Question

orbits of satellites quick check
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 low earth orbit? (1 point)
( \boldsymbol{v = sqrt{\frac{gm_p}{(200 \text{km})}}} )
( \boldsymbol{v = \frac{1}{(7,000 \text{km})} sqrt{gm_p}} )
( \boldsymbol{v = \frac{1}{(200 \text{km})} sqrt{gm_p}} )
( \boldsymbol{v = sqrt{\frac{gm_p}{(7,000 \text{km})}}} )

Explanation:

Step1: Recall orbital velocity formula

The formula for the orbital velocity \( v \) of a satellite in a circular orbit is derived from equating gravitational force to centripetal force: \( \frac{Gm_pm}{r^2}=\frac{mv^2}{r} \), where \( m \) is the satellite's mass, \( r \) is the orbital radius (distance from the planet's center). Canceling \( m \) and solving for \( v \), we get \( v = \sqrt{\frac{Gm_p}{r}} \).

Step2: Determine orbital radius for low Earth orbit

For a low Earth orbit, the orbital radius \( r \) is approximately the radius of the Earth plus the altitude. The Earth's radius is about \( 6370\ \text{km} \), and a typical low Earth orbit altitude is a few hundred kilometers, so the total orbital radius \( r\approx 7000\ \text{km} \) (when considering approximate values).

Step3: Match with given options

Using the formula \( v = \sqrt{\frac{Gm_p}{r}} \) and \( r\approx 7000\ \text{km} \), the correct equation is \( v = \sqrt{\frac{Gm_p}{(7,000\ \text{km})}} \), which is the fourth option. The other options either have incorrect denominators (like \( 200\ \text{km} \) which is just the altitude, not the total radius) or incorrect algebraic manipulations (the second and third options have incorrect factorizations of the formula).

Answer:

\( v = \sqrt{\frac{Gm_p}{(7,000\ \text{km})}} \) (the fourth option)