Key Equations

Definition of momentum	$\vec{p} = m \vec{v}$
Impulse	$\vec{J} \equiv \int_{t_{i}}^{t_{f}} \vec{F} (t) d t or \vec{J} = {\vec{F}}_{ave} Δ t$
Impulse-momentum theorem	$\vec{J} = Δ \vec{p}$
Average force from momentum	$\vec{F} = \frac{Δ \vec{p}}{Δ t}$
Instantaneous force from momentum (Newton’s second law)	$\vec{F} (t) = \frac{d \vec{p}}{d t}$
Conservation of momentum	$\frac{d {\vec{p}}_{1}}{d t} + \frac{d {\vec{p}}_{2}}{d t} = 0 or {\vec{p}}_{1} + {\vec{p}}_{2} = constant$
Generalized conservation of momentum	$\sum_{j = 1}^{N} {\vec{p}}_{j} = constant$
Conservation of momentum in two dimensions	$\begin{matrix} p_{f, x} = p_{1,i, x} + p_{2,i, x} \\ p_{f, y} = p_{1,i, y} + p_{2,i, y} \end{matrix}$
External forces	${\vec{F}}_{ext} = \sum_{j = 1}^{N} \frac{d {\vec{p}}_{j}}{d t}$
Newton’s second law for an extended object	$\vec{F} = \frac{d {\vec{p}}_{CM}}{d t}$
Acceleration of the center of mass	${\vec{a}}_{CM} = \frac{d^{2}}{d t^{2}} (\frac{1}{M} \sum_{j = 1}^{N} m_{j} {\vec{r}}_{j}) = \frac{1}{M} \sum_{j = 1}^{N} m_{j} {\vec{a}}_{j}$
Position of the center of mass for a system of particles	${\vec{r}}_{CM} \equiv \frac{1}{M} \sum_{j = 1}^{N} m_{j} {\vec{r}}_{j}$
Velocity of the center of mass	${\vec{v}}_{CM} = \frac{d}{d t} (\frac{1}{M} \sum_{j = 1}^{N} m_{j} {\vec{r}}_{j}) = \frac{1}{M} \sum_{j = 1}^{N} m_{j} {\vec{v}}_{j}$
Position of the center of mass of a continuous object	${\vec{r}}_{CM} \equiv \frac{1}{M} \int \vec{r} d m$
Rocket equation	$Δ v = u ln (\frac{m_{i}}{m})$