How do we know that the escape velocity formula is what it is? How was it determined?
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It’s all about energy. Consider an object at infinite distance from a star; define that as zero potential energy. Any closer object will be deeper in the star’s gravity well, so it will have negative potential energy.
But an object may also have kinetic energy, from having a speed relative to that of the star. Kinetic and potential can be converted into one another, but the sum will be constant. So what’s the kinetic energy an object needs to have zero total energy?
Kinetic: 0.5 m v^2 , where v is the speed of the object, and m is its mass.
[Potential](https://en.wikipedia.org/wiki/Gravitational_energy): –GMm/r , where G is the gravitational constant, M is the mass of the star, and r is the distance from the star to the object.
Total: 0.5mv^2 – GMm/r. If that = 0, then
0.5 mv^2 = GMm/r. Solve for v and get
v_esc = sqrt{2GM/r}
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