## Is anyone good at special relativity? the "twin paradox?"

1. The problem statement, all variables and given/known data

In the problem a pilot accelerates away from earth for 10 years, then decelerates the same way for 10 years, then turns around and repeats the whole process to return to earth. His acceleration is given by 2g(1-exp(-rt)) Where g=9.8 and r=1/(3 years). Now I'm stuck finding the pilots (1) time and (2) distance as seen from EARTH.

2. Relevant equations

{c*sinh[(2g/c)(t'+(1/r)exp(-rt')-(1/r)]}/{2g-2gexp(-rt')} All evaluated from 0 to t'

and:

l=int[v(t)dt] where t={c*sinh[(2g/c)(t'+(1/r)exp(-rt')-(1/r)]}/{2g-2gexp(-rt')} and v(t')=ctanh[(2g/c)(t'+(1/r)exp(-rt)-(1/r)]

3. The attempt at a solution

(1)This one I've mostly gotten right I believe, my teacher told me to plug in all times as seconds, so for t' i put in 10 years=3.156*10^8sec, then do I need to put r in as seconds too? that would make it 1/(94670777.9 sec)? After evaluating the integral I got that it took 582016 years, which I multiplied by 4 to get the entire trip to be 2.33*10^6 years. Is this right? I'm almost certain that is right but I'm just checking ;)

(2) Now to get the distance as seen by earth i need l=int[v(t)dt], but I only have v in terms of t'. So I used t={c*sinh[(2g/c)(t'+(1/r)exp(-rt')-(1/r)]}/{2g-2gexp(-rt')} to try to get v(t')=ctanh[(2g/c)(t'+(1/r)exp(-rt)-(1/r)] in terms of t alone. But no matter how hard I try, there is still at least ONE t' leftover in my v(t) equation, always in the form of exp(-rt') How do I get rid of this?? I'm so lost! :( Any help I would be eternally grateful for! Thanks so much in advance! :)

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 Tags dilation, paradox, rocket, special relativity, twin