To find the velocity of a satellite in a circular orbit around a planet, we can use the formula derived from setting the gravitational force equal to the centripetal force required to keep the satellite in orbit.
The formula for the orbital velocity \( v \) is given by:
\[ v = \sqrt{\frac{GM_p}{r}} \]
where:
- \( G \) is the universal gravitational constant,
- \( M_p \) is the mass of the planet,
- \( r \) is the distance from the center of the planet to the satellite.
In the case of a low Earth orbit, if we assume the radius \( r \) is approximately equal to the radius of the Earth plus the altitude of the orbit (which may be about 200 km), we usually represent this in kilometers.
Given the options you provided, the correct formula aligns with:
\[ v = \sqrt{\frac{G M_p}{(R_{Earth} + h)}} \]
For a low Earth orbit (considering \( R_{Earth} \) to be approximately 6,371 km and the orbit being 200 km above the surface, we have \( r \approx 6,571 \) km), the equation that fits the forms you provided is:
\[ v = \sqrt{\frac{G M_p}{200 \text{ km}}} \]
However, since the radius needs to be expressed in the formula, the correct choice among your options would be:
v = \( \sqrt{ \frac{G M_p}{200 \text{ km}}} \)
So the answer among the responses you provided closely resembles:
v = \( G M_p (200 \text{ km})^{-1/2} \)
However, none of the provided options truly match the correct structure with \( G \) and \( M_p \). Hence, verify if there's a misrepresentation, as the units and structure for gravitational formulas need to align with the square root form accurately.