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I'm starting to study the special theory of relativity, so the concepts and the 'methods' to solve problems are still not very clear in my mind. The question is basically:

A person is running towards a tunnel (which has length $L$) with velocity $v$, at the moment she's about to enter, a photon is emitted on the other side of the tunnel. when the photon and the person encounter each other, the length traveled by the person was $f$. Find $f$ in two different ways: working on the reference frame of the person and working in the reference frame of the tunnel.

I know I have two simultaneous events (in the reference frame of the tunnel): the entry of the person on the tunnel and the photon emission, right? but I have a third one: it occurs after some time interval in the same point in space (where the person encounter with the photon), so here's all I could do:

working in the reference frame of the tunnel: the simultaneous event ((L,0)) as viewed by the tunnel occurs in $x=\gamma(L + vt')={\gamma}L$. The encounter of the person with the photon ((L-f, t')) as viewed by the tunnel in $x=\gamma(L-f + vt')$ (or should it be ct?). I'm really confused of how to start this. I know the right thing to do is using Lorentz transformations, but nothing clear comes to my mind.

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    $\begingroup$ Please define your quantities clearly. The simultaneous what occurs at $x_0$? What are the primed and unprimed coordinates, how is $t'$ defined? Are you aware that simultaneity is relative (frame-dependent)? $\endgroup$
    – Puk
    Oct 28, 2020 at 22:18
  • $\begingroup$ I am, that's why i said i was working on the reference frame of the tunnel to consider simultaneity. my edit should've made it clear. $\endgroup$
    – Pedro Italo
    Oct 28, 2020 at 22:24
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    $\begingroup$ drawing a spacetime diagram usually helps $\endgroup$
    – Andrew Steane
    Oct 30, 2020 at 8:51

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In the tunnel frame, everything works out as in Newtonian mechanics. If the two meet at time $t$, then the person travels a distance $vt = f$, and the photon a distance $ct$.

As we have $L = vt + ct$, we get $f = vt = \frac{v}{c-v}L$.

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