With "normal" gravity, we used a potential energy of mgh. now with the gravity that is more accurate over longer distances we use a different potential energy. in you own words compare and contrast the two situations. what are the important differences?

A general solution is and a particualr case is mgh, it is just to distance around the radius Earth.

Explanation:

We can use a general equation of the potential energy to understand the particular and general case:

The potential energy is defined as , we know that the gravitational force is , so we could find the potential energy taking the integral of F.

(1)

We can find the particular case, just finding the gravitational potential energy difference:

. Here Uf is the potential evaluated in r+Δh and Ui is the potential evaluated in r.

Using (1) we can calculate ΔU.

Simplifying and combining terms we have a simplified expression.

(2)

Let's call . It is the acceleration due to gravity on the Earth's surface, if r is the radius of Earth and M is the mass of the Earth and we can write (2) as ΔU=mgh, but if we have distance grader than r we should use (2), otherwise, we could get incorrect values of potential energy.

I hope i hleps you!

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