The correct equation to find the velocity of a satellite in a low Earth orbit can be derived from the formula for orbital velocity:
\[ v = \sqrt{\frac{G \cdot m_p}{r}} \]
where
- \( v \) is the orbital velocity,
- \( G \) is the universal gravitational constant,
- \( m_p \) is the mass of the planet (Earth in this case), and
- \( r \) is the distance from the center of the planet to the satellite.
For "low Earth orbit," you typically consider an altitude of around 200 km above the surface of the Earth. The radius \( r \) would then be the radius of the Earth (approximately 6,371 km) plus the altitude. If we assume a low Earth orbit at 200 km, the effective radius \( r \) would be:
\[ r \approx 6,371 , \text{km} + 200 , \text{km} = 6,571 , \text{km} \]
However, if just considering the gravitational force at the altitude (and using a simplification or the direct 200 km approximation for the orbital altitude), we can use the following formula, corresponding to the closest options provided:
\[ v = \sqrt{\frac{G \cdot m_p}{200 , \text{km}}} \]
Therefore, among the given options, the response that most closely matches this formulation is:
v = Gmp(200 km)−−−−−−√.