# Basic Linear Momentum and Conservation

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!

\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} 