Conservação do momento

Momentum is a quantity that describes how easy or hard it is for one to stop an object. As an equation, $\vec{p} = m\times\vec{v}$. Momentum is usually transferred between objects via collisions. During a collision, the total momentum is conserved for all dimensions, which can also be expressed as: $$\sum m_{ai}\vec{v}_{ai} = \sum m_{af}\vec{v}_{af}$$ Note that momentum in each dimension is conserved, this means that the vector sum of all momentum vectors is conserved. Do not add momentum like scalars, they are vector quantities.

An example problem: Block A has a mass of 10 kg and is moving due East at 5 m/s. Block B has the mass of 3 kg and is moving $40^{\circ}$ South of West at 10 m/s. They collide, sending block A moving due West at 3 m/s. Calculate the velocity of block B after collision.

We split scenarios like this into components. Let East be the positive x direction and North be the positive y direction. Let's tackle the x components first, let the final x-component of B's velocity be $v_{fx}$, we get: $$(10 \, \text{kg})(5 \, \text{m/s}) + (3 \, \text{kg})(-10 \, \text{m/s})(\cos{40^\circ}) = (10 \, \text{kg})(-3 \, \text{m/s}) + (3 \, \text{kg})(v_{fx})$$ Solve, we get: $$v_{fx} \approx 19 \, \text{m/s}$$ Then we do the same thing for the y components. A look at the problem shows that Block A experienced no change of velocity in the y component, therefore, block B retains is y component velocity, which is: $$v_{fy} = (-10 \, \text{m/s})\sin{40^\circ} \approx -6.43 \, \text{m/s}$$ Thus, the final velocity is: $$v_{f} = \sqrt{v_{xf}^2 + v_{yf}^2} \approx 20 \, \text{m/s}$$ That was the magnitude of the final velocity, its direction can be calculated by: $$\theta = \tan^{-1}\left({\frac{-6.43}{19}}\right) \approx -18.7^\circ$$ Therefore, B's final velocity has a magnitude of 20 m/s and the direction of approximately 18.7 degrees south of east.

Similar to energy, momentum is sometimes conserved. But when exactly does this happen?

Noether's theorem tells us that (in loose terms) that a symmetry gives rise to a conservation law. In this case, the conservation of momentum is born from spatial invariance. But what exactly is spatial invariance?

This video explains it quite well (10 minute but actually very interesting and worth watching).

https://www.youtube.com/watch?v=CxlHLqJ9I0A