To find the velocity of a satellite in a geostationary orbit, we use the formula derived from the gravitational force and centripetal force, which leads to the relationship:
\[ v = \sqrt{\frac{G \cdot m_p}{r}} \]
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 a geostationary orbit, the satellite is positioned at a specific altitude where it orbits the Earth once per day (24 hours), matching the rotation period of the Earth.
The radius \( r \) in the context of a geostationary orbit specifically is approximately 42,164 km from the center of the Earth. This includes the Earth's radius (approximately 6,371 km) plus the altitude of the geostationary orbit (approximately 35,786 km).
Given the choices provided in your question, we can identify that the correct equation to find the velocity of the satellite is:
\[ v = \sqrt{\frac{G \cdot m_p}{42,164 \text{ km}}} \]
Thus, the correct response is:
v=Gmp(42,164 km)−−−−−−−−√