Can anyone explain this derivation of the variable mass equation?

[etotheipi]

This question stems from one of the recent homework threads. I'm familiar with the derivation given here regarding mass accretion and ejection, where the general idea is to define a system around body and all of the incoming/leaving mass so that we can once again apply NII to the whole thing.

I came across another derivation here (the top-voted answer) and struggle to see how it fits together.

Another equation may be derived from this one ($\mathbf{F} = \frac{d\mathbf{p}}{dt}$) applied to the whole super-system (system + incoming/leaving parts). This can be done because the super-system does not lose or gain particles. The new equation is $$\mathbf{F}_{ext} + \mathbf{F}_{parts} = \sum_k m_k \mathbf{a}_k$$ where $\mathbf{F}_{parts}$ is force on the system due to parts no longer inside the control volume and summation is to be done over all particles in the control volume. It can also be written$$\mathbf{F}_{ext} + \mathbf{F}_{parts} = \frac{d\mathbf{p}}{dt} + \frac{d\mathbf{p}_{lost}}{dt}$$In the simplest case, where the lost particles all leave in direction same or opposite to the body velocity $\mathbf{v}$ (idealized rocket), this can be further simplified. Let the boundary of the control volume be far from the rocket, so that velocity of particles crossing the boundary (relatively to the rocket) is constant $\mathbf{c}$ and $\mathbf{F_{parts}}$ is negligible. Then the lost momentum per unit time is $$\frac{d\mathbf{p}_{lost}}{dt} = -\frac{dm}{dt}(\mathbf{v} + \mathbf{c})$$

I have a few questions about all of this. First of all, where do the first two equations come from? He says he applied Newton II to the 'super-system' however I wonder then why we have this strange $\mathbf{F}_{parts}$ term (surely this is an internal force to that super-system?). Furthermore, this only corresponds to the sum over particles in the 'control volume'.

And I'm not sure how he obtains the second equation from the first. It's not clear to me in any of the steps which choice of system he is using. I wondered whether anyone could shed some light on this? Thanks!

 

[vanhees71]

You have to take the momentum-conservation equation. Maybe this old post helps?

https://www.physicsforums.com/threa...v-and-galilean-invariance.953956/post-6069532

 

[etotheipi]

You have to take the momentum-conservation equation. Maybe this old post helps?

https://www.physicsforums.com/threa...v-and-galilean-invariance.953956/post-6069532

Thanks for the reply. I understand the derivation based around conserving momentum for the entire (rocket + fuel) system, I'm just not sure about this derivation (particularly about $\mathbf{F}_{parts}$ and which systems are being used in each line).

I'll have a closer read through that thread however and see if I can see any parallels!

 

[etotheipi]

I've now read through the whole thread and yes your derivation of $dt(m\dot{v} + (v-u)\dot{m}) = F_{ext}dt$ is the one that I am familiar with.

I now wonder how we obtain the relationship (with $k$ running over the particles in the rocket system only) $$\mathbf{F}_{ext} + \mathbf{F}_{parts} = \sum_k m_k \mathbf{a}_k = \frac{d\mathbf{p}}{dt} + \frac{d\mathbf{p_{lost}}}{dt}$$ At first glance it would seem like an application of $\mathbf{F} = \frac{d\mathbf{p}}{dt}$ to the rocket sub-system only, however we know this cannot be true since that system is not closed. I'm very confuzzled!

 

[etotheipi]

I think I've worked it out. At time $t$, he considers a control volume containing the rocket and particles which haven't yet crossed the boundary, and the surroundings of this control volume consist of the particles that have already crossed the boundary.

He now considers the system containing everything within the control volume including the particles which cross out of the control volume in the time interval $[t, t+ dt]$. This system is of constant mass, so Newton II applies. And furthermore, $\mathbf{F}_{parts}$ is just a part of the the total external force, exerted on this system by particles already outside the control volume.

The key thing is that the total momentum of this newly defined system $\mathbf{p}_{sys}$ can be split into the momentum of the particles which cross the boundary $\mathbf{p}_{lost}$ plus the momentum of those that remain in the control volume $\mathbf{p}$. It follows that $$\mathbf{F}_{ext} + \mathbf{F}_{parts} = \frac{d\mathbf{p}_{sys}}{dt} = \frac{d\mathbf{p}}{dt} + \frac{d\mathbf{p}_{lost}}{dt}$$

And $\frac{d\mathbf{p}_{lost}}{dt}$ is then just $-\frac{dm}{dt} (\mathbf{v} + \mathbf{c})$, and if the control volume is sufficiently large we do indeed get $\mathbf{F}_{parts} \rightarrow 0$. This allows us to deduce the variation of the momentum of the control volume, without ever applying NII directly.

So in hindsight it's pretty similar to the other derivation, I was just having trouble deciphering what his variables and equations were referring to!