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Wikipedia defines thrust as $$T=v\frac{dm}{dt}$$

Is this something fundamental or can this be derived ?

I was not able to find it on the internet so I thought of asking if here .If proof questions are not allowed a link would be very helpful.

Thank you .

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Thrust ($T$) is the force one receives by "expelling" some mass $dm$.

We start by conservation of momentum for a mass $m$ travelling in one dimension at speed $v$ which expels a smaller mass $dm$ and changes thus its velocity by $dv$. The expelled mass travels at speed $-c$ with respect to the moving mass, so that it has a speed $(-c+v)$ in the observer's frame of reference.

We have $$m v = (m - dm)(v+dv)+dm (-c+v)$$ which becomes

$$m v = mv +mdv -dmv -dmdv -dmc + dm v$$

and simplifies to

$$ 0 = mdv-dmdv -dm c $$ we neglect $dmdv$ as is the product of two small quantities and are left with

$$mdv = c dm$$

we divide everything by $dt$ and get

$$m {dv \over dt} = c {dm \over dt}$$

Finally, using Newton's second law, $$m {dv \over dt}=ma_T=T$$ is a force and indeed it is the force that "accelerated" the mass forward by a quantity $dv$ due to mass expulsion, which is indeed the thrust, so

$$T= c {dm \over dt}$$

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