This question already has an answer here:
Say you have two planes flying next to each other at the same speed and one decides to pick up speed by burning a tank of rocket fuel.
If someone on the ground wanted to know that plane's new speed after burning the fuel and knew the plane's mass, initial speed, and energy stored in the fuel, he could calculate it using conservation of energy.
If the pilot of the other plane wanted to calculate the same thing but didn't know his own speed, he could calculate the speed of the first plane relative to himself using conservation of energy--the initial relative speed is 0 and all the work done by the rocket fuel is converted into the plane's new kinetic energy.
He could then relay this relative speed to someone on the ground who knew his plane's speed, but adding the speed of the second plane to the calculated speed of the first relative to the second would give a different value for the first plane's speed than the first guy on the ground would have calculated.
I see why this doesn't work mathematically (a^2+b^2 isn't (a+b)^2), but why can't the second pilot calculate the new speed of the first plane relative to himself by converting the stored fuel energy into kinetic energy? Why when on the ground do we not consider our motion relative to, say, the moon when calculating the first plane's speed relative to us, but the second pilot must consider his speed relative to the ground before calculating the first plane's speed relative to himself? Or maybe to put it better, why is having no movement relative to the Earth instead of relative to the plane's initial speed (the second plane) or anything else the correct way to approach this problem?