[walwathi] Unreliable interstellar engines

At low speeds of interstellar travel, starships need to function reliably for extremely long periods of time.

However, at high speeds, relativistic time dilation decreases the perceived time on board, so ship systems don't need to function that long.

Assuming a ship accelerates at 1g (measured in the ship's frame) for half the voyage and decelerates for the other half, what is the distance (measured in the origin's frame) it can travel as a function of time measured on the ship? I suspect exponential or super-exponential.

Answer (without proof): x(tau) = c^2/a * cosh (a/c*tau)-c^2/a \approx c^2/(2*a)*exp(a/c*tau), where tau = on-board clock; t(tau) = c/a * sinh(a/c*tau).