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##\Delta{}E = E_{in} - E_{out}##

I am limiting myself to rectilinear motion.

Suppose we are in a RF moving with a constant velocity ##V##.

Let the system consist of a mass ##m##. The absolute and relative velocities of ##m## are ##V_a## and ##V_r## with ##V_a = V + V_r##.

We burn a fuel releasing energy ##Q## which is completely utilized to do work on the mass ##m##. All this work goes into increasing the velocity of ##m##.

The absolute velocity of ##m## changes from ##V_{a1}## to ##V_{a2}##. The relative velocity of ##m## changes from ##V_{r1}## to ##V_{r2}##.

The change in kinetic energy (KE) as measured in the fixed and moving RFs are:

##\Delta{}KE_a = \frac{1}{2}m\left(V^2_{a2} - V^2_{a1}\right)## and ##\Delta{}KE_r = \frac{1}{2}m\left(V^2_{r2} - V^2_{r1}\right)##

Thus we have: ##\Delta{}KE = Q##.

I think that we measure identical ##Q## in both stationary and moving RFs. But ##\Delta{}KE_a \ne \Delta{}KE_r##.

So where am i going wrong?