To find the velocity of a satellite in a geostationary orbit, we can use the balance between gravitational force and the centripetal force required to keep the satellite in circular motion.
The gravitational force \( F_g \) experienced by the satellite due to the planet is given by Newton's law of gravitation:
\[ F_g = \frac{G \cdot m_p \cdot m_s}{r^2} \]
where:
- \( G \) is the universal gravitational constant,
- \( m_p \) is the mass of the planet,
- \( m_s \) is the mass of the satellite, and
- \( r \) is the distance from the center of the planet to the satellite.
The centripetal force \( F_c \) required to keep the satellite in circular motion is given by:
\[ F_c = \frac{m_s v^2}{r} \]
where \( v \) is the orbital velocity of the satellite.
For a geostationary orbit, these two forces must be equal:
\[ \frac{G \cdot m_p \cdot m_s}{r^2} = \frac{m_s v^2}{r} \]
We can cancel \( m_s \) from both sides (assuming it's not zero), leading to:
\[ \frac{G \cdot m_p}{r^2} = \frac{v^2}{r} \]
Multiplying both sides by \( r \) gives:
\[ \frac{G \cdot m_p}{r} = v^2 \]
Taking the square root of both sides results in the formula for the velocity \( v \):
\[ v = \sqrt{\frac{G \cdot m_p}{r}} \]
In the case of a geostationary orbit, \( r \) is the distance from the center of the planet to the satellite, which is equal to the radius of the planet plus the altitude of the satellite above the planet. However, for the equation to determine velocity in a geostationary orbit, the key relationship is:
\[ v = \sqrt{\frac{G \cdot m_p}{r}} \]
This equation can be used to find the velocity of the satellite placed in a geostationary orbit.
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