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)

Question

Responses

v=1(200 km)Gmp−−−−√
v=1(200 km)Gmp−−−−√

v=Gmp(200 km)−−−−−−√
v=Gmp(200 km)−−−−−−√

v=1(7,000 km)Gmp−−−−√
v=1(7,000 km)Gmp−−−−√

v=Gmp(7,000 km)−−−−−−−√

Answer 1

To find the velocity of a satellite in a low Earth orbit, we can use the formula derived from the balance of gravitational force and centripetal force. The orbital velocity \( v \) of a satellite in a circular orbit can be calculated using the following equation:

\[ v = \sqrt{\frac{G m_p}{r}} \]

Where:

• \( G \) is the universal gravitational constant.
• \( m_p \) is the mass of the planet (Earth in this case).
• \( r \) is the distance from the center of the Earth to the satellite.

For a satellite in low Earth orbit, the altitude is approximately 200 km above the Earth's surface, and the average radius of the Earth is about 6,371 km. Therefore, the distance \( r \) from the center of the Earth would be:

\[ r \approx 6,371 \text{ km} + 200 \text{ km} = 6,571 \text{ km} \]

The correct equation for the velocity of a satellite in low Earth orbit would thus be:

\[ v = \sqrt{\frac{G m_p}{6571 \text{ km}}} \]

However, none of the provided options directly reflect this formulation. The closest option from the responses you provided would be:

v = \sqrt{\frac{G m_p}{7,000 \text{ km}}}

This estimation is reasonable and likely falls within the correct range for calculations regarding a low Earth orbit. Therefore, this response would be the best choice.