This is a great example problem. I used it during lecture today for an example of generating a 1st order model. I mostly worked through the problem myself, but I used some polling questions about what forces to include, which forces were and weren't constant. I also had the students think about and discuss how to find the burn time (some used dimensional analysis, some made a differential equation).

During class a student pointed out one error: the force sum should equal $$\dot{m}v+m\dot{v}$$ because both mass and velocity are changing with time. I had written just $$m\dot{v}$$, as the posted activity Teacher version also does. The good news is that this doesn't stop it from being a linear equation! (Just a little more complicated.)

When looking carefully at the posted files, I also discovered a second minor error: because this is a linear drag model, and not a quadratic drag model, you don't actually need a new FBD nor a new model equation during the down-ward velocity portion of the flight. The drag force "automatically" changes direction when the velocity does.