Definition: Impulse and momentum

We define the momentum of an object to be \begin{equation} \vec{p}=m\vec{v} \end{equation}such that we can write Newton's law in an alternate form: \begin{equation} F=\frac{\Delta p}{\Delta t}\equiv\frac{dp}{dt} \end{equation}We can rearrange the law to define impulse as \begin{equation} I=F\Delta t\equiv\int F dt \end{equation}

Using the definitions of impulse and momentum, we can define conservation of momentum just as we did with energy.

Theorem: Conservation of momentum

We can say the momentum of a system of objects is conserved if the net external force acting on the objects over a given time is 0 (i.e. if $F=0$ in a given time, $\Delta p=0$). By Newton's Third Law, all internal forces between the objects (i.e. impulses from collisions) will cancel.

Since we're discussing motion between multiple objects, it's helpful to define the center of mass of the system.

Definition: Center of mass

If we have a collection of $n$ masses, each with mass $m_i$ and placed at a position $x_i$, their center of mass is the weighted average of their positions: \begin{equation} x_{\text{cm}}=\frac{m_1x_1+\cdots +m_nx_n}{m_1+\cdots +m_n} \end{equation}We can replace the $x$'s with $y$'s to extend to other dimensions, and replace them with $v$'s to find the velocity of the center of mass.

In some types of force fields, we can assume that the combined effect of the force on multiple objects is equivalent to if the force acted solely on the center of mass. We can do this similarly with larger bodies that we can't treat as point masses. In other words, in these force fields, the center of mass acts as a point mass in the field. There are two notable force fields that follow this property:

A completely uniform force field, such as in gravity close to the Earth's surface
Centripetal forces (radial forces proportional to $r$)

Finally, we will discuss collisions, both elastic and inelastic. Going by the definition of conservation of momentum, all collisions (if no external force acts on the objects) conserve momentum. We say a collision is elastic if it also conserves energy, and it is inelastic if it doesn't.

Theorem: Elastic collisions

Elastic collision problems can be solved with the conservation of energy equation, but can be replaced with \begin{equation} v_{1i}-v_{2i}=v_{2f}-v_{1f} \end{equation}

Tip: Center of mass frame

When analyzing an elastic collision (or with collisions in general), it's helpful to shift into the frame of reference of the center of mass. For elastic collisions in particular, the incoming objects will collide and rebound at their initial speeds in the COM frame, but may scatter at some angle. To keep total momentum equal to 0, their final trajectories will still lie on the same line.