A spaceship departs from the origin of the LCF Σ and moves along the $x$-axis with constant proper acceleration $a = 10 m/s^2$ until it attains the speed $v = 0.5 c$. Calculate the time the spaceship moved in Σ and in its proper frame.
Now,for constant accelerations we have $$u = c \sinh(a \tau /c)$$ So at first i thought that the answer for proper time would simply be $$\tau = 3*10^7* \text{arcsinh}(0.5) \approx 0.45 \text{yr} $$
But this is, apparently, wrong. The answer is, actually, $0.52 \text{yr}$. What exactly have i done wrong?
