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=1(7,000 km)Gmp−−−−√
v=1(7,000 km)Gmp−−−−√
v=Gmp(200 km)−−−−−−√
v=Gmp(200 km)−−−−−−√
v=1(200 km)Gmp−−−−√
v=1(200 km)Gmp−−−−√
v=Gmp(7,000 km)
Responses
v=1(7,000 km)Gmp−−−−√
v=1(7,000 km)Gmp−−−−√
v=Gmp(200 km)−−−−−−√
v=Gmp(200 km)−−−−−−√
v=1(200 km)Gmp−−−−√
v=1(200 km)Gmp−−−−√
v=Gmp(7,000 km)
Answers
GPT-4o mini
To find the orbital velocity \( v \) of a satellite in a low Earth orbit, you can use the formula derived from equating the gravitational force to the centripetal force acting on the satellite. The correct formula for the velocity \( v \) is:
\[
v = \sqrt{\frac{G m_p}{r}}
\]
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 satellite in low Earth orbit, if we take the average radius of the Earth as approximately 6,370 km, and then add the altitude of the satellite (for example, if the satellite is at an altitude of about 200 km), the total radius \( r \) would be about \( 6,570 \, \text{km} \).
Given your choices, the closest valid equation appears to be:
\[
v = \sqrt{\frac{G m_p}{(7,000 \, \text{km})}}
\]
However, this specific choice isn't explicitly listed, but the general idea is that you would take the total radius (the radius of the Earth plus the altitude of the satellite) to calculate the velocity.
From the provided responses, if you choose one that represents the format of \( v = \sqrt{\frac{G m_p}{r}} \), you may opt for:
\[
v = \sqrt{\frac{1}{(200 \, \text{km})} G m_p} \quad \text{(if the assumption is that this is the distance from the mass to the satellite)}
\]
But this is not typical since \( r \) should be the total distance from the center of the Earth.
So technically, none of the options are completely correct based on typical physics principles for calculating orbital velocity, as the radius used should be based on the Earth's radius plus the altitude of the orbit above the Earth's surface. However, the response that positions itself closest to the intended formula would be the one with the correct format.
\[
v = \sqrt{\frac{G m_p}{r}}
\]
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 satellite in low Earth orbit, if we take the average radius of the Earth as approximately 6,370 km, and then add the altitude of the satellite (for example, if the satellite is at an altitude of about 200 km), the total radius \( r \) would be about \( 6,570 \, \text{km} \).
Given your choices, the closest valid equation appears to be:
\[
v = \sqrt{\frac{G m_p}{(7,000 \, \text{km})}}
\]
However, this specific choice isn't explicitly listed, but the general idea is that you would take the total radius (the radius of the Earth plus the altitude of the satellite) to calculate the velocity.
From the provided responses, if you choose one that represents the format of \( v = \sqrt{\frac{G m_p}{r}} \), you may opt for:
\[
v = \sqrt{\frac{1}{(200 \, \text{km})} G m_p} \quad \text{(if the assumption is that this is the distance from the mass to the satellite)}
\]
But this is not typical since \( r \) should be the total distance from the center of the Earth.
So technically, none of the options are completely correct based on typical physics principles for calculating orbital velocity, as the radius used should be based on the Earth's radius plus the altitude of the orbit above the Earth's surface. However, the response that positions itself closest to the intended formula would be the one with the correct format.