Honestly, I made that comment without fully thinking through how to implement my suggestion, and now I realize that I've stumped myself. However, let me explain what I meant anyway:
First of all, you're right that this won't really impact your answer at insignificant fractions of light speed. You mentioned that using the 4.72% growth rate, the equation tells you to wait until you've passed the speed of light, and I thought it might be interesting to more accurately model the energy required at relativistic speeds.
So the same way that you used the classical mechanics equation for kinetic energy, E=mv^2/2, ignoring mass and solving for v, to get v=sqrt(2E) and approximating to v=sqrt(E), I thought you could manipulate the relativistic equation similarly.
Now, having gotten a solution from WA, I'm starting to think that I overestimated the effect on accuracy that changing the equation would have. I want to approximate the solution by ignoring some terms or changing an nE to an E^2 or something, but I think that might negate any gain in accuracy.
So to answer your questions more directly, I was attempting to address the error you get when the calculation tells you to wait until you can travel above or near the speed of light, and I only included mass in my equation to try to communicate it to you more accurately, with the assumption that when actually using it you would ignore the mass.
Anyway, I hope I at least clarified my previous comment, even if it turned out not to be very useful! If anyone has a better understanding of how to better model relativistic speeds I'd love to hear their explanation.
Btw, if you measure the time as experienced by the traveller, the classic equation will give you exactly the right answer:
Pouring more energy into acceleration won't make you move faster (even subjectively) but it will shorten the way. (From the outside, it looks like time dilation.)
First of all, you're right that this won't really impact your answer at insignificant fractions of light speed. You mentioned that using the 4.72% growth rate, the equation tells you to wait until you've passed the speed of light, and I thought it might be interesting to more accurately model the energy required at relativistic speeds.
So the same way that you used the classical mechanics equation for kinetic energy, E=mv^2/2, ignoring mass and solving for v, to get v=sqrt(2E) and approximating to v=sqrt(E), I thought you could manipulate the relativistic equation similarly.
Now, having gotten a solution from WA, I'm starting to think that I overestimated the effect on accuracy that changing the equation would have. I want to approximate the solution by ignoring some terms or changing an nE to an E^2 or something, but I think that might negate any gain in accuracy.
So to answer your questions more directly, I was attempting to address the error you get when the calculation tells you to wait until you can travel above or near the speed of light, and I only included mass in my equation to try to communicate it to you more accurately, with the assumption that when actually using it you would ignore the mass.
Anyway, I hope I at least clarified my previous comment, even if it turned out not to be very useful! If anyone has a better understanding of how to better model relativistic speeds I'd love to hear their explanation.