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)Responsesv=Gmp(7,000 km)−−−−−−−√v is equal to square root of the fraction with numerator cap g m sub p and denominator left bracket 7 comma 000 kilometers right bracket end rootv=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 rootv=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 rootv=Gmp(200 km)−−−−−−√
Answers
GPT-4o mini
To find the velocity of a satellite in a low Earth orbit, we can use the formula for the orbital velocity \( v \) of an object in a circular orbit around a planet:
\[
v = \sqrt{\frac{G m_p}{r}}
\]
where:
- \( G \) is the universal gravitational constant,
- \( m_p \) is the mass of the planet,
- \( r \) is the distance from the center of the planet to the satellite.
In a low Earth orbit, the altitude is typically around 200 km to 700 km above the Earth's surface. However, when using the formula, \( r \) must be the distance from the center of the Earth, which is approximately \( 6371 \text{ km} + \text{altitude} \).
For example, if the satellite is at 7000 km from the center of the Earth (which would be 7000 km - the approximate radius of the Earth), we are simplifying this as:
\[
r \approx 7000 \text{ km}
\]
With this context, the correct equation among the given options for a satellite at approximately 7000 km from the center would be:
\[
v = \sqrt{\frac{G m_p}{7000 \, \text{km}}}
\]
Therefore, the response that reflects this equation format (which can be rearranged to match the given choices) would be:
**v is equal to square root of the fraction with numerator \( G m_p \) and denominator \( 7000 \, \text{km} \)**.
Thus, if the provided options include:
1. \( v = \sqrt{G m_p/(7000 \text{ km})} \)
This would be the correct representation, matching the structure of:
- **v = G m_p(7,000 km)−−−−−−−√**.
Make sure to apply the correct formatting in your response to ensure clarity.
\[
v = \sqrt{\frac{G m_p}{r}}
\]
where:
- \( G \) is the universal gravitational constant,
- \( m_p \) is the mass of the planet,
- \( r \) is the distance from the center of the planet to the satellite.
In a low Earth orbit, the altitude is typically around 200 km to 700 km above the Earth's surface. However, when using the formula, \( r \) must be the distance from the center of the Earth, which is approximately \( 6371 \text{ km} + \text{altitude} \).
For example, if the satellite is at 7000 km from the center of the Earth (which would be 7000 km - the approximate radius of the Earth), we are simplifying this as:
\[
r \approx 7000 \text{ km}
\]
With this context, the correct equation among the given options for a satellite at approximately 7000 km from the center would be:
\[
v = \sqrt{\frac{G m_p}{7000 \, \text{km}}}
\]
Therefore, the response that reflects this equation format (which can be rearranged to match the given choices) would be:
**v is equal to square root of the fraction with numerator \( G m_p \) and denominator \( 7000 \, \text{km} \)**.
Thus, if the provided options include:
1. \( v = \sqrt{G m_p/(7000 \text{ km})} \)
This would be the correct representation, matching the structure of:
- **v = G m_p(7,000 km)−−−−−−−√**.
Make sure to apply the correct formatting in your response to ensure clarity.