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"A space probe is sent to the vicinity of star Capella, which is $D = 42.2 \ light \ years $ from earth. The probe travels with a speed of 0.9930c. An astronaut recruit on board is 19 years old when the probe leaves the earth. What is her biological age when the probe reaches Capella?"

Here is the thing, by working with Lorentz transformations I found that her total biological age at the end of the trip is approximately 24 years old. The answer in my book says 42.5 years, which seems strange.

Thoughts

The first thing I did when I encountered this problem is to define biological time. I assumed (I think reasonably) that biological time is the time observed (Δt') by observers inside the moving reference frame (the probe). And therefore the total biological time would be 19 years + Δt'.

Results and Working

We will consider the classical picture of a stationary reference frame S and one S' (the probe's frame) moving with $ u = 0.9930c $. The Lorentz transformations that connect events between the two reference frames are: $$ x'=γ(x-ut) $$ $$ t'=γ(t-\frac{ux}{c^2}) $$ $$ z'=z,\ \ y'=y $$

By working with the second equation we find: $$ Δt'=γ(Δt-\frac{uΔx}{c^2}) $$

And therefore the total biological time is: $$ 19 \ years + γ(\frac{D}{u}-\frac{uD}{c^2})=19 \ years + \frac{D}{γu} \approx 24 \ years$$ Where am I wrong ? Did I assume something that I shouldn't?

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closed as off-topic by John Rennie, Kyle Kanos, enumaris, ZeroTheHero, Jon Custer Dec 18 '18 at 14:18

This question appears to be off-topic. The users who voted to close gave this specific reason:

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If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Hi Andrew and welcome to the Physics SE! Please note that we don't answer homework or worked example type questions. Please see this page in the site help for more on what topics you can ask about here. $\endgroup$ – John Rennie Dec 17 '18 at 17:07
  • $\begingroup$ It's not homework. I just want to discuss with people where my mistake might be. That's why i presented my workings and line of thought. $\endgroup$ – Unstoppable Tachyon Dec 17 '18 at 17:22
  • $\begingroup$ I see nothing wrong with your reasoning and math, assuming constant velocity. Are you sure there are no extra conditions in the question? $\endgroup$ – Cuspy Code Dec 17 '18 at 17:25
  • $\begingroup$ Thanks for the answer. No, I copied the question exactly from the book (University Physics, Fourteenth edition by Hugh D. Young and Roger A. Freedman, p.1273, problem 37.47). $\endgroup$ – Unstoppable Tachyon Dec 17 '18 at 17:30
  • $\begingroup$ Please read this Meta post for "check my work problems". $\endgroup$ – Kyle Kanos Dec 17 '18 at 17:32