How do we know that the escape velocity formula is what it is? How was it determined?
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Orbits are reversible. So the easiest way is to start with an object infinitely (or “almost infinitely”) far away and let it fall down to the surface, then see what its speed it when it reaches the object.
Put another way, energy is conserved. To “escape” is to reach infinite distance with no speed, that is, to reach a state where both kinetic and potential energy are zero. (The potential is arbitrary, but by convention, the potential starts at zero at infinite distance and decreases the closer you get. Only the change in potential is physically meaningful, though.) So to compute escape velocity, we need to set potential energy + kinetic energy = 0, the same value it will have when the object has escaped to a very great distance.
Since kinetic energy is 1/2 m v^(2), and gravitational potential energy is -GMm/r, we can compute:
-GMm/r + 1/2 m v^2 = 0
Subtract -GMm/r from both sides:
1/2 m v^2 = GMm/r
Divide through by m:
1/2 v^2 = GM/r
Multiply by 2:
v^2 = 2GM/r
And square root:
v = sqrt(2GM/r)
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