I fully accept that kinetic energy is not invariant between Galilean frames of reference, velocity is. So the same change in velocity requires different work (i.e. change in kinetic energy) done in different frames.
But what I'm not clear on is where the energy is "lost" when considering the source of the work, like fuel.
For example, consider two rockets of mass $M$ flying side-by-side towards an observer with a speed $v$. Vacuum, spherical cows, all that jazz. So their $KE = Mv^2$ for the observer, $0$ for eachother.
Now one of those rockets changes it's speed by $v$ again, exchanging mass of fuel $m$ into work. That mass of fuel has some intrinsic energy to be extracted that everyone can agree on when everything is at rest, like for ex. rocket fuel.
From the point of view of the other rocket, the accelerated rocket's $W_1 = KE = (M-m)v^2$ (at least I hope that's correct).
From the observer's point of view that's now $W_2 = KE_2 - KE_1 = (M-m)4v^2 - Mv^2 = (3M-4m)v^2$
Even if we assume the m is small but non-negligable, like 5%$M$, the ratio in work done still approaches 3. Even at $m=0.5M$, $W_2$ is two times $W_1$. But there was non-variable amount of energy in the fuel. So where is this fuel energy "wasted" between the frames of reference? Propulsion efficiency? Somewhere else?
