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/(7,000 km)−−−−−−−√

v=1/(200 km)Gmp−−−−√

v=1/(7,000 km)Gmp−−−−√

v=Gmp/(200 km)−−−−−−√

Answer 1

To find the velocity (\(v\)) of a satellite in a low Earth orbit, you can use the formula for orbital velocity given by:

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

where:
- \(G\) is the universal gravitational constant,
- \(m_p\) is the mass of the planet (Earth in this case),
- \(r\) is the distance from the center of the planet to the satellite.

For a low Earth orbit (LEO), the altitude is typically about 200 km to 2,000 km above Earth's surface. The radius \(r\) from the center of the Earth can be calculated as:

\[
r = R + h
\]
- where \(R\) is the Earth's radius (approximately 6,371 km) and \(h\) is the altitude of the orbit.

If we want to find the velocity for a satellite at 200 km above the Earth's surface, the total radius \(r\) would be approximately:

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

In the given options, only one option takes a related form to this requirement. The correct equation should have the form:

\[
v = \sqrt{\frac{G m_p}{r}} \quad (where \ r \text{ is the distance from the center})
\]

Given your provided responses, the equation that most closely resembles this format (while interpreting \(7,000 \text{ km}\) as an approximation for \(r\)) would be:

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

So the correct answer is:

**v = Gmp/(7,000 km)−−−−−−−√**