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
=
𝐺
𝑚
𝑝
(
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 root
v=Gmp(200 km)−−−−−−√
𝑣
=
𝐺
𝑚
𝑝
Responses
v=Gmp(7,000 km)−−−−−−−√
𝑣
=
𝐺
𝑚
𝑝
(
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 root
v=Gmp(200 km)−−−−−−√
𝑣
=
𝐺
𝑚
𝑝
Answers
GPT-4o mini
To find the velocity of a satellite in a low Earth orbit, you can use the formula for orbital velocity derived from the gravitational force. The correct formula to calculate the orbital velocity \( v \) of a satellite in a circular orbit 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 (Earth, in this case), and
- \( r \) is the distance from the center of the planet to the satellite.
For a low Earth orbit, if you are considering an altitude of \( 200 \, \text{km} \) above the Earth's surface, the orbital radius \( r \) would be:
\[
r = R_{Earth} + h
\]
where \( R_{Earth} \) is the radius of the Earth (approximately \( 6,371 \, \text{km} \)) and \( h \) is the altitude of the orbit (e.g., \( 200 \, \text{km} \)). Thus, \( r \) for a \( 200 \, \text{km} \) orbit would be:
\[
r = 6,371 \, \text{km} + 200 \, \text{km} = 6,571 \, \text{km}
\]
Given the context of your question and the typical approximation used in orbital mechanics, the option closest to this would therefore be:
\[
v = \sqrt{\frac{G m_p}{7,000 \, \text{km}}}
\]
However, \( 7,000 \, \text{km} \) is more representative of the total radius of the Earth plus the satellite altitude for an orbit of approximately \( 200 \, \text{km} \). So, for the choices given, the appropriate choice to find the orbital velocity at low Earth orbit would be:
**If you are considering the \( 200 \, \text{km} \) altitude, you should indeed be using:**
\[
v = \sqrt{\frac{G m_p}{r}} \text{ with } r \approx 7,000 \, \text{km}
\]
Thus, the response that most closely resembles this is likely the first one, given the approximations used in many physics problems:
\[
v = \sqrt{G m_p (7,000 \, \text{km})^{-1}}
\]
(Note when typing equations, remember to check for brackets or any misinterpretations.) If you are to select the option reflecting low Earth orbit with respect to mass and radius, make sure it aligns with the expectations of simplifications made in academic discussions on such topics.
\[
v = \sqrt{\frac{G m_p}{r}}
\]
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, if you are considering an altitude of \( 200 \, \text{km} \) above the Earth's surface, the orbital radius \( r \) would be:
\[
r = R_{Earth} + h
\]
where \( R_{Earth} \) is the radius of the Earth (approximately \( 6,371 \, \text{km} \)) and \( h \) is the altitude of the orbit (e.g., \( 200 \, \text{km} \)). Thus, \( r \) for a \( 200 \, \text{km} \) orbit would be:
\[
r = 6,371 \, \text{km} + 200 \, \text{km} = 6,571 \, \text{km}
\]
Given the context of your question and the typical approximation used in orbital mechanics, the option closest to this would therefore be:
\[
v = \sqrt{\frac{G m_p}{7,000 \, \text{km}}}
\]
However, \( 7,000 \, \text{km} \) is more representative of the total radius of the Earth plus the satellite altitude for an orbit of approximately \( 200 \, \text{km} \). So, for the choices given, the appropriate choice to find the orbital velocity at low Earth orbit would be:
**If you are considering the \( 200 \, \text{km} \) altitude, you should indeed be using:**
\[
v = \sqrt{\frac{G m_p}{r}} \text{ with } r \approx 7,000 \, \text{km}
\]
Thus, the response that most closely resembles this is likely the first one, given the approximations used in many physics problems:
\[
v = \sqrt{G m_p (7,000 \, \text{km})^{-1}}
\]
(Note when typing equations, remember to check for brackets or any misinterpretations.) If you are to select the option reflecting low Earth orbit with respect to mass and radius, make sure it aligns with the expectations of simplifications made in academic discussions on such topics.