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 velocity of the satellite if it is placed in a geostationary orbit?
V=square root gmp/(48,115 km)
V=square root gmp/(15,522 km)
V=square root gmp/(42,164 km)
V= square root gmp/(7,324 km)
V=square root gmp/(48,115 km)
V=square root gmp/(15,522 km)
V=square root gmp/(42,164 km)
V= square root gmp/(7,324 km)
Answers
Answer
Are you sure
Answered by
GPT-4o mini
To find the velocity of a satellite in a geostationary orbit around a planet, you can use the formula for orbital velocity, which is given by:
\[
V = \sqrt{\frac{G \cdot m_p}{r}}
\]
Where:
- \( V \) is the orbital velocity,
- \( G \) is the universal gravitational constant,
- \( m_p \) is the mass of the planet, and
- \( r \) is the distance from the center of the planet to the satellite.
For a geostationary orbit, the satellite must orbit at a specific radius that allows it to appear stationary relative to the Earth's surface. The radius for geostationary orbit can be calculated as approximately 42,164 kilometers from the center of the Earth.
Given the options you provided, the appropriate equation for the velocity of a satellite in a geostationary orbit is:
\[
V = \sqrt{\frac{g m_p}{42,164 \text{ km}}}
\]
Thus, the correct answer is:
\[
V = \sqrt{\frac{g m_p}{42,164 \text{ km}}}
\]
\[
V = \sqrt{\frac{G \cdot m_p}{r}}
\]
Where:
- \( V \) is the orbital velocity,
- \( G \) is the universal gravitational constant,
- \( m_p \) is the mass of the planet, and
- \( r \) is the distance from the center of the planet to the satellite.
For a geostationary orbit, the satellite must orbit at a specific radius that allows it to appear stationary relative to the Earth's surface. The radius for geostationary orbit can be calculated as approximately 42,164 kilometers from the center of the Earth.
Given the options you provided, the appropriate equation for the velocity of a satellite in a geostationary orbit is:
\[
V = \sqrt{\frac{g m_p}{42,164 \text{ km}}}
\]
Thus, the correct answer is:
\[
V = \sqrt{\frac{g m_p}{42,164 \text{ km}}}
\]
Answered by
GPT-4o mini
Yes, I am confident in the response I provided, but let me clarify the reasoning for your understanding.
In the context of geostationary orbits, the relevant formula for calculating the orbital velocity \( V \) of a satellite is:
\[
V = \sqrt{\frac{G m_p}{r}}
\]
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.
For a geostationary orbit around the Earth, the radius \( r \) is approximately 42,164 kilometers from the center of the Earth (this includes the Earth's radius, which is about 6,378 km).
Thus, for a geostationary satellite orbiting Earth, the correct distance to use is \( r = 42,164 \text{ km} \).
As such, the relevant equation to find the velocity of a satellite in geostationary orbit is:
\[
V = \sqrt{\frac{G m_p}{42,164 \text{ km}}}
\]
If you find a misinterpretation or if the problem relates specifically to another planet, the principle remains that we look for the specific orbital radius for a geostationary orbit around that particular planet. But in the context of Earth and the provided choices, the correct equation corresponds to the third option.
If you have any further clarifications or need assistance, feel free to ask!
In the context of geostationary orbits, the relevant formula for calculating the orbital velocity \( V \) of a satellite is:
\[
V = \sqrt{\frac{G m_p}{r}}
\]
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.
For a geostationary orbit around the Earth, the radius \( r \) is approximately 42,164 kilometers from the center of the Earth (this includes the Earth's radius, which is about 6,378 km).
Thus, for a geostationary satellite orbiting Earth, the correct distance to use is \( r = 42,164 \text{ km} \).
As such, the relevant equation to find the velocity of a satellite in geostationary orbit is:
\[
V = \sqrt{\frac{G m_p}{42,164 \text{ km}}}
\]
If you find a misinterpretation or if the problem relates specifically to another planet, the principle remains that we look for the specific orbital radius for a geostationary orbit around that particular planet. But in the context of Earth and the provided choices, the correct equation corresponds to the third option.
If you have any further clarifications or need assistance, feel free to ask!
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