How you can put an unlimited amount of momentum into something without using any energy.

As a child I once stood on a dock and pushed on the side of a large fishing boat and noticed it move about a foot away from the dock, after I had pushed for about a minute.

Momentum is proportional to mass times velocity, while kinetic energy is proportional to mass times velocity squared.

I realized at some point that I had put a lot of momentum into the boat, because change of momentum is given by force times time, but very little kinetic energy, because change of kinetic energy is given by force times distance. The time of pushing was great, while the distance was very little.

In fact if the mass of the boat had been infinite, and I had pushed for years, I would have put a vast amount of momentum into the boat, but no energy into it at all.

Had I placed a strong spring between the dock and the boat, all of the above would still apply. If the mass of the boat were large enough the spring would sit there putting momentum into the boat without putting in any energy.

Momentum is conserved because the spring puts an equal and opposite amount of momentum into the dock and whatever it is rigidly attached to, in this case the Earth. Newton’s third law of motion requires that the force exerted by the spring on the boat is equal and opposite to the force it exerts on the dock (this ignores the mass of the spring and a few other things we don’t need to worry about here).

My favorite activity is learning new things.