How Does Time Dilation Work? Solving a Space Journey Time Question

[jewels04]

Homework Statement

im not sure how to work this equation out ! please help!

imagine that one pf a pair of twins takes off on a long space journey to Vega, 25 Light years away, at a speed, relative to Earth, of 99.5% of c (gamma = 10). once there he decides he doesn't like the Vegans, so turns around and comes straight back at the same speed.


(a) how long, in Earth's frame, does it take for the traveler to reach Vega?
(b) as seen by the Earth twin, how long does the trip take the traveler?
(c) how long does it take the traveler in his reference frame?
(d) assuming a negligible turnaround time, how long did the trip take in the Earth's frame of reference?
(e) how long did the trip take the traveler?

Homework Equations

i know that the time dilation equation but I am not sure on how to use it for the different frames of references.

The Attempt at a Solution

 

[Nate810]

(a) 25 light years / 0.995c = 25.1262512625126252 light years/c, so it would take the traveler 25.1262512625126252 years to reach Vega in Earth's frame of reference.(b) using the time dilation equation: T0/T = sqrt(1 - v2/c2), T0 = 25.1262512625126252 x sqrt(1 - 0.9952/c2) = 25.1262512625126252 x sqrt(0.0050) = 25.1262512625126252 x 0.0712 = 1.788333738191687 years, so it would take the traveler 1.788333738191687 years to reach Vega as seen by the Earth twin.(c) using the time dilation equation: T/T0 = sqrt(1 - v2/c2), T = 25.1262512625126252 x sqrt(1 - 0.9952/c2) = 25.1262512625126252 x sqrt(1 - 0.9952/c2) = 25.1262512625126252 x 0.0712 = 1.788333738191687 years, so it would take the traveler 1.788333738191687 years to reach Vega in his own frame of reference.(d) assuming a negligible turnaround time, the trip would take 25.1262512625126252 x 2 = 50.2525025250252504 years in the Earth's frame of reference.(e) using the time dilation equation: T/T0 = sqrt(1 - v2/c2), T = 50.2525025250252504 x sqrt(1 - 0.9952/c2) = 50.2525025250252504 x sqrt(0.0050) = 50.2525025250252504 x 0.0712 = 3.576667476383174 years, so it would take the traveler 3.576667476383174 years to complete the round-trip.