linguistic:paraphrasing
Derivation

For circular orbits around a body, the centripetal force required to maintain the orbit (F) is equal to the gravitational force acting on the satellite (F):

From Isaac Newton 's universal law of gravitation,

where F is the gravitational force acting between two objects, M is the mass of the Earth, 5.9736 × 10 kg, m is the mass of the satellite, r is the distance between the centers of their masses, and G is the gravitational constant, (6.674 28 ± 0.000 67) × 10 m kg s.

The magnitude of the acceleration, a, of a body moving in a circle is given by:

where v is the magnitude of the velocity (i.e. the speed) of the satellite. From Newton's second law of motion, the centripetal force F is given by:

As F = F,

so that

Replacing v with the equation for the speed of an object moving around a circle produces:

where T is the orbital period (i.e. one sidereal day), and is equal to 86 164.090 54 s. This gives an equation for r:

The product GM is known with much greater precision than either factor alone; it is known as the geocentric gravitational constant μ = 398 600.4418 ± 0.0008 km s. Hence

The resulting orbital radius is 42,164 kilometres (26,199 miles). Subtracting the Earth's equatorial radius, 6,378 kilometres (3,963 miles), gives the altitude of 35,786 kilometres (22,236 miles).

The orbital speed is calculated by multiplying the angular speed by the orbital radius: