homework and exercises - Variable mass dynamics: Particle and Rigid Body

I'm encountering some issues in the understanding of some basic concepts about the dynamics of variable-mass particles and rigid bodies.

For what I found, for example reading On the use and abuse of Newton's Second Law for Variable Mass Problems (Plastino,Muzzio) and also Lectures On Theoretical Physics: Mechanics (Sommerfield -- p28) the second law of Dynamics is not suitable for a variable mass particle; instead you should use the momentum conservation:

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e.g. Rocket:

Applying the conservation of momentum for an isolated system:

$$p_t=mv$$ $$p_{t+\Delta t}=(m-\text{d}m)(v+\text{d}v) +\text{d}m\,(v-u_e)$$

$$\frac{\text{d}p}{\text{d}t}=\frac{mv + mdv -vdm +v\text{d}m -u_edm - mv}{\text{d}t}=0$$ $$0 =\frac{mdv-u_edm}{\text{d}t}$$ $$dv=-u_e\frac{dm}{m}$$

that is the classical Tsiolkovsky rocket equation, that can be integrated $\int_{t_0}^t$:

$$\boxed{\Delta v = u_e\ln\frac{m_0}{m}}$$

Where $u_e$ is the velocity of the gases that are coming out from the nozzle.

In other books the same equation is obtained from Newton's Law:

$$m\frac{\text{d}v}{\text{d}t} = \sum_i F^{\text{ext}}_i$$ where the external forces are basically just the thrust (simplest case) that is given by: $$T = \dot{m}u_e + A_e(p_e - p_a) = \dot{m}c$$

where $A_e$ is the area of outflow for the nozzle, $p_e$ the outflow pressure, $p_a$ the ambient pressure (hence $c=u_e + \frac{A_e(p_e - p_a)}{\dot{m}}$ is the equivalent velocity)

substituting:

$$m\frac{\text{d}v}{\text{d}t} = \dot{m}c$$ being $\dot{m} = -\frac{\text{d}m}{\text{d}t}$ because of the mass loss we get: $$\text{d}v = -c\frac{\text{d}m}{m}$$

that is basically the same equation but with the equivalent velocity instead of the real convective gas velocity. This is the first confusing passage...

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And what about rigid bodies?

The equation would be:

$$\frac{\text{d}\mathbf{Q}}{\text{d}t} =\sum_i \mathbf{F_i^\text{ext}}$$ so should I do: $$\frac{\text{d}\mathbf{Q}}{\text{d}t} = m\mathbf{\dot{v}} + \mathbf{v}\dot{m}=\sum_i \mathbf{F_i^\text{ext}}$$

or not? I'm really confused about that.