It's been a while since I played with special relativity, I seem to have forgotten most of it. And right now, I'm having a problem that has acceleration thrown in, and it's doing weird things to my brain.

The first question - constant acceleration. Let's assume a spacecraft constantly accelerates with 1G until it reaches 0.9c. Obviously it's the guy in the spaceship that wants to accelerate with one G, for comfort and stuff. But... as time slows down for him, those 9.81 m/s^2 wouldn't be 9.81 m/s^2 to the observer, because his time is running slower. Except there's the lorentz contraction from the spaceship's point of view, which I rather unsure how to calculate after all this time. Do the length contraction and the time dilation cancel each other out, so the acceleration is determined to be the same to both the observer and the observed, or do they actually measure different accelerations?

Second question - and a rather troubling one, I'm afraid: How

Can somebody give me a function t=f(v), where v is a given velocity that was reached by accelerating with 1G from 0, and t is the relative time elapsed on the ship?

That is, if my above assumption is correct that both measure the same acceleration. If not, this is becoming nightmarish...

Huh... Looks like the atomic rockets website has quite a comprehensive collection of equations for this. Because of course it has, why didn't I check that first?

The first question - constant acceleration. Let's assume a spacecraft constantly accelerates with 1G until it reaches 0.9c. Obviously it's the guy in the spaceship that wants to accelerate with one G, for comfort and stuff. But... as time slows down for him, those 9.81 m/s^2 wouldn't be 9.81 m/s^2 to the observer, because his time is running slower. Except there's the lorentz contraction from the spaceship's point of view, which I rather unsure how to calculate after all this time. Do the length contraction and the time dilation cancel each other out, so the acceleration is determined to be the same to both the observer and the observed, or do they actually measure different accelerations?

Second question - and a rather troubling one, I'm afraid: How

*do*I calculate how much time passes during the acceleration phase on the space ship? It's easy enough for constant velocity, but for an acceleration phase, it seems to be rather difficult. I found this paper here, but I can't handle integrals: https://www2.math.uconn.edu/~bridgeman/posts/acceleration.pdfCan somebody give me a function t=f(v), where v is a given velocity that was reached by accelerating with 1G from 0, and t is the relative time elapsed on the ship?

That is, if my above assumption is correct that both measure the same acceleration. If not, this is becoming nightmarish...

Huh... Looks like the atomic rockets website has quite a comprehensive collection of equations for this. Because of course it has, why didn't I check that first?

### Nearlight Starships - Atomic Rockets

www.projectrho.com

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