I want to visualize the energy required to continue accelerating near the speed of light. If I use some amount of energy to accelerate to 0.5c, what speed will I be going if I use the same amount of energy to accelerate again? I tried taking a stab at calculating this, but I'm stuck by the fact that I don't know enough calculus (which I think I need). I'm making some assumptions based on the Lorentz factor chart I've seen (the blue line in the image below).
Lets say I've traveled from a stationary position where my Lorentz factor is 1, to a velocity where it's 1.2, which is some segment I'll call "a". If I accelerate with the same amount of energy again, I'll arrive at the end of segment "b". Does it take double the energy to reach the end of "b" as it does to reach the end of "a" from a standstill?
If this is all correct, then I think I need help coming up with the calculation for getting the velocity using lengths along the line defined by the Lorentz factor equation.
If this is not correct, then hopefully what I'm trying to accomplish makes sense at least!
Note I'm not interested in the actual energy requirements or the mass of the ship. Assume a perfect ship that accelerates without losing mass.