When I think of energy, it seems intuitive to me that an object moving has energy; kinetic energy to be specific. But mass X velocity does not give the units of energy, mass X velocity^2 does. Now, considering the equation for kinetic energy:

KE = 1/2mv^2

I understand (hopefully correctly) that the "1/2" is used to average the initial and final states of something, though I'm not sure exactly what is being averaged. Anyway, this is likely an elementary question to you guys, but what exactly does "v^2" represent, and a related question is, why does "v" alone not define energy (aside from the obvious reason that the units don't add up).

Don't know if this could help you.

Let's say you have two steel balls moving frictionless on a plane, they have the same mass m but different speeds: v and 2v, and you want to stop them with the same force F.

If you need a time t for the one with speed v, then you need a time 2t to stop the one with speed 2v, because V = a*t and a is the same: a = F/m.

Now, why the work you make on the second ball is not 2 times the work you make on the first, but it's 4 times instead?

Work is F*S where S is the space covered. In a uniformly accelerated motion, S = (1/2)*a*t

^{2} so if time doubles, space becomes 4 times.

So, F is the same but S is 4 times and for this reason the work you have to make to stop the second ball is 4 times the one required for the first.