If I understand correctly, the system you mean here will includes both the rocket part and all the expelled fuel part, right? But my question is to consider only the rocket part and work out the force exerted on the rocket part, whose mass varies with time.
And m1 and m2 in my formula are not two mass of two objects but the mass of the same object at two different moment: at t1, the mass of the rocket part is m1; at t2, the mass of the rocket part is m2, because the rocket part mass varies with time, right?