I have a question regarding the meaning of the force in the equation of motion for a variable mass system. I will begin by deriving the equation, in order to better explain my reasoning.
Thus consider an arbitrary control volume, and let the mass inside this volume be $m$ and let the total momentum of the particles inside the control volume be $\mathbf{p}$. Suppose that at time $t+\Delta t$ a parcel of mass $\Delta m_o$ ('o' for out) leaves the system with absolute velocity $\mathbf{v}_o$ in some inertial frame. Similarly, assume that at time $t$ a parcel of mass $\Delta m_i$ ('i' for in) enters the system with absolute velocity $\mathbf{v}_i$.
By the impulse equation, we have $$(\mathbf{p}(t+\Delta t)+\Delta m_o\mathbf{v}_o)-(\mathbf{p}(t)+\Delta m_i\mathbf{v}_i) = \int_{t}^{t+\Delta t}\mathbf{F}dt. $$ Here, the force $\mathbf{F}$ must be the force not just on the control volume, but on all the particles, even those which may leave or enter the volume, in order that we may use Euler's first law.
Dividing through by $\Delta t$, taking the limit $\Delta t \to 0$, invoking the mean value theorem, putting $q_o = \lim_{\Delta t \to 0} \frac{\Delta m_o}{\Delta t}$, similarly putting $q_i = \lim_{\Delta t \to 0} \frac{\Delta m_i}{\Delta t}$ and noting that $\dot{m}=q_i-q_o$, we obtain
$$\mathbf{F}+q_i\mathbf{v}_i-q_o\mathbf{v}_o = \dot{\mathbf{p}} $$
or, if you will,
$$\mathbf{F}+q_i(\mathbf{v}_i-\mathbf{v})-q_o(\mathbf{v}_o-\mathbf{v})=m\dot{\mathbf{v}},$$ in order to get an expression that's explicitly Galielei invariant.
My question then is about the force $\mathbf{F}$. In the derivation, we remarked that this has to be the total external force on all the particles under consideration, not just the particles inside the control volume itself. However, this seems quite unreasonable from another perspective. If I fire a machine gun in space, I will experience some recoil and accelerate as a result of it. Neglecting gravitational influence, we might be tempted to set $\mathbf{F}=\mathbf{0}$. But now suppose that someone puts a plate in the way of my bullets, so that the bullets eventually hit this plate and experience some recoil as a result. This would amount to a nonzero average force on the bullets and thereby on the entire system under consideration, affecting my acceleration according to the equation of motion. However this seems quite absurd. The plate might be pretty far away, and my acceleration is evidently not going to change when the bullets hit the plate.
What am I missing here?
EDIT
We can see the problem even easier in the following way. Say we are still in space and let out collection of particles be, say some nice asteroid. As the control volume, pick a region of empty space, thus with $\mathbf{p}=\mathbf{0}$ and such that no mass passes through it. Our equation of motion is then $\mathbf{F}=\mathbf{0}$. Again, this is absurd since e.g. gravitational forces will be acting on the asteroid, which is part of our collection of particles.
It seems then, that the equation of motion only makes sense if $\mathbf{F}$ refers to the net external force on the control volume. However, in using the impulse law and thereby Euler's first law, we must assume that $\mathbf{F}$ is the net external force on the entire collection of particles.
EDIT
Alright, so I guess the issue is in the original impulse equation. If there are forces acting one the mass outside of the control volume it is not correct, since the LHS does not account for the momentum of this mass. I suppose then that we can only use the above equation for variable mass systems if there is no force on particles which have left the control volume.
