Time Dilation and Length Contraction Problem

[ddobre]

Homework Statement

Anna and Bob are both born just as Anna's spaceship passes Earth at 0.9c. According to Bob on Earth, Planet Z is a fixed 30 ly away. As Anna passes Planet Z on her continuing outward journey, what will be (a) Bob's age according to Bob, (b) Bob's age according to Anna, (c) Anna's age according to Anna, and (d) Anna's age according to Bob?

Homework Equations

L = L₀/γ_v
Δt = γ_vΔt₀
γ_v = 1/√1-v²/c²

The Attempt at a Solution

- Anna experiences length contraction in her frame: L = 30√.19 = 13.08 ly to reach planet Z

a) to Bob, Anna takes Δt_bob = 30/0.9c = 33.33 yrs to reach planet Z
- Δt_anna = 13.08/0.9c = 14.5 yrs

b) Bob's age according to Anna: Δt = 14.5/√.19 = 33.3 yrs

c) Anna's age according to Anna: 14.5 yrs

d) Anna's age according to Bob: Δt = 33.33/√.19 = 76.5 yrs

I think my issue is that I'm not taking into account the time dilation effect experienced by Bob, but the problem confuses me because I'm not quite used to reference frames and the relative effects. Some guidance would be appreciated

 

[Orodruin]

b) Bob's age according to Anna: Δt = 14.5/√.19 = 33.3 yrs

This is not correct. In Anna’s rest frame, Bob is moving and should therefore be time dilated.

d) Anna's age according to Bob: Δt = 33.33/√.19 = 76.5 yrs

Same problem. The proper time of a moving observer is less than the coordinate time.

Also note that in Anna’s rest frame, she does not reach Z, it is Z that is reaching her. Anna is not moving in her rest frame.