Do you know of example advances that have been made or problems that have been solved as a consequence of applying the concept implied in that formula?
Rockets
Care to elaborate?
Rockets have an exhaust with constant velocity, and they expel mass in the form of fuel. It's this expulsion of mass that generates the force that propels the rocket. Taking a force balance and assuming no external forces (e.g. gravity), you can integrate to get the rocket equation, which tells us how much fuel mass we need to reach a given velocity (the quantity we care about when talking about stellar distances).