Question

Let G be the universal gravitational constant and mp 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)
Responses

v=Gmp(200 km)−−−−−−√
v is equal to square root of the fraction with numerator cap g m sub p and denominator right paran 200 kilometers left paran end root

v=1(7,000 km)Gmp−−−−√
v is equal to the fraction with numerator 1 and denominator left bracket 7 comma 000 kilometers right bracket end fraction square root of cap g m sub p end root

v=1(200 km)Gmp−−−−√
v is equal to the fraction with numerator 1 and denominator right paran 200 kilometers left paran end fraction square root of cap g m sub p end root

v=Gmp(7,000 km)−−−−−−−√

Answer 1

To find the velocity of a satellite in low Earth orbit, we can use the formula for the circular orbital velocity, which is given by:

\[
v = \sqrt{\frac{G m_p}{r}}
\]

where \( G \) is the universal gravitational constant, \( m_p \) is the mass of the planet, and \( r \) is the distance from the center of the planet to the satellite.

For low Earth orbit, if we consider a satellite at an altitude of 200 km above the Earth's surface, we need to add the Earth's radius (approximately 6,371 km) to this altitude to get the value of \( r \). Thus, \( r \) would be approximately:

\[
r \approx 6,371 \text{ km} + 200 \text{ km} = 6,571 \text{ km}
\]

Thus, the correct equation to find the velocity at that orbit should look similar to:

\[
v = \sqrt{\frac{G m_p}{6,571 \, \text{km}}}
\]

Among the provided options, the closest one is:

\[
v = \sqrt{\frac{G m_p}{7,000 \, \text{km}}}
\]

This can serve as a good approximation (depending on the context), since 7,000 km is close to the calculated 6,571 km.

Thus, the best choice according to the provided options is:

**v=1(7,000 km)Gmp−−−−√**