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Why kinetic energy is $m v^{2} / 2$ ?

why the square: physics.stackexchange.com/questions/535/why-does-kinetic-energy-increase-quadratically-not-linearly-with-speed on Physics Stack Exchange. Ron Maimon's answer is great, as it relies only on the following staring points:
- energy is conserved
- Galilean invariance
He also offers a symmetry argument considering the case of potential energy.
why the half: physics.stackexchange.com/questions/27847/why-is-there-a-frac-1-2-in-frac-1-2-mv2 on Physics Stack Exchange

Others:

physics.stackexchange.com/questions/156696/why-is-kinetic-energy-defined-as-1-2m-v2

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Ron Maimon