Physics Cup – TalTech 2023 – Problem 4

By Jaan Kalda (TalTech).

A spaceship takes off from the Earth at $t=0$, and henceforth keeps the modulus of its proper acceleration equal to $g$; here and in what follows, $t$ denotes the spaceship’s proper time. Thus, the astronauts on board will always feel a constant free fall acceleration $g$. However, the direction of the proper acceleration is changed four times at equal intervals, by turning the engines counterclockwise by $90^\circ$. So, the proper acceleration is:
parallel to the $x$-axis when $0\le t< \tau$;
parallel to the $y$-axis when $\tau\le t< 2\tau$;
antiparallel to the $x$-axis when $2\tau\le t< 3\tau$;
antiparallel to the $y$-axis when $3\tau\le t< 4\tau$.
It turns out that the spaceship’s speed relative to the Earth takes exactly the same value $v$ at $t=\tau$, and $t=4\tau$. Find this value $v$.

Please submit the solution to this problem via e-mail to physcs.cup@gmail.com. The first intermediate results for Problem 4 will be published on 5th March, 13:00 GMT. The first hints will appear here on 12th March 2023, 13:00 GMT. After the publication of the first hint, the base score is reduced to 0.9 pts. For full regulations, see the “Participate” tab.