To find the velocity of a satellite in a low Earth orbit, we can use the formula for circular orbital velocity:
\[ v = \sqrt{\frac{G m_p}{r}} \]
where:
- \( v \) is the orbital velocity,
- \( G \) 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.
In a low Earth orbit, the altitude is typically around 200 km above the Earth's surface. Since the radius \( r \) is the distance from the center of the Earth, we must add the Earth's radius (approximately 6370 km) to the altitude to get the total distance from the center of the Earth.
Thus, for an orbit at 200 km altitude, \( r \) would be:
\[ r = 6370 , \text{km} + 200 , \text{km} = 6570 , \text{km} \]
Additionally, since we are looking for a formula that represents this setup with respect to the altitude given, we can substitute the total radius into the orbital velocity formula:
So, the correct formula in the options that uses the proper altitude would be:
\[ v = \sqrt{\frac{G m_p}{6570 , \text{km}}} \]
Looking through the choices provided, it seems they are incorrectly formatted or based on notations. However, we would choose the formula based on the altitude referenced:
For an altitude of 200 km, the actual velocity equation would be:
Choice 1: \( v = \sqrt{\frac{G m_p}{(6370 , \text{km} + 200 , \text{km})}} \)
Unfortunately, no options accurately reflect this equation directly with correct notation and units, but we recognized that you need the total distance from the center of the planet (which combines the radius of the planet and the altitude).
If we are strictly limited to the given choices, and assuming we're using the altitude in a simplified form without adding Earth's radius, then the closest choice for low orbit (not considering Earth's radius) could be:
Response: \( v = \sqrt{(200 , \text{km}) G m_p} \)
But remember, it needs to be adjusted with the radius included for true calculations.