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My questions mostly concern the history of physics. Who found the formula for kinetic energy

##E_k =\frac{1}{2}mv^{2}##

and how was this formula actually discovered? I've recently watched Leonard Susskind's lecture where he proves that if you define kinetic and potential energy in this way, then you can show that the total energy is conserved. But that makes me wonder how anyone came to define kinetic energy in that way.

My guess is that someone thought along the following lines:

Energy is conserved, in the sense that when you lift something up you've done work,

but when you let it go back down you're basically back where you started.

So it seems that my work and the work of gravity just traded off.

But how do I make the concept mathematically rigorous? I suppose I need functions ##U## and ##V##, so that the total energy is their sum ##E=U+V##, and the time derivative is always zero, ##\frac{dE}{dt}=0##.

But where do I go from here? How do I leap to either

a) ##U=\frac{1}{2}mv^{2}##

b) ##F=-\frac{dV}{dt}?##

It seems to me that if you could get to either (a) or (b), then the rest is just algebra, but I do not see how to get to either of these without being told by a physics professor.