A space station is at a height equal to the radius of the Earth. If `V_E` is the escape velocity on the surface of the Earth, the same on the space station is ___ times `V_E`,

Options

A. `1/2`
B. `1/4`
C. `1/sqrt2`
D. `1/sqrt3`

Answer

To solve the problem, we need to determine the escape velocity at a space station located at a height equal to the radius of the Earth, and express it in terms of the escape velocity at the surface of the Earth.

### Step-by-Step Solution:

1. **Understanding Escape Velocity**:
The escape velocity \( V \) from a distance \( r \) from the center of the Earth is given by the formula:
\[ V = \sqrt{\frac{2GM}{r}} \]
where \( G \) is the gravitational constant and \( M \) is the mass of the Earth.

2. **Escape Velocity at the Surface of the Earth**:
At the surface of the Earth, the distance \( r \) is equal to the radius of the Earth \( R \). Thus, the escape velocity at the surface \( V_E \) is:
\[ V_E = \sqrt{\frac{2GM}{R}} \]

3. **Escape Velocity at the Space Station**:
The space station is at a height equal to the radius of the Earth, which means the distance from the center of the Earth to the space station is:
\[ r = R + R = 2R \]
Therefore, the escape velocity \( V_{E'} \) at the space station is:
\[ V_{E'} = \sqrt{\frac{2GM}{2R}} = \sqrt{\frac{2GM}{2} \cdot \frac{1}{R}} = \sqrt{\frac{GM}{R}} = \frac{1}{\sqrt{2}} \sqrt{\frac{2GM}{R}} = \frac{V_E}{\sqrt{2}} \]

4. **Expressing Escape Velocity at the Space Station in Terms of \( V_E \)**:
From the above calculations, we find that:
\[ V_{E'} = \frac{V_E}{\sqrt{2}} \]

5. **Final Answer**:
Thus, the escape velocity at the space station is \( \frac{1}{\sqrt{2}} \) times the escape velocity at the surface of the Earth.

### Conclusion:
The escape velocity at the space station is \( \frac{1}{\sqrt{2}} V_E \).