How much energy in return force?

Question

Assuming I had a magic device able to discharge its stored energy in a directional force, and absorb the energy from the resulting return force (for every action, there is an equal and opposite reaction) which it could then discharge... What laws of physics are being violated? (basically, energy that would have moved the device, ignoring obstacles like ground, instead just gets stored as magic. This device can freely exchange between kinetic and magic energy, so 1KE absorbed is 1KE of discharge. And it is selective in what it converts to magic, so it is affected by gravity, it does not absorb that).

Looking at it from conservation of energy, this would be 100% efficient energy transfer (from kinetic, to magic to kinetic) so the device should have no power left after the discharge, but since there is energy in the return force... is there no return force or is the return force a fraction of the discharge force? If the later, what is that fraction?

The device just converts kinetic energy, to magic, to kinetic energy with 100% effecency. I just want a reality check on how this device works given that the energy absorb/discharge is the only exception to physics happening.

Just as an example, you charge the device by striking it with a hammer, than place a ball bearing next to the device (physically touching it), the device than discharges its energy into the ball as kinetic energy, and stores the recoil energy from the ball, which can than be used to launch another ball bearing. What would be the speed of each ball bearing? (assuming whatever makes the math easier/cleaner and doesn't violate physics)

Answer 1

There are two important but distinct concepts here: conservation of momentum and conservation of energy. When you talk about absorbing the return force, you're referring to momentum (force is a change in momentum over time).

The momentum of a moving object is $\vec{p} = m \vec{v}$, that is, momentum is mass multiplied by velocity. The arrow over velocity means the direction is important here; "momentum going east" is different from "momentum going west".

Conservation of momentum means that the total momentum of a closed system should remain constant. If one object loses momentum, another should gain it. This leads to Newton's third law: if I apply a change in momentum to something, it applies an equal and opposite change in momentum to me. Add $+p$ to one thing, add $-p$ to another, and the sum stays the same.

Kinetic energy, on the other hand, doesn't have a direction. $E_k = mv^2$. No matter which direction I'm moving, so long as I'm moving, I have positive kinetic energy.

Energy is also conserved, but it's much easier to lose it. In fact, the Second Law of Thermodynamics says you have to lose it. Energy can't be destroyed, but it can be turned into a form that's useless to you, and there's no practical way to prevent this from happening. (Though your magic very well could.)

For an example: suppose I have two metal balls, each weighing 1 kg. I put them on a straight track, then roll one of them east at 1 m/s, and the other one west at 1 m/s.

The total momentum is $p = 1 kgm/s + (-1) kgm/s = 0$. The total energy is $E_k = 1 kg mm/ss + 1 kgmm/ss = 2$.

Now the balls collide with each other, and stick together (a fully inelastic collision). So their velocities are both zero.

The total momentum afterward is $p = 0 + 0 = 0$, same as before. The total energy is $E_k = 0 + 0 = 0$. Where did all that energy go? It still exists, but it went out into the environment: some of it caused the air to vibrate, for example, making sound.

Now, for your magic box. Let's assume the magic box is already charged up with some amount of energy. You take one of my 1kg metal balls, and set it in front of the box. $p = 0$ and $E_k = 0$ since nothing is moving.

The box does its magic, and now the ball is flying east at 1 m/s. It's gained momentum of 1 kg meter per second east. But we know the total has to be zero, so something else needs to gain momentum of 1 kg meter per second west. The easiest solution is to give that momentum to the Earth, where it has no measurable effect because the Earth is so heavy. This is what happens to the equal-and-opposite momentum when you walk.

But the ball has also gained energy. It now has 1 kg meter-squared per second-squared of kinetic energy. Where did this come from? The obvious answer is: the box. But since this energy doesn't have a direction, the box can't "gain" anything in response. It now has a little bit less magic energy in it than before.

So you've expended some stored energy to move an object; in other words, you have a magic gun.

A converter of kinetic energy is a wonderful idea, but alas, there is no energy to be regained from recoil. The most you can get is momentum.