To find the velocity of a satellite in a geostationary orbit, we can use the formula for the orbital velocity of a satellite, which is derived from the balance of gravitational force and centripetal force. The formula is given by:
\[ v = \sqrt{\frac{G m_p}{r}} \]
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.
For a geostationary orbit, the satellite orbits at a specific altitude such that it has a period of 24 hours (the same as the rotation period of the Earth). This means that \( r \) is the Earth's radius plus the altitude of the geostationary orbit.
The radius of a geostationary orbit is approximately 42,164 km from the center of the Earth (the Earth's radius is about 6,371 km, and the altitude of the geostationary orbit is approximately 35,786 km above the surface of the Earth).
Thus, the correct equation to use for finding the velocity of a satellite placed in a geostationary orbit, from your provided options, would be:
\[ v = \sqrt{\frac{G m_p}{42,164 \text{ km}}} \]
Therefore, the correct response is:
v=Gmp(42,164 km)−−−−−−−−√
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