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I have read a bit on this topic and have come to the conclusion that this is one of the things in physics we just accept (like it it a "tested" fundamental law (from the beginnings of time etc etc)).

So instead of asking where does it come from, I want to know how conservation of momentum can be connected to other concepts such as newtons laws.

Or a logical thought process to answer how conservation of momentum comes about.

Qmechanic
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Muffin
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    conservation of momentum is not a physical principle. Momentum balance, or Newton's second law, is the physical principle. Conservation of momentum comes from Newton's second law applied to a system with no net external force acting on it – basics Feb 28 '23 at 09:59
  • Could you explain the application of newtons second law – Muffin Feb 28 '23 at 10:13
  • Take a look at the as the answers below. If needed, I'll give you another answer. If you're asking how to mathematically prove momentum balance, the answer is that you can't prove it, since it is a principle, i.e. a law that agrees with experiments – basics Feb 28 '23 at 10:32

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Conservation of momentum follows immediately from Newton's 2nd and 3rd law. For simplicity, I will assume a 2-particle system but the logic can be easily generalised to many particles. For the purpose of the derivation I will also assume that the only force acting on particles is the intermolecular forces between them.

Given two particles (1 and 2) in 3-d, the equations of motion (via Newton's 2nd Law) is given by

\begin{align} \frac{d\vec{p_1}}{dt} = \vec{F_1},\hspace{0.5cm} \frac{d\vec{p_2}}{dt} = \vec{F_2} \end{align} where $\vec{F}_1$ is the force acted on the particle 1 by particle 2 and vice versa for $\vec{F}_{2}$. By Newton's Third Law, we know $\vec{F}_1 = - \vec{F_2} $ so we have \begin{align} \frac{d\vec{p_1}}{dt} + \frac{d\vec{p_2}}{dt} = \vec{F_1} +\vec{F_2} = \vec{F_1} - \vec{F_1} = 0 \end{align} Collecting things up, we get \begin{align} \frac{d}{dt}(\vec{p_1} + \vec{p_2}) = 0 \end{align} meaning the total momentum of the system doesn't change.

emir sezik
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