I'm not a physicist but I was attempting to write a story that includes some special relativity shenanigans that I hoped someone could verify.
The gist of it is that people in a spaceship with a 60 day head start need to outrun a signal sent from their starting point, by accelerating for $t_1$ time (as perceived from within the ship) at $5g$ and then coasting at the resulting speed so that they are only reached by the signal after 80 years (as perceived from within the ship).
I found this answer very helpful: What is the proper way to explain the twin paradox? but I believe those trajectories are measured from a rest frame outside of the ship.
It seems to me that you could only start coasting once you were $c*(80*365*24*60*60)$ meters away from the source of the signal, a distance which I believe I can calculate from the above linked answer, but would always take longer than 80 years, from the frame of reference of the source of the signal.
But I'm struggling to understand how long it would take to get that far away when measured from within the ship.
Is it possible to outrun a light-speed signal if you have a head start, accelerate for a while and then coast? Or do you have to keep accelerating, never coasting, to that the signal never reaches you?