Escape velocity on the surface of earth is 11.2 km/s . Escape velocity from a planet whose mass is the same as that of earth and radius 1/4 that of earth is

Options

A. 2.8 km/s
B. 15.6 km/s
C. 22.4 km/s
D. 44.8 km/s

Answer

To find the escape velocity from a planet whose mass is the same as that of Earth but with a radius that is 1/4 of Earth's radius, we can use the formula for escape velocity:

### Step-by-Step Solution:

1. **Understand the Escape Velocity Formula**:
The escape velocity (V) from the surface of 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.

2. **Identify Given Values**:
- Escape velocity on Earth (\( V_E \)) = 11.2 km/s
- Mass of the new planet (\( M \)) = Mass of Earth (\( M_E \))
- Radius of the new planet (\( R \)) = \( \frac{1}{4} R_E \) (where \( R_E \) is the radius of Earth)

3. **Write the Escape Velocity for the New Planet**:
For the new planet, we can express the escape velocity (\( V_P \)) as:
\[ V_P = \sqrt{\frac{2GM}{R}} \]
Substituting \( R = \frac{1}{4} R_E \):
\[ V_P = \sqrt{\frac{2GM}{\frac{1}{4} R_E}} = \sqrt{\frac{2GM \cdot 4}{R_E}} = \sqrt{\frac{8GM}{R_E}} \]

4. **Relate the Escape Velocity of the New Planet to Earth's Escape Velocity**:
We know that:
\[ V_E = \sqrt{\frac{2GM_E}{R_E}} \]
Therefore, we can express \( V_P \) as:
\[ V_P = \sqrt{4} \cdot \sqrt{\frac{2GM_E}{R_E}} = 2 \cdot V_E \]

5. **Substitute the Known Value of \( V_E \)**:
Now substituting \( V_E = 11.2 \, \text{km/s} \):
\[ V_P = 2 \cdot 11.2 \, \text{km/s} = 22.4 \, \text{km/s} \]

6. **Conclusion**:
The escape velocity from the new planet is \( 22.4 \, \text{km/s} \).

### Final Answer:
The escape velocity from the planet is **22.4 km/s**.

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