How do "geosynchronous satellites" stay in place while vortexing around the universe at millions of miles per hour?
Simple Newtonian dynamics. The same laws that cause the moon to orbit the earth. Or do you deny that the moon orbits the earth? The same laws that cause you to fall to the ground when you jump out of a 2nd floor window. The same laws that NASA uses to send a vehicle to the moon.
I have already given examples where you can personally experience travelling at 500mph but it feels and looks like you are stationary. You can swing something on the end of a string around your head while standing still on the earth. You could do the same while standing in a train doing 125mph. Or on a plane flying at 500mph.
Objects orbiting other objects (eg moon around the earth, earth around the sun): Here's the maths:
The gravitational force between two bodies of mass m1 and m2, at a distance r apart is given by:
F = G . m1 . m2 / r^2
and where G is the universal gravitational constant, = 6.673E-11 N m^2/kg^2.
For an object of mass m1 to move in a circle of radius r, and going at a speed of v, needs a centripetal force, F, given by:
F = m1 . v^2 / r
If m1 is your satellite and m2 is the earth, when the gravitational force between matches the centripetal force, then the satellite will orbit the earth. If the satellite is going faster, it will spiral away. If slower, it will spiral towards the earth.
To orbit the earth, the satellite must be moving at a speed of:
v = sqrt( G . m2 / r)
(here, m2 is the mass of the earth, r the distance from the satellite to the centre of the earth)
This explains the movement of the moon and the movement of GPS equally well.
Ah, I hear you say, the earth could be stationary, and these laws would still work. True. So far, none of this tells you anything about the earth moving or rotating or being stationary. A satellite could just as well be orbiting a stationary earth.
A geosynchronous satellite is a special case. Its height above the earth is such that it takes 24 hours to make one orbit around the earth. Closer objects will orbit faster (eg GPS in about 12 hours). Objects further away will take longer, eg the moon (about 29 days). If you put a geosynchronous satellite directly above the equator, then it appears to be stationary when viewed from the earth because as it travels around the earth, the earth also rotates at exactly the same angular speed. It is geostationary.
If the earth was stationary, a geosynchronous would still take the same 24 hours to make one circuit. No change. However, it would not appear to be stationary when viewed from the earth. But geostationary satellites exist. They are used for satellite TV transmission, and is why satellite dishes point in a fixed direction instead of having to move to track a moving satellite.
I have answered you question. Now, your turn. If the earth is stationary, please explain how a satellite can just hang 35,786 kilometres above the equator without falling to the earth.