How do I find the satellite's altitude above the earth's surface

When in orbit, a communication satellite attracts the Earth with a force of F and the earth-satellite gravitational potential energy (relative to zero at infinite separation) is - U. The radius of Earth is assumed to be r.

How do I find the satellite's altitude above the Earth's surface and the mass of the satellite?

Obtain an expression for r²/m from the gravitation law of force between the Earth and your satellite, and obtain an expression for r/m from the potential energy of the system. Plug the two expressions together.

I found r = -U/F and m = U²/FGM

(meaning r= \nabla F/F.. kinda cool.)

To find the satellite's altitude above the Earth's surface, we can use the formula for gravitational potential energy: U = -GmM/r, where G is the gravitational constant, m is the mass of the satellite, M is the mass of the Earth, and r is the distance between the satellite and the center of the Earth.

We can rearrange this formula to solve for r, which will give us the distance between the satellite and the center of the Earth. Once we have this distance, we can subtract the radius of the Earth (r) to find the altitude of the satellite above the Earth's surface.

To find the mass of the satellite, we can use the force of attraction between the satellite and the Earth: F = GmM/r^2. Again, we can rearrange this formula to solve for m, which will give us the mass of the satellite.

It is important to note that these calculations assume a circular orbit, and the satellite's actual altitude and mass may vary depending on the shape and characteristics of its orbit. Additionally, these calculations do not take into account other factors such as atmospheric drag and the gravitational influences of other celestial bodies.