Question
Orbits of Satellites Quick Check
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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=Gmp(7,000 km)−−−−−−−√
v=Gmp(200 km)−−−−−−√
v=Gmp(200 km)−−−−−−√
v=1(200 km)Gmp−−−−√
v=1(200 km)Gmp−−−−√
v=1(7,000 km)Gmp−−−−√
Answers
GPT-4o mini
To find the velocity of a satellite in a low Earth orbit, you can use the formula derived from the law of universal gravitation and centripetal force. The orbital velocity \( v \) of a satellite can be expressed as:
\[
v = \sqrt{\frac{G m_p}{r}}
\]
where \( G \) is the universal gravitational constant, \( m_p \) is the mass of the planet (Earth in this case), and \( r \) is the distance from the center of the planet to the satellite.
For a low Earth orbit, the radius \( r \) would be the Earth's radius plus the altitude of the orbit. Assuming the radius of the Earth is approximately 6,371 km, for a low Earth orbit typically around 200 km, the total radius would be:
\[
r \approx 6,371 \text{ km} + 200 \text{ km} = 6,571 \text{ km} \approx 7,000 \text{ km}
\]
Thus, the correct choice from your options, which reflects this formula, would be:
\[
v = \sqrt{\frac{G m_p}{(7,000 \text{ km})}}
\]
However, based on your options, the most accurate formula that relates to low Earth orbit and closest to our calculation is:
\[
v = \sqrt{G m_p (200 \text{ km})^{-1}}
\]
or
\[
v = \sqrt{\frac{G m_p}{(200 \text{ km})}}
\]
Since none of the options provided mirror the exact correct formula clearly, depending on slight differences, you may want to choose:
**v = Gmp(200 km)−−−−−−√**, which indicates the use of altitude in relation to gravitational force.
The velocity-related options should be reflecting the inverse of distances accurately.
\[
v = \sqrt{\frac{G m_p}{r}}
\]
where \( G \) is the universal gravitational constant, \( m_p \) is the mass of the planet (Earth in this case), and \( r \) is the distance from the center of the planet to the satellite.
For a low Earth orbit, the radius \( r \) would be the Earth's radius plus the altitude of the orbit. Assuming the radius of the Earth is approximately 6,371 km, for a low Earth orbit typically around 200 km, the total radius would be:
\[
r \approx 6,371 \text{ km} + 200 \text{ km} = 6,571 \text{ km} \approx 7,000 \text{ km}
\]
Thus, the correct choice from your options, which reflects this formula, would be:
\[
v = \sqrt{\frac{G m_p}{(7,000 \text{ km})}}
\]
However, based on your options, the most accurate formula that relates to low Earth orbit and closest to our calculation is:
\[
v = \sqrt{G m_p (200 \text{ km})^{-1}}
\]
or
\[
v = \sqrt{\frac{G m_p}{(200 \text{ km})}}
\]
Since none of the options provided mirror the exact correct formula clearly, depending on slight differences, you may want to choose:
**v = Gmp(200 km)−−−−−−√**, which indicates the use of altitude in relation to gravitational force.
The velocity-related options should be reflecting the inverse of distances accurately.