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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?

Yes, for a constant mass object, the change in momentum does not depend on the frame, as your formula proves. But what about a variable mass object: Frame 1: the change in momentum is m2v2- m1v1. In frame 2, moving at a velocity u relative to frame 1: the change in momentum is m2(v2+u) - m1(v1+u) = m2v2-m1v1+u(m2-m1) where the mass of the object changes from m1 to m2. You can then see that the change in momentum depends on u, the relative velocity of the frame you have chosen. If you mean P the momentum of the entire system (including the rocket part and all expelled fuel part to ensure no mass crosses the boundary of the composed system), F here you write will be the net force on the whole system, the sum of the force on the rocket part and the force on the expelled fuel part. Is my understanding right?

If the force that you mean is defined by F=ma=mdv/dt for a constant-mass object, then the force does not depend on the inertial frame you choose because the change in velocity does not depend on the inertial frame you choose. However, for a variable-mass object, the force has a new definition: F=dp/dt, and the force now does depend on the inertial frame you choose because the change in momentum really depends on the inertial frame you choose. I think the main point is that the definition that we use here to define force does matter, since a conclusion or claim about the force should be derived from its definition.

According to F= dp/dt, the force is now defined by the change in momentum. For a variable-mass object, its change in momentum indeed does depend on the inertial frame you choose. So I think it's OK.

I recently learnt the rocket problem: how to derive rocket equation. I accidently saw the problem about the force that is exerted on rocket part (rocket and unused fuel). My problem is: F(t) fore force that is exerted on rocket part (rocket and unused fuel) at time t should be F(t)=mdv/dt+vdm/dt, where m is the mass of rocket part at time t and v is the velocity of rocket part at time t, according to F=dp/dt=mdv/dt+vdm/dt which is true as the foundation of established physics, right? If we use the velocity function with respect to mass derived from the rocket equation v=v0+ve*ln(m0/m) and assume rocket expels gas mass at a constant mass flow rate R: m(t)=m0-Rt, the final result of force F(t)=mdv/dt+vdm/dt=ve*R+R*(v0+ve*ln(m0/m0-Rt)) where mdv/dt=ve*R which is called the thrust of rocket in many materials, and vdm/dt=R*(v0+ve*ln(m0/m0-Rt)). Thus, the force on the rocket is not only just the thrust but acutally the thrust plus vdm/dt=R*(v0+ve*ln(m0/m0-Rt)), right?