14.13:
Circular Orbits and Critical Velocity for Satellites
A ball pushed from a tower with a certain horizontal velocity v0 will land at a point A under the influence of the Earth's gravitational force.
If a ball is pushed with higher horizontal velocity, it will land further, at point B. With a further increase in velocity, it will go farther than point B and land at point C.
However, at a certain critical velocity vc, it will follow a perfectly circular orbit around the Earth. If the velocity increases above the critical value, the ball will follow an elliptical orbit.
Therefore, if a satellite orbits the Earth with critical velocity, it will follow a perfectly circular path around the Earth, under the influence of the Earth's gravitational force. Hence, the centripetal acceleration of a satellite would be equal to its acceleration due to gravity.
Substituting for gravitational acceleration and rearranging the terms, the critical velocity of a satellite equals the square root of the product of the gravitational constant and the Earth’s mass divided by its total distance from the Earth's center.
14.13:
Circular Orbits and Critical Velocity for Satellites
The Moon orbits around the Earth. In turn, the Earth (and other planets) orbit the Sun. The space directly above our atmosphere is filled with artificial satellites in orbit. One can examine the circular orbit, the simplest kind of orbit, to understand the relationship between the speed and the period of planets and satellites with respect to their positions and the bodies that they orbit.
Nicolaus Copernicus (1473-1543) first suggested that the Earth and all other planets orbit the Sun in circles. He further noted that orbital periods increased with distance from the Sun. Later, an analysis by Johannes Kepler (1571-1630) showed that these orbits are actually ellipses, although the orbits of most planets in the solar system are nearly circular. A circular orbit is a result of a tangential velocity such that the Earth's surface curves away at the same rate that the object falls towards the Earth. The Earth's orbital distance from the Sun varies by a mere 2%. An exception is the eccentric orbit of Mercury, whose orbital distance varies by nearly 40%. Determining the orbital speed and orbital period of a satellite is much easier for circular orbits.
This text is adapted from Openstax, University Physics Volume 1, Section 13.4: Satellite Orbits and Energy.