# Time Dilation for spaceship

## Homework Statement

The nearest star to Earth is Proxima Centauri, 4.3 light-years away. At what constant velocity must a spacecraft travel from Earth if it is to reach the star in 3.7 years, as measured by travelers on the spacecraft? How long does the trip take according to Earth observers?

## Homework Equations

Earth observer: Δt=x/(c*√(1-v2/c2))=Δto/√(1-v2/c2) where Δto=time observed by person on spacecraft

Δto=3.7 yr=1.164e8 s

x=4.3 ly=4.068e16 m

c=3.0e8 m/s

## The Attempt at a Solution

I tried using the two versions of the equation above with the known quantities plugged in such that (1.164e8/√(1-v2/c2)=4.068e16/(c*√(v2/c2)), but this is a false statement, so there must be something wrong with my equations or the way I am using them because that method does not yield an answer. Any help is appreciated, thanks.

Doc Al
Mentor
Earth observer: Δt=x/(c*√(1-v2/c2))=Δto/√(1-v2/c2) where Δto=time observed by person on spacecraft
I don't quite understand the first part of that equation. Try:
v = Δx/Δt = Δx/(γΔt0)

Thanks for the equation corrections! How do I figure out v without knowing Δt though? (or vice versa?)

Doc Al
Mentor
Thanks for the equation corrections! How do I figure out v without knowing Δt though? (or vice versa?)
You know Δt0. Express γ as a function of v. Then you can solve that equation for v, the only unknown.

Ok, working that out I got v=0.759c and Δt according to the Earth observers is 5.68 years, both of which are correct. Thanks for the help.