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I have a question regarding how to calculate the energy required to move objects within the frame of special relativity. To that extent, I understand the mass of an object as it approaches the speed of light increases, which I would assume necessitates an increase in energy. How would one calculate the energy required to move an object at a constant proper acceleration? For example, if an object had a rest mass of 10 kg and a constant proper acceleration of 10 ms^-2 and the goal was to accelerate it to 99% of the speed of light, how would one calculate the energy it would require? I believe it has something with the relationship of the rate change in energy over space being equivalent to the relativistic force, but I could definitely be wrong.

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Just calculate the KE at the speed desired. By the work-energy theorem, that's the work necessary to accelerate the object to that speed. The only tricky thing is to use the correct relativistic formula for KE, namely (gamma - 1) * mc^2.

The beauty is that this avoids all complications regarding how you did the accelerating -- that's the power of energy methods.

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