I\'m working on a simple time dilation problem: Astronomers discover a planet or
ID: 1965559 • Letter: I
Question
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]/v(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/v(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!
Explanation / Answer
L = L0 (1 - v^2/c^2)^1/2 v = L / T v^2 = L0^2 / t^2 (1 - v^2 / c^2) c^2 T^2 v^2 = Lo^2 c^2 - L0^2 v^2 v^2 = L0 c^2 / (c^2 t^2 + L0^2) after collecting terms in v^2 Now c t = L0 since the total distance is 40 L-y v = c / 2^1/2 = .707 c You must have erred in the algebra since your units are not consistent T0 / L = time / distance which is not consistent with distance / time in the rest of your equation.