# Problem 4 – Hint 3

Here we´ll provide more detailed instructions on how to make use of Hint 2. To begin with, we introduce the following reference frames and 4-velocities:
Frame (A), the laboratory frame, and its 4-velocity $u$;
Frame (B), the spaceship after the first acceleration stage, and its 4-velocity $v$;
Frame (C), the spaceship after the second acceleration stage, and its 4-velocity $w$.
As a first step, we derive the 4-components of $u$ in frame (C), and the components of $w$ in frame (A). To that end, we make use of the equalities relating the expressions of the 4-invariants in one frame to the their expressions in another frame, e.g. $u^A_tv^A_t-u^A_xv^A_x-u^A_yv^A_y=u^B_tv^B_t-u^B_xv^B_x-u^B_yv^B_y$ (we have dropped the $z$-components as we assume motion in the $x-y$-plane). As a second step, we introduce one more frame;
Frame (D), the spaceship after the fourth acceleration stage, and its 4-velocity $q$. Note that the expression we derived during the first step can also be used to express the components of $q$ in the frame (C) (be careful, it is easy to make a mistake here). The 4-invariants relating $q$ and $u$ can be used to obtain a closed equation for the speed $V$ of the spaceship after the first acceleration stage.

Please submit the solution to this problem via e-mail to physcs.cup@gmail.com. The next hint for the Problem 4 will be published at 13:00 GMT, 26th March 2023, together with the next updates of the intermediate results.