If `v_(0)` be the orbital velocity of an articial satellite orbital velocity of the same satellite orbiting at an altitude equal to earth's radius is

Options

A. `v_(0)sqrt((2)/(3))`
B. `v_(0)sqrt((3)/(2))`
C. `v_(0)sqrt(2)`
D. `(v_(0))/(sqrt(2))`

Answer

To solve the problem, we need to find the orbital velocity of a satellite at an altitude equal to the Earth's radius, given the orbital velocity \( v_0 \) of the satellite at the Earth's surface.

### Step-by-Step Solution:

1. **Understanding Orbital Velocity**:
The orbital velocity \( v \) of a satellite in a circular orbit is given by the formula:
\[ v = \sqrt{\frac{GM}{r}} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( r \) is the distance from the center of the Earth to the satellite.

2. **Orbital Velocity at Earth's Surface**:
For a satellite orbiting at the Earth's surface (where \( r = R \), the radius of the Earth), the orbital velocity \( v_0 \) can be expressed as:
\[ v_0 = \sqrt{\frac{GM}{R}} \]

3. **Orbital Velocity at an Altitude Equal to Earth's Radius**:
When the satellite is at an altitude equal to the Earth's radius, the total distance from the center of the Earth to the satellite becomes:
\[ r = R + R = 2R \]
Thus, the orbital velocity \( v \) at this altitude is:
\[ v = \sqrt{\frac{GM}{2R}} \]

4. **Relating the Two Velocities**:
Now, we can relate \( v \) to \( v_0 \):
\[ \frac{v_0}{v} = \frac{\sqrt{\frac{GM}{R}}}{\sqrt{\frac{GM}{2R}}} \]
Simplifying this expression:
\[ \frac{v_0}{v} = \sqrt{\frac{GM}{R} \cdot \frac{2R}{GM}} = \sqrt{2} \]

5. **Finding \( v \)**:
Rearranging the equation gives us:
\[ v = \frac{v_0}{\sqrt{2}} \]

6. **Conclusion**:
Therefore, the orbital velocity of the satellite at an altitude equal to the Earth's radius is:
\[ v = \frac{v_0}{\sqrt{2}} \]

### Final Answer:
The orbital velocity of the satellite at an altitude equal to the Earth's radius is \( \frac{v_0}{\sqrt{2}} \).