* why the square: https://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: https://physics.stackexchange.com/questions/27847/why-is-there-a-frac-1-2-in-frac-1-2-mv2 on <Physics Stack Exchange>

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

Why kinetic energy is $mv^2/2$?

Why kinetic energy is mv2/2?

Discussion on "Why kinetic energy is mv2/2?" by Ciro Santilli (@cirosantilli, 34)

Discussion on "Why kinetic energy is $m v^{2} / 2$ ?" by Ciro Santilli (@cirosantilli, 34)