To find the velocity of a satellite in a low Earth orbit, we can use the formula derived from the gravitational force and centripetal force. The velocity \( v \) of a satellite in 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,
- \( r \) is the distance from the center of the planet to the satellite.
For a low Earth orbit, the radius \( r \) can be approximated as the radius of the Earth plus the altitude of the satellite. The average radius of the Earth is approximately 6,371 km. Therefore, a low Earth orbit at an altitude of 200 km would have:
\[ r \approx 6,371 km + 200 km \approx 6,571 km = 6,571,000 m \]
So the velocity would be calculated using this \( r \).
Among the options provided, the closest to the correct formulation for a satellite at about 7,000 km total radius from the center of the Earth is:
\[ v = \sqrt{\frac{G m_p}{7,000 km}} \]
None of the provided options may show the equation in the right format directly, but if they intended to represent the correct relationship using the effective radius for a low Earth orbit:
The correct form would look like
\[ v = \sqrt{\frac{G m_p}{r}} \]
So based on the provided options, if \( r \) is interpreted as the whole distance from the center of the Earth to the satellite in orbit, the closest match would be:
v=Gmp(7,000 km)−−−−−−−√
It's important to note that the equation format in the options might not match the traditional representation found in textbooks, but conceptually, you're considering the effective radius during the orbit.
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