If `v_(e)` is escape velocity and `v_(0)`, is orbital velocity of satellite for orbit close to the earth's surface. Then are related by

Options

A. `v_(0)=sqrt(2)v_(e)`
B. `n_(0)=n_(e)`
C. `v_(e)=(v_(0))/2`
D. `v_(e)=sqrt(2)v_(0)`

Explanation

To find the relationship between escape velocity (\( v_e \)) and orbital velocity (\( v_0 \)) for a satellite in an orbit close to the Earth's surface, we can follow these steps:

### Step 1: Write the formula for escape velocity
The escape velocity (\( v_e \)) from the surface of a planet 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,
- \( R \) is the radius of the planet.

### Step 2: Write the formula for orbital velocity
The orbital velocity (\( v_0 \)) for a satellite in a circular orbit close to the surface of the planet is given by the formula:
\[ v_0 = \sqrt{\frac{GM}{R}} \]

### Step 3: Relate the two velocities
Now, we can relate the escape velocity to the orbital velocity. We can express the escape velocity in terms of the orbital velocity:
\[ v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2} \cdot \sqrt{\frac{GM}{R}} = \sqrt{2} \cdot v_0 \]

### Conclusion
Thus, the relationship between escape velocity and orbital velocity is:
\[ v_e = \sqrt{2} \cdot v_0 \]