If you push a ball, you are using energy to do so, and you are also increasing the momentum of the ball. So the pusher now has less energy and the ball has more momentum, so wouldn't momentum be the same as kinetic energy? Why is this wrong?

The big difference between momentum and kinetic energy is that if you double the velocity, you double the momentum, but you quadruple the kinetic energy. At three times the velocity you have triple the momentum but the kinetic energy has increased ninefold. In other words when your vehicle hits the wall at 30 mph it will have 9 times the impact that it would at 10 mph. On the other hand, it would only coast three times as far.

Not exactly wrong, but a little mixed up. The ball, when shoved, gains not only momentum, but also kinetic energy. If the law of conservation of energy is true (and there's little reason to doubt it), the kinetic energy gained by the ball is precisely equal to that lost (in the form of what? stored up fuel?) by the pusher in his push. That is, of course, neglecting energy lost due to friction and air resistance and so on.

[This message was edited by maiku on 08-15-02 at 06:53 PM.]

Energy is the ability to do work, which is force times distance (F*ds).
Momentum is defined as mass times velocity (m*v).
Mr Isaac Newton discovered in his second law of motion that force is equal to the rate of momentum change with time, F=m*dv/dt, for constant mass. Now since distance ,ds, = v*dt, and incremental KE of motion = F*ds, then substituting, KE= (m*dv/dt)v*dt = m*v*dv. Integrating , KE=1/2m*v squared.