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A star is 95.0 year lights away from Earth. How much time does it take to a cosmic ship, which moves with speed 0.96 c, to reach the star, if it is measured from a watcher from a) Earth a)ship c)What is the distance of trip, based on the viewer from the ship?

a) t=L/v=99 years

c) $Lr=L*sqrt(1-v^2/c^2)=26.6 years$

b) tr=Lr/v=27.7 years.

I'm not sure if for this I should use $tr=t/sqrt(1-v^2/c^2)$. Which one is the best to use because the results are different. But if I use $tr=t*sqrt(1-v^2/c^2)$ it has the exact same result. If 27.7 years is the correct answer, then why should the length contract and time not dilate?

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So when the spaceship moves fast relative to us, there are two effects going on:

  1. They really see the star as closer than we see it, and
  2. We really see their clocks appear to be ticking slow.

These two effects need to perfectly balance out, because I can either measure the time for the spaceship to travel to the star directly, or I can measure it by looking at their clock and then correcting for time dilation.

So just to make these numbers a little more easy than what you've got, if in one reference frame the start and end points stay a constant 90 light-years away from each other, and you travel with a $\gamma = 3$ (roughly 94.28% the speed of light) relative to that reference frame, due to length contraction you will see the distance of the journey as only 90/3 = 30 light-years, which you will cross in a little over 30 of your years (really 30/0.9428 = 31.82). Someone who sees the start and end points at rest will say it instead takes you a little over 90 of their years to travel (really 90/0.9428 = 95.46) which can be calculated by taking your travel-time and dilating it by this factor of 3.

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  • $\begingroup$ We have learnt two formulas Δtp=Δt/γ and Lp=L*γ, but now I am understanding that for the same problem tp and Lp are two different systems. Am I getting this right? In this problem I see that Lp is the system related to Earth and tp is the system related to the spaceship. Please correct me if I am wrong $\endgroup$
    – prishila
    Commented Feb 12, 2017 at 17:01
  • $\begingroup$ @prishila For this case yes, proper length and proper time are measured by the two different reference frames. In general a "proper time" between two events is the time seen by an observer who thinks that both happened at the same place. So for the travelling spaceship it says "both of those stars were in the same place, 'right here next to me'." Similarly a "proper distance" between two objects which are not moving relative to each other is measured by an observer who is not moving relative to either object. $\endgroup$
    – CR Drost
    Commented Feb 12, 2017 at 22:55
  • $\begingroup$ I understood the proper distance, but not the proper time. In the problem there is only one star. Why did you say two stars? $\endgroup$
    – prishila
    Commented Feb 13, 2017 at 15:49
  • $\begingroup$ @prishila One of them is the Sun; the spaceship travels from our solar system to the other one. In general I like stars because they're basically at a well-defined "point" of space that everyone can agree on (more properly, a well-defined worldline of spacetime that everyone can agree on), even if they're moving relative to the star. $\endgroup$
    – CR Drost
    Commented Feb 13, 2017 at 16:00
  • $\begingroup$ So if an object is moving with a speed relative to c, then the time he measures is the proper time? If not, please give me an example when this doesn't happen. $\endgroup$
    – prishila
    Commented Feb 13, 2017 at 16:27

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