I have a projectile in 2D space $r=(x, y)$ at time $t_0=0$, which has an initial velocity $v_0$, a launch angle $\theta$ from $(1, 0)$ and which accelerates with a constant $a_0$ until time $t_1$ in the current flight direction of the projectile as well as a constant $g$ downwards $(0, -1)$.
This is, for example, a simplified model of a rocket with a short-lived motor, ignoring changes in mass from the propellant and any air drag.
I'm looking for a definition of the flight trajectory, so that I can determine functions describing the angle $\theta$ to hit a point $(x, y)$, the time to get there, and similar. I only found https://cnx.org/contents/--TzKjCB@8/Projectile-motion-on-an-incline so far. I planned to use the given formulas there to piece together a case distinction based on whether the time to target is smaller or larger than $t_1$, but I am not really sure how to connect the "ends" of the two cases and with the acceleration vector changing over time, I don't know if this can even be done in this way.