Stuck on Algebra in Time Dilation Problem.

[erincaldwell]

I'm working on a simple time dilation problem: Astronomers discover a planet orbiting around a star similar to our sun that is 20 LY away. How fast must a rocket ship go if the round trip is to take no longer than 40 years in time for the astronauts aboard?

I have set up the problem:

T= L/v=T[o]/√(1-v^2/c^2)
where v is velocity and c is the speed of light and T[o] is proper time.
So:

= [2*(20LY)*(9.5*10^15 m/LY)]/v = 40 years/√(1-v^2/c^2).

Now I need to solve for v. I don't know how to get v alone. I tried squaring both sides and ended up with an equation like T[o]^2/L^2 + c^2 = v^2 , but that doesn't get me the right answer.
Please help! I'm going crazy!

 

[Slinkey]

This is the basic equation that you have written out.

$$\frac{d}{v} = \frac{t}{\sqrt{1- \frac{v^2}{c^2}}}$$

Solving for v gives:

$$v = \frac{cd}{\sqrt{d^2+c^2t^2}}$$

where
c = speed of light constant
d = distance in metres
t = time (on spaceship) in seconds
v = velocity in m/s

At least by my reckoning!

 

[Slinkey]

If you want me to run through the procedure that I used to get the new equation in v then I'd be happy to run through it for you.