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)

( \bigcirc ) ( v = sqrt{\frac{gm_p}{(42,164 \text{km})}} )

( \bigcirc ) ( v = sqrt{\frac{gm_p}{(15,522 \text{km})}} )

( \bigcirc ) ( v = sqrt{\frac{gm_p}{(7,324 \text{km})}} )

( \bigcirc ) ( v = sqrt{\frac{gm_p}{(48,115 \text{km})}} )

Explanation:

Step1: Recall Geostationary Orbit Radius

The radius of a geostationary orbit (distance from the center of the Earth, for Earth - orbiting satellites) is approximately \(42164\space km\). For a satellite in geostationary orbit, the centripetal force required for circular motion is provided by the gravitational force. The formula for the orbital velocity \(v\) of a satellite is derived from equating the gravitational force \(\frac{Gm_pm}{r^{2}}\) (where \(m\) is the mass of the satellite and \(r\) is the orbital radius) to the centripetal force \(\frac{mv^{2}}{r}\). Canceling out the mass of the satellite \(m\) from both sides, we get \(v = \sqrt{\frac{Gm_p}{r}}\), where \(r\) is the radius of the geostationary orbit.

Step2: Match with the Given Options

We know that the radius \(r\) for geostationary orbit is approximately \(42164\space km\). So the formula for the velocity of a satellite in geostationary orbit should be \(v=\sqrt{\frac{Gm_p}{(42,164\space km)}}\)

Answer:

\(v = \sqrt{\frac{Gm_p}{(42,164\space km)}}\) (the first option)