Why does accelerating in two frames use less energy than accelerating once?

[rcgldr]

I never claimed that. $\Delta E_1$ just meant the total change in kinetic energy in the first frame (for the first half of the acceleration). Similarly, $\Delta E_2$ was the total change in kinetic energy over the second half of the acceleration calculated using the second reference frame.

Ok, I understand now, I mistook E1 and E2 to be Ecar and Eearth.

 

[jacobrhcp]

I've analysed my original problem and have come to the following answer:

The thing with this paradox is that it assumes you can change frames in the middle of a problem. You can look at things in one frame, or another. And this does not mean that because they both are valid that it's still valid if you look at it in two different ways.

You can summarize all kinds of ways to change the frames legally, and make it two separate problems. But the real thing is just that physics is not in such a way you can think of it from two frames at the same time, and still see something sensible. You will need conversions. And that's okay, because no one ever said you could do without.