{e, \text{planet}} = \sqrt{\frac{2G(100 M_{\text{Earth}})}{4 R_{\text{Earth}}}} \]

### Step 3: Simplify the expression
We can simplify the expression:

\[ v_{e, \text{planet}} = \sqrt{\frac{100 \cdot 2GM_{\text{Earth}}}{4 R_{\text{Earth}}}} \]

This can be further simplified to:

\[ v_{e, \text{planet}} = \sqrt{\frac{100}{4}} \cdot \sqrt{\frac{2GM_{\text{Earth}}}{R_{\text{Earth}}}} \]

\[ v_{e, \text{planet}} = \sqrt{25} \cdot v_{e, \text{Earth}} \]

### Step 4: Calculate the escape velocity
Since \( \sqrt{25} = 5 \), we can substitute the known escape velocity of Earth:

\[ v_{e, \text{planet}} = 5 \cdot 8 \text{ km/s} = 40 \text{ km/s} \]

### Conclusion
The escape velocity for the planet is **40 km/s**.

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