Is anyone good at special relativity? the twin paradox?

[khfrekek92]

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

Homework Statement

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.

Homework 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)]

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! :)

 

[mark726]

Yes, I am familiar with the twin paradox and special relativity. I have worked on similar problems and I would be happy to assist you with your questions.

Firstly, for your first question about the pilot's time as seen from Earth, your approach seems to be correct. However, I would recommend using consistent units throughout your calculations, so it would be better to convert the r value to seconds as well.

For the second question about the distance as seen from Earth, you are on the right track by trying to get v(t) in terms of t alone. One way to do this is by using the identity: exp(-rt') = 1/exp(rt), which can help eliminate the t' term in your v(t) equation. You can also try using the chain rule to manipulate the equation and eliminate the t' term.

I hope this helps. Let me know if you have any further questions or if you need more clarification. Good luck with your calculations!