I can't ELI5 (don't like such things anyway) but if you know calculus, the v^2 follows directly out of trying to integrate momentum (or quantity of motion, a fu

I can't ELI5 (don't like such things anyway) but if you know calculus, the v^2 follows directly out of trying to integrate momentum (or quantity of motion, a fun phrasing) with respect to velocity. Momentum is mv, units are kilogram * meter/second. Integrate with dv, you get 1/2 mv^2, units are kilogram * meter^2/second^2. (Double-check, take the derivative of that with respect to v, and you get mv.) This 1/2 thing times otherthing^2 relationship actually shows up all over the place in math and physics, it's quite beautiful, and incidentally another reason to prefer using tau=2pi instead of pi...