Special relativity, calculate velocity Lets assume that a spaceship travels to a star 8 light-years away, in a time its crew considers to be 8 years. How do we calculate the speed of the ship? Is at as simple as saying: since the star is 8 light-years away and it takes 8 years, they should travel at $v=c$? Or is there any other way to do this?
 A: Let's assume that they go with the speed $v$ wrt their earthy friends. So for them (now onwards 'they' refere to spaceship crew unless otherwise mentioned) the initial distance is $l_0 \sqrt{1-v^2/c^2}$ and the planet is coming towards them at the speed of $v$
. Where $l_0$ is 8 light years. Now in their frame the planet takes 8 years to reach them. 
Therefore, 8 = 8$(c/v)\sqrt{1-v^2/c^2}$
So, $v$ = $c/\sqrt2$.
A: Answer is right, and @Dvij did some nice reasoning. 
Just to explain it, which is easy math (but can be confusing because of the two 8's , which will confuse most people, though one is time and the other distance. They could have been unequal numbers), his equation with the two 8's is conceptually:
8 years = time in their frame = distance/v in their frame
Or 8 y = 8 ly/c = distance/v in their frame 
This is 8 = c x $l_0 \sqrt{1-v^2/c^2}$/v  where we have used Dvij's equation for the Lorentz contracted distance, which is of course right.
That leads to this last equation, and if we set $l_0 = 8$ ly, we get
$\sqrt{1-v^2/c^2}$ x c/v = 1
And then easily v = c/$\sqrt{2}$
That is his answer and it is correct, conceptually and mathematically. He did some mental unit translations between years and ly. 
