Proof of Conservation of Momentum

By Newton's Third Law of Motion,

\[\mathbf{F}_{12}=-\mathbf{F}_{21}\]

By Newton's Second Law of Motion,

F12=F21p˙1=p˙2t1t2p˙1dt=t1t2p˙2dtΔp1=Δp2Δ(mv1)=Δ(m2v2)m1Δ(v1)=m2Δ(v2)m1(v1u1)=m2(v2u2)m1v1m1u1=m2u21m2v2m1v1+m2v2=m2u2+m1u1pf=pi\begin{aligned} \mathbf{F}_{12}&=-\mathbf{F}_{21}\\ \dot{\mathbf{p}}_{1}&=-\dot{\mathbf{p}}_{2}\\ \int_{t_1}^{t_2}\dot{\mathbf{p}}_{1}\,\mathrm{d}t&=-\int_{t_1}^{t_2}\dot{\mathbf{p}}_{2}\,\mathrm{d}t\\ \Delta\mathbf{p}_{1}&=-\Delta\mathbf{p}_{2}\\ \Delta(m\mathbf{v}_{1})&=-\Delta(m_2\mathbf{v}_{2})\\ m_1\Delta(\mathbf{v}_{1})&=-m_2\Delta(\mathbf{v}_{2})\\ m_1(\mathbf{v}_{1}-\mathbf{u}_{1})&=-m_2(\mathbf{v}_{2}-\mathbf{u}_{2})\\ m_1\mathbf{v}_{1}-m_1\mathbf{u}_{1}&=m_2\mathbf{u}_{21}-m_2\mathbf{v}_{2}\\ m_1\mathbf{v}_{1}+m_2\mathbf{v}_{2}&=m_2\mathbf{u}_{2}+m_1\mathbf{u}_{1}\\ \sum\mathbf{p}_{f}&=\sum\mathbf{p}_{i}\quad\blacksquare \end{aligned}

Note by Gandoff Tan
4 months, 1 week ago

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What happens when there are 33 objects?

Josh Silverman Staff - 4 months, 1 week ago

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Alright, let's give this a try:

Let's say we have a bunch of masses, in a system with only internal forces. where mass mim_i exerts a force on mjm_j denoted by FijF_{ij}. Thus, the net force is a sum of all the internal forces:

Fnet = i, j FijF_{net} \ = \ \displaystyle\sum_{i, \ j} \ F_{ij}

By Newton's Third Law, we will have Fij = FjiF_{ij} \ = \ -F_{ji} (if mim_i exterts FijF_{ij} on mim_i, then it follows that mjm_j exerts an equal an opposite force on mim_i. We also know that:

Fnet = dptotaldtF_{net} \ = \ \frac{d p_{total}}{dt}

Thus, we can write our sum as a bunch of Fij + Fji = Fij  Fij = 0F_{ij} \ + \ F_{ji} \ = \ F_{ij} \ - \ F_{ij} \ = \ 0 terms, which just equates to 00, giving us:

Fnet = dptotaldt = 0F_{net} \ = \ \frac{d p_{total}}{dt} \ = \ 0

This implies that total momentum does not change with time, thus momentum is conserved when there are only internal forces acting on the system.

I probably missed a lot of nuances, how did I do @Josh Silverman? :)

Jack Ceroni - 4 months, 1 week ago

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It looks like a nose.

Joseph Dalton - 3 months, 3 weeks ago

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