Special Relativity - Length Contraction Problem

[DougD720]

Homework Statement

The radius of our galaxy is 3x10^20m (30000 lightyears).

(part a): Can a person in principle travel from the center to the edge of the galaxy in a normal lifetime? Explain using Time-Dilation, then Length-Contraction.

Homework Equations

d\tau = dt \sqrt{1 -\beta²}

L = \acute{L} / \gamma

The Attempt at a Solution

Okay so I've done part a - 1 which uses time-dilation and that went fine, part b asked for the velocity required to make the trip in 30 years, did that, no problems, but this length contraction is killing me. I've got a stack of (now) scrap legal-pad pages trying to figure this thing out. I think what I really need to know is where to plugin the radius of the galaxy (i tried using that as both the proper length and the observed length, but I'm not sure of which, both are giving me non-sensical answers), and short of using the beta I calculated from the time-dilation (0.9999499) i don't know how to get beta, I tried plugging in values for v such as replacing v with (displacement/time) but that didn't work because I don't know whether to use the radius of the galaxy as the displacement, or what. I don't need anyone to do the math for me, just someone to point me in the right direction of the values to plugin where.

Also, I considered a normal lifetime 100 years.

Thanks for the help!

 

[kuruman]

If you did the time-dilation part and it went fine then you have gamma. Now imagine that you are in the spaceship traveling with the speed implied by this gamma relative to the galaxy. As far as you are concerned, the galaxy is shorter than 30,000 light years by that same factor gamma.

 

[DougD720]

Thank you! I just didn't know if I could use the gamma from the time-dilation part, but yeah, duh, that makes perfect sense. Thanks!