# Special Relativity: Time Dilation Prob

1. Feb 4, 2017

### ChrisJ

It has been 2.5years since I last did any special relativity so am rather rusty on it, I have a simple time dilation problem and its making my head hurt which way around it should be. Any help much appreciated!

1. The problem statement, all variables and given/known data

Bob leaves Sarah on earth and travels in a spaceship at 0.8c in a straight line to planet Bongo and then turns around and travels now at 0.9c back to Earth. To Sarah on Earth, planet Bongo is 1lyr away. Ignoring any acceleration effects, by how much have Bob and Sarah aged when Bob returns to Earth.

2. Relevant equations
$\gamma = \frac{1}{\sqrt{1-v^2/c^2}}$
$\Delta t ' = \gamma \Delta t$

3. The attempt at a solution

Originally, I had Sarah's frame as the rest frame, and therefore if travelling at 0.8c it should take 1/0.8=1.25 years to get to Bongo and 1/0.9=1.1 years to get back.

The gamma factor for the outbound journey is $\gamma_o = \frac{1}{\sqrt{1-0.8^2}} = 1.6$ and for the return journey is $\gamma_r = \frac{1}{\sqrt{1-0.9^2}} = 2.29$

Without looking at the equations, or drawing a spacetime diagram, I reminded myself of the twin paradox and similar problems, that if one travels on a space ship at close to light sped and returns, everyone on earth is older than expected. Remembering this I thought, well that means

that for Bob it has been $1.25+1.1=2.35$ years whilst for Sarah its been $(1.25)(1.6) + (1.1)(2.29) = 4.5$years

But then after drawing a space-time diagram, I think that was wrong and that it is actually that for Sarah it has been $1.25+1.1=2.35$ years whilst for Bob it has only been $\frac{1.25}{1.6}+\frac{1.1}{2.39} = 1.26$ years.

And now I am pretty certain its the latter, but am self doubting. As I said its been almost 3 years since I have I had to think about SR or done any SR problems. Any help is much appreicated.

Last edited: Feb 4, 2017
2. Feb 4, 2017

### PeroK

If you analyse the problem in Sarah's frame, then SR doesn't enter into her calculations. It's a simple $t = d/v$ problem!

3. Feb 4, 2017

### ChrisJ

Yeah that is how I started, as you can see that is what I did to get the 1.25yrs and 1.1yrs, but then somehow I got confused. But I did realise my mistake and by the time I had finished writing this post was certain that my second attempt was correct, but as I spent the time to write it out I thought I may as well post it.