2.

# Momentum

Momentum

\(\vec p\)

Ns $$\vec p=m\vec v$$

It has the same SI units as impulse does (Ns, which can also be written as $\u{kg\cdot m/s}$). The momentum is a vector which points in the direction that the object is moving, and there is a relationship between the momentum and the impulse: when an impulse acts on an object, it changes its momentum according to

$\vec J=\Delta\vec p=\vec p_f-\vec p_i$

$\text{or }\vec p_f = \vec p_i + \vec J$

$\text{or }\vec p_f = \vec p_i + \vec J$

This is incredibly useful, as we will see in the following pages.

### Optional

*We can actually prove this relationship, if you're interested:*

According to Newton's Second Law, | $$\RED{\vec F_{avg}} = \RED{m\vec a_{avg}}$$ |

The average acceleration is related to the change in velocity | $$\begin{align} \vec a_{avg}&=\frac{\vec{v}_f-\vec v_i}{\Delta t}\\ \implies \BLUE{\vec a_{avg}\Delta t} &=\BLUE{\vec v_f-\vec v_i}\\ \end{align}$$ |

The Impulse is equal to | $$\begin{align} \vec J&=\RED{\vec F_{avg}}\Delta t\\ &=(\RED{m\vec a_{avg}})\Delta t\\ &=m(\BLUE{\vec a_{avg}\Delta t})\\ &=m(\BLUE{\vec v_f-\vec v_i})\\ &=m\vec v_f-m\vec v_i\\ \end{align}$$ |

And since momentum is $\vec p=m\vec v$, | $$\vec J=\vec p_f-\vec p_i$$ |