# 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?

L = L0v
Δt = γvΔt0
γv = 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 Δtbob = 30/0.9c = 33.33 yrs to reach planet Z
- Δtanna = 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

Staff Emeritus