Useful Formulas - Exam II

Motion definitions

displacement:	$\Delta\textbf{r}\equiv\textbf{r}_f-\textbf{r}_i$
velocity:	average, $\bar{\textbf{v}} \equiv \frac{\Delta \textbf{r}}{\Delta t}$	instantaneous, $\textbf{v} \equiv \frac{d \textbf{r}}{dt}$
acceleration:	average, $\bar{\textbf{a}} \equiv \frac{\Delta \textbf{v}}{\Delta t}$	instantaneous, $\textbf{a} \equiv \frac{d \textbf{v}}{dt}$
radial acceleration:	$a_r=\frac{v^2}{r}$
tangential acceleration:	$a_t=\frac{d\vert\textbf{v}\vert}{dt}$

Kinematic equations ($\mathbf{a}$ is constant)

$\textbf{v}_f=\textbf{v}_i+\textbf{a}\,t$	$v_f^2=v_i^2+2\,\textbf{a}\cdot\textbf{r}$
$\textbf{r}_f=\textbf{r}_i+\textbf{v}_0\,t+\frac{1}{2}\textbf{a}\,t^2$	$\textbf{r}_f=\textbf{r}_i+\frac{1}{2}(\textbf{v}_i+\textbf{v}_f)\,t$

Trigonometric formulas

$\sin(\theta) =\frac{opp}{hyp}$	Law of sines: $\frac{\sin (\alpha )}{a}=\frac{\sin (\beta )}{b}=\frac{\sin (\gamma )}{c}$
$\cos (\theta )=\frac{adj}{hyp}$	Law of cosines: $c^2=a^2+b^2-2\,a\,b\cos(\gamma)$
$\tan (\theta )=\frac{opp}{adj}$
Identities
$\sin ^{2}(\theta )+\cos ^{2}(\theta )=1$	$\sin (A\pm B)=\sin (A)\cos (B)\pm \cos (A)\sin (B)$
$\sin (2\theta )=2\sin (\theta )\cos (\theta )$	$\cos (A\pm B)=\cos (A)\cos (B)\mp \sin (A)\sin (B)$
$\cos (2\theta )=\cos ^{2}(\theta )-\sin ^{2}(\theta )$