Question

let g be the universal gravitational constant and ( m_p ) be the mass of the planet a satellite is orbiting. which equation could be used to find the velocity of the satellite if it is placed in a geostationary orbit? (1 point)
( \boldsymbol{v = sqrt{\frac{gm_p}{(42,104 \text{km})}}} )
( \boldsymbol{v = sqrt{\frac{gm_p}{(48,115 \text{km})}}} )
( \boldsymbol{v = sqrt{\frac{gm_p}{(15,522 \text{km})}}} )
( \boldsymbol{v = sqrt{\frac{gm_p}{(7,324 \text{km})}}} )

Explanation:

Step1: Recall geostationary orbit radius

The radius of a geostationary orbit (from the center of the Earth, approximately) is about 42,104 km. The formula for the orbital velocity \( v \) of a satellite is derived from equating gravitational force to centripetal force: \( \frac{Gm_p m}{r^2}=\frac{mv^2}{r} \), simplifying to \( v = \sqrt{\frac{Gm_p}{r}} \), where \( r \) is the orbital radius.

Step2: Match the radius in options

We check the options for the radius \( r \) equal to the geostationary orbit radius (42,104 km). The first option has \( r = 42,104 \) km, so its formula \( v=\sqrt{\frac{Gm_p}{(42,104\ \text{km})}} \) is the correct one.

Answer:

\( v = \sqrt{\frac{Gm_p}{(42,104\ \text{km})}} \) (the first option)