A slight quibble that maybe is an option for further work. The "escape velocity" as reported on space launches refers to a scenario where a rocket is expending fuel and at some point the commentator says: "we have reached escape velocity and ...".

Thus your example, though extensive, and I had to work forwards & backwards needs to consider such a scenario. Further, this relates to a question my father asked which was: "Surely if the rocket went at 1m/s continuously then it would not need to reach "escape velocity".