Definition: Momentum

The momentum of a mass $m$ moving at seped $\vec v$ has a momentum $\vec p$ defined as $$\vec p = m\vec v$$ Momentum doesn't have fancy units like energy, so we measure in units $\unit{kg\cdot m/s}$. However, notice that momentum is a vector, unlike energy which is a scalar. When adding momentum make sure to take care of positive and negative directions as well as vector addition.

Definition: Impulse

When force $F$ is applied in time $\Delta t$, then the impulse exerted by the force is $$I=F\Delta t$$ (Compare this to work, which is defined as $W=F\Delta x$). Impulse is equal to the change in momentum $\Delta p$ of the object the force is exerting on.

Idea: Newton's Second Law

Given our definition of impulse, we can write Newton's Second Law in the alternate form $$F=\frac{\Delta p}{\Delta t}=\frac{\Delta (mv)}{\Delta t}$$ which is actually the way he originally wrote it (not $F=ma$)!

Definition: Conservation of Momentum

If there is no force acting on a system of objects, then $F=0$, and so $\Delta p$ is 0. Thus, we have conservation of momentum. If there is no external force acting on a collection of objects, then their momentum will always be the same. Internal forces of interaction between the objects will cancel out via Newton's Third Law.

Idea: Inelastic Collisions

An inelastic collision is one where two objects collide and stick together (move together as one object). In any collision, so long as there is no external force on the objects, momentum is conserved. Thus, if I have a mass $m_1$ moving at velocity $\vec v_1$ and a mass $m_2$ moving at velocity $\vec v_2$ which collide inelastically, they form a final mass of $m_1+m_2$ with some velocity $\vec v$ given by $$m_1\vec v_1+m_2\vec v_2=(m_1+m_2)\vec v\Rightarrow \vec v = \frac{m_1v_1+m_2v_2}{m_1+m_2}$$