Calculating Time Dilation for a Spacecraft Traveling to Proxima Centauri

[grouper]

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))=Δt_o/√(1-v²/c²) where Δt_o=time observed by person on spacecraft

Δt_o=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]

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

I don't quite understand the first part of that equation. Try:
v = Δx/Δt = Δx/(γΔt₀)

 

[grouper]

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

 

[Doc Al]

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.

 

[grouper]

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.