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I know it's a big and confusing title, if someone could improve it I would appreciate.

My question is a thought experiment.
If kinetic energy is relative to other objects and the total energy of a system doesn't change, what would happen if in a completely empty universe, besides the existence of a simple rocket, this simple rocket starts it's engine?
Well, the combustion process would turn the potential energy of the fuel into mechanical energy, pushing combustion products to one side and, through Newton's third law, the rocket to the other. So the rocket gained energy, and this was supposed to be kinetic energy, but the hypothesis is that there are no other objects in this universe, so there's no relative position to compare the rocket's position. So where did this missing energy went?
What am I missing?

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Rockets work by expelling exhaust gas at a high velocity, which, by conservation of momentum, causes the rocket to move in the opposite direction. Since there are no external forces acting on the rocket-exhaust system, its centre of mass gives an inertial frame of reference. This is the reference frame in which we can measure the change in kinetic energy of the rocket (and exhaust).

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When you fire a gun, you feel recoil, because the momentum imparted upon the bullet must have an equal and opposite momentum imparted upon the body of the gun itself. In short, you pull the trigger, the tiny bullet shoots forward, and you and the gun together as a single system gain an equal momentum distributed over the far larger mass pushing you backward.

A rocket operates on the same principle, except in reverse. Instead of applying force to the gun as a side-effect of applying force to the projectile (the bullet), you apply force to the projectile, in this case exhaust gases, as a side-effect of applying force to the rocket. The gases shoot downward with a momentum equal and opposite to that of the rocket, which shoots upward, assuming you're trying to get your rocket to space, that is.

This loss of exhaust mass is what allows a rocket to move, without which you would be correct, and there would be some difficulty explaining the motion of the rocket within the confines of Newtonian mechanics.

As for frame of reference, you can attach your frame to the rocket itself, in which case you have an inertial frame of reference that needs to take account of the loss of mass to the exhaust. Or you can be a bit more adventurous and attach your frame to the exhaust gases, which is valid if done for small slices of time, but far more difficult if you want to account for the total length of time the rocket is in motion. Or you can attach your frame a point in space where the rocket is at a given motion, and you'll notice two equal and opposite momenta being applied to the rocket going one way, and the gases going the other, though clearly the gases will be traveling at a much higher speed. Up to you as to how far you want to take it!

Finally, to address your last question, your empty universe starts out with a rocket, complete with the fuel that it will then burn and expel as exhaust gases. These gases don't disappear from the universe as soon as they leave the rocket, and you're left with an almost empty universe, containing a rocket, and a bunch of fast-moving gas. That's what you're missing.

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It went to the kinetic energy of the rocket exhaust

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  • $\begingroup$ But the rocket exhaust is attached to the rocket body. So you would say that the whole rocket gained kinectic energy, right? But how does it have kinectic energy if there's no referential? $\endgroup$
    – Teodoro
    Aug 16, 2020 at 1:04
  • $\begingroup$ The rocket exhaust gases are “thrown” out in the opposite direction. The kinetic energy of rocket exhaust gases are “equal and opposite” to the kinetic energy gained by the rocket. $\endgroup$ Aug 16, 2020 at 1:07
  • $\begingroup$ Right, but can we talk about kinectic energy of the rocket if there's no referential? $\endgroup$
    – Teodoro
    Aug 16, 2020 at 1:12
  • $\begingroup$ Once the engines fired there would be the rocket AND it's exhaust gasses , also kinetic energy is frame dependent $\endgroup$ Aug 16, 2020 at 1:14
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    $\begingroup$ The referential in this case is the exhaust gases of the rocket. The exhaust gases, taken all together as a whole, are travelling in the opposite direction with a “backward” momentum equal to the rockets “forward” momentum. $\endgroup$ Aug 16, 2020 at 1:16

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