Why does kinetic energy increase with the square of velocity?

I was thinking kinetic energy would behave similarly to momentum. For example a smaller projectile could have much more kinetic energy than a larger one but less momentum. I don't know that doesnt make much sense to me. If anyone could help me understand it would be much appreciated.

Public Comments

1. It's just the definition of kinetic energy, which is just a mathematic formula, not some magical quality of motion.
2. Actually you are not correct (speaker to first responder). The expression for kinetic energy is derived using calculus. It is a simple derivation and I cannot show it here, because there is no way to write integrals on this keyboard. But, you take momentum and do the integral from Vi to Vf and you get Integral(mv) = (1/2)m*v^2 Calculus: It comes in handy sometimes. Recall that you use the integral because kinetic energy is simply summing up all the little bits of momentum vectors at differential steps.
3. You don't need calculus, just Newton's equations of motion:- f=ma and v^2 = u^2 + 2as, along with the definition of the joule. Work done = energy transferred f x s = gain in KE u = 0 so v^2 = 2as so s = v^2 / 2a so f x s = ma x v^2 /2a = mv^2 / 2