Explanation based upon Newtonian mechanics - using Newton's Law of Gravity
Over short distances, compared to the earths radius, the gravitational field spreads out very little. The Earth's radius is about 6,400 Km so for distances of a few tens of kilometres we can calculate the energy to lift a mass simply by:
Energy = mass x gravitational field strength x height
or E = mgh
without a significant loss of accuracy because the change in the strength of the Earth's gravitational field is small.
We cannot use this simple equation when distances are large, that is similar to or greater than the radius of the Earth, because the force of gravity reduces to a fraction of its value when the distances are large.
At a height from the surface equal to the Earth's radius the force of attraction is only one quarter of that near the surface. If we rise to twice that height, that is three radii from the centre, the force will only be one ninth.
To calculate the energy change when something is lifted or falls a large distance we need to find the gravitational potential at the start and at the end, subtract the two and then multiply by the mass moved.
As the height increases the force gradually decreases.
F is force, m is a mass we are raising from the earths surface to a height of 2r from the ground, that is 3r from the centre. Me is the mass of the Earth and g is the gravitational field strength in N/Kg
Since F = mg = -GMem/ r2
then cancelling m, g = -GMe / r2
Work done equals mgh but we have to take the changing value of g into account.
To calculate the work done we have to integrate the expression between the limits of the two distances, in this case between r and 3r, so:
The expression circled -GMe /r is the gravitational potential.
to calculate the value of the change in gravitational energy over a large distance:
Mass moved x (gravitational potential at first height - gravitational potential at second height) which is:
-GMe m/r + GMem/3r so the total work done in lifting the satellite mass is -2GMe m/3r