To find the velocity of a satellite in geostationary orbit, we can start with the formula for the orbital velocity of a satellite:
\[ v = \sqrt{\frac{G m_p}{r}} \]
where:
- \( v \) is the orbital velocity,
- \( G \) 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 geostationary orbit, the satellite is located at an altitude above the equator where it matches the rotation period of the Earth. The radius \( r \) for a geostationary satellite is approximately 42,164 km from the center of the Earth (which includes the Earth's radius of about 6,371 km plus the altitude of the satellite).
Therefore, the correct equation for the velocity of a satellite in a geostationary orbit is:
\[ v = \sqrt{\frac{G m_p}{42,164 , \text{km}}} \]
From the options provided, the correct response would be:
v = Gmp(42,164 km)−−−−−−−−√
(Note: There seems to be a typographical error in the units of the provided options. The units should be \(\text{m}\) instead of \(\text{km}\) in the context of gravitational calculations.)