Derivation
The conservation can be “derived” from Newton’s 3rd law. Personally, “derive” is a strong word, and I rather consider them to be equivalent statements. Anyway, imagine two particles a and b that interact with each other only. Then, and using the Third law (Newton’s laws)
\begin{gather} F_{ab} = \frac{dp_a}{dt}\\ \int_0^t F_{ab} \cdot dt = \int_{p_1}^{p_2}{dp_a}\\ p_{a}(t) - p_{a}(0) = \int_0^t F_{ab} \cdot dt \\ \int_0^t F_{ab} \cdot dt = - \int_0^t F_{ba} \cdot dt = - (p_b(t) - p_b(0))\\ p_a(t) - p_a(t) = - p_b(t) + p_b(0) \\ p_a(t) + p_a(t) = p_a(0) + p_b(0) \end{gather} $$This is the conservation law, which also holds for 3,4 and n particles. --- #### Rocket movement It turns out that the rocket movement is easy to derive. Assume that an exhaust has a constant relative speed $u$ relative to the rocket. Now, imagine a rocket at some time $t$ with a mass $m$ and velocity $v$. At a later time $t+dt$, an exhaust with mass $dm$ was emitted. Using the conservation, we law we can write:\begin{gather} mv = (m-dm)(v+dv)+(dm)(v-u)\ mv = mv + mdv-vdm -dmdv +vdm-udm\ mdv= -udm \ dv = -u \frac{dm}{m}\ v_2-v_1 = u\ln(\frac{M}{m}) \end{gather}