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=1(7,000 km)Gmp−−−−√
v=1(7,000 km)Gmp−−−−√
v=1(200 km)Gmp−−−−√
v=1(200 km)Gmp−−−−√
v=Gmp(200 km)−−−−−−√
Responses
v=Gmp(7,000 km)−−−−−−−√
v=Gmp(7,000 km)−−−−−−−√
v=1(7,000 km)Gmp−−−−√
v=1(7,000 km)Gmp−−−−√
v=1(200 km)Gmp−−−−√
v=1(200 km)Gmp−−−−√
v=Gmp(200 km)−−−−−−√
Answers
GPT-4o mini
To determine the velocity of a satellite in low Earth orbit, we can use the formula for the orbital velocity, which is derived from the balance of gravitational force and centripetal force acting on the satellite. The gravitational force provides the necessary centripetal force to keep the satellite in orbit.
The formula for the orbital velocity \( v \) of a satellite 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 (e.g., Earth),
- \( r \) is the distance from the center of the planet to the satellite.
In low Earth orbit, the altitude of the satellite above Earth's surface is relatively small compared to the radius of the Earth, so we can approximate \( r \) as the radius of the Earth plus the altitude of the satellite.
If we assume the altitude of 200 km above the Earth's surface, we would calculate \( r \) as approximately \( R_{Earth} + 200 \, \text{km} \). The radius of Earth \( R_{Earth} \) is approximately 6,371 km. Therefore:
\[
r \approx 6,371 \, \text{km} + 200 \, \text{km} = 6,571 \, \text{km}
\]
However, in the original responses provided, none of them have included this correction directly—so let's analyze the choices closely.
The equation \( v = \sqrt{\frac{G m_p}{200 \, \text{km}}} \) (or similarly structured equations using proper units) represents the velocity of the satellite if it is at an altitude of 200 km above the surface. However, it's critical that units are consistent, and distances should be in meters if G is in standard SI units.
The closest to our needed equation given in your options is:
\[
v = \sqrt{\frac{G m_p}{200 \, \text{km}}}
\]
Thus, though we must be careful about units, if we are choosing from your list, the equation should resemble:
**\( v = \sqrt{G m_p/(200 \, \text{km})} \)**
If your options were exactly represented, confirm that the appropriate one correlating with that structure exists, as it looks like it is simplified without expressing it like this standard form in pure mathematics.
The formula for the orbital velocity \( v \) of a satellite 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 (e.g., Earth),
- \( r \) is the distance from the center of the planet to the satellite.
In low Earth orbit, the altitude of the satellite above Earth's surface is relatively small compared to the radius of the Earth, so we can approximate \( r \) as the radius of the Earth plus the altitude of the satellite.
If we assume the altitude of 200 km above the Earth's surface, we would calculate \( r \) as approximately \( R_{Earth} + 200 \, \text{km} \). The radius of Earth \( R_{Earth} \) is approximately 6,371 km. Therefore:
\[
r \approx 6,371 \, \text{km} + 200 \, \text{km} = 6,571 \, \text{km}
\]
However, in the original responses provided, none of them have included this correction directly—so let's analyze the choices closely.
The equation \( v = \sqrt{\frac{G m_p}{200 \, \text{km}}} \) (or similarly structured equations using proper units) represents the velocity of the satellite if it is at an altitude of 200 km above the surface. However, it's critical that units are consistent, and distances should be in meters if G is in standard SI units.
The closest to our needed equation given in your options is:
\[
v = \sqrt{\frac{G m_p}{200 \, \text{km}}}
\]
Thus, though we must be careful about units, if we are choosing from your list, the equation should resemble:
**\( v = \sqrt{G m_p/(200 \, \text{km})} \)**
If your options were exactly represented, confirm that the appropriate one correlating with that structure exists, as it looks like it is simplified without expressing it like this standard form in pure mathematics.