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I have begun learning about Momentum and the Conservation of Momentum. For some reason, I have really struggled with understanding this topic.

Right, so I understand momentum is given by $$ p = mv $$

However, I fully don't understand the following statement:

"If no external forces are acting on our system, the total momentum of the system remains constant"

if this true, apparently the following is $$ m_1v_1 + m_2v_2 = m_1v'_1 + m_2v'_2 $$

I understand the formulaic approach of showing this $$ \displaystyle \frac{dp}{dt} = m\frac{dv}{dt} = \frac{d(mv)}{dt}=ma $$ $$ \therefore \frac{dp}{dt} = F_{net} $$

The first thing that throws me off with the above statement is the external forces part. Why doesn't it hold if internal forces are acting on our system? Furthermore, what is a external force? Something like gravity right?

The second part throws me off even more! Why does the total momentum remain constant??

I understand this question may be very vague but it's very hard to describe what you don't understand!

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"External" and "internal" is relative to the system you are working with. If your system is made of, for example, two bodies A and B, then the internal forces are: the force A makes over B, and the force made by B over A. Internal forces, by action and reaction principle, always comes in pairs and furthermore, they are opposite! so they cancel when you calculate the total force. Then only external forces appear, the ones that other bodies different from A and B in the example apply to the system.

The complete demonstration is:

\begin{equation} \frac{dp}{dt} = F_{net} \end{equation} if we separate the sum in external and internal forces, we have \begin{equation} F_{net} = \sum F = \sum F_{int} + \sum F_{ext} \end{equation} Then, for the third Newton Law $\sum F_{int} = 0$, and so \begin{equation} F_{net} = \sum F = \sum F_{ext} \rightarrow \frac{dp}{dt} = \sum F_{ext} \end{equation}

In this image, there are 4 bodies. If your system is composed by the red balls then the internal forces are the grey arrows and the external forces are the black ones. Note that in that system the black ones doesn't have a pair.

internal and external forces

About the second question, you'll see that if the derivative of p over time is zero, then p is constant ;)

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