Say I have a bucket of fuel that can produce 150J of energy by combustion. No matter what frame of reference an observer or the bucket of fuel is in, since the configuration of molecules stay the same, the amount of energy produced by complete combustion of the fuel would be 150J.
Say I have a spaceship traveling at 10ms-1 relative to an outside observer A. The spaceship then burns all the fuel in the bucket, converting all the energy produced into kinetic energy without regard to thermal energy loss. Since it originally has 50J of kinetic energy, the additional kinetic energy of the fuel would increase its kinetic energy to 200J which would mean that it is now traveling at 20ms-1, so says A.
However, from the frame of reference moving at 10ms-1 in the same direction as the spaceship relative to A, the spaceship was originally moving at 0ms-1, and after burning all the fuel, now has 150J of kinetic energy, meaning that it is moving at 17.32ms-1. That means that relative to A, the spaceship should be traveling at 27.32ms-1. This is different from the value predicted by A, which seems like a contradiction.
What's wrong with my reasoning?
Answer
The thing is (whatever the exact numbers are) that you forget about the energy of the burnt fuel coming out of the spaceship. That, of course, has bigger energy for observer B than for observer A, and that is where your extra energy goes.
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