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(200 km)−−−−−−√
𝑣
=
𝐺
𝑚
𝑝
(
200
km
)
v=Gmp(200 km)−−−−−−√
𝑣
=
𝐺
𝑚
𝑝
(
200
km
)
v=1(7,000 km)Gmp−−−−√
𝑣
=
1
(
7
,
000
km
)
𝐺
𝑚
𝑝
v=1(7,000 km)Gmp−−−−√
𝑣
=
1
(
7
,
000
km
)
𝐺
𝑚
𝑝
v=Gmp(7,000 km)−−−−−−−√
𝑣
=
𝐺
𝑚
𝑝
(
7
,
000
km
)
v=Gmp(7,000 km)−−−−−−−√
𝑣
=
𝐺
𝑚
𝑝
(
7
,
000
km
)
v=1(200 km)Gmp−−−−√
Responses
v=Gmp(200 km)−−−−−−√
𝑣
=
𝐺
𝑚
𝑝
(
200
km
)
v=Gmp(200 km)−−−−−−√
𝑣
=
𝐺
𝑚
𝑝
(
200
km
)
v=1(7,000 km)Gmp−−−−√
𝑣
=
1
(
7
,
000
km
)
𝐺
𝑚
𝑝
v=1(7,000 km)Gmp−−−−√
𝑣
=
1
(
7
,
000
km
)
𝐺
𝑚
𝑝
v=Gmp(7,000 km)−−−−−−−√
𝑣
=
𝐺
𝑚
𝑝
(
7
,
000
km
)
v=Gmp(7,000 km)−−−−−−−√
𝑣
=
𝐺
𝑚
𝑝
(
7
,
000
km
)
v=1(200 km)Gmp−−−−√
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 Newton's law of gravitation and the centripetal force requirement. The formula for 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 (in this case, the Earth), and
- \( r \) is the distance from the center of the planet to the satellite.
For a low Earth orbit (LEO) at an altitude of approximately 200 km, the value for \( r \) would be the Earth's radius (about 6,371 km) plus 200 km, which totals about 7,000 km. Therefore, you should convert \( r \) into kilometers for the computation.
From your provided responses, the correct equation to use, considering the radius for a low Earth orbit, is:
\[
v = \sqrt{\frac{G m_p}{7,000 \text{ km}}}
\]
This implies the correct form from your options would be:
\[
v = \frac{1}{(7,000 \text{ km})} \sqrt{G m_p}
\]
Thus, the option that corresponds to the orbital velocity formula is:
\[
v = \frac{1}{(7,000 \text{ km})} G m_p^{\frac{1}{2}}
\]
Make sure to check the specific forms given in your options. If the version without the reciprocal is present, it may not be formatted clearly as mentioned in specific options but would still fundamentally represent the orbital velocity relationship.
\[
v = \sqrt{\frac{G m_p}{r}}
\]
where:
- \( G \) is the universal gravitational constant,
- \( m_p \) is the mass of the planet (in this case, the Earth), and
- \( r \) is the distance from the center of the planet to the satellite.
For a low Earth orbit (LEO) at an altitude of approximately 200 km, the value for \( r \) would be the Earth's radius (about 6,371 km) plus 200 km, which totals about 7,000 km. Therefore, you should convert \( r \) into kilometers for the computation.
From your provided responses, the correct equation to use, considering the radius for a low Earth orbit, is:
\[
v = \sqrt{\frac{G m_p}{7,000 \text{ km}}}
\]
This implies the correct form from your options would be:
\[
v = \frac{1}{(7,000 \text{ km})} \sqrt{G m_p}
\]
Thus, the option that corresponds to the orbital velocity formula is:
\[
v = \frac{1}{(7,000 \text{ km})} G m_p^{\frac{1}{2}}
\]
Make sure to check the specific forms given in your options. If the version without the reciprocal is present, it may not be formatted clearly as mentioned in specific options but would still fundamentally represent the orbital velocity relationship.