Key Equations

Key Equations

Position vector	$\overrightarrow{r}(t)=x(t)\widehat{i}+y(t)\widehat{j}+z(t)\widehat{k}$
Displacement vector	$\text{Δ}\overrightarrow{r}=\overrightarrow{r}(t_2)-\overrightarrow{r}(t_1)$
Velocity vector	$\overrightarrow{v}(t)=\underset{\text{Δ}t\to 0}{\text{lim}}\frac{\overrightarrow{r}(t+\text{Δ}t)-\overrightarrow{r}(t)}{\text{Δ}t}=\frac{d\overrightarrow{r}}{dt}$
Velocity in terms of components	$\overrightarrow{v}(t)=v_x(t)\widehat{i}+v_y(t)\widehat{j}+v_z(t)\widehat{k}$
Velocity components	$v_x(t)=\frac{dx(t)}{dt}{v}_y(t)=\frac{dy(t)}{dt}{v}_z(t)=\frac{dz(t)}{dt}$
Average velocity	${\overrightarrow{v}}_{\text{avg}}=\frac{\overrightarrow{r}(t_2)-\overrightarrow{r}(t_1)}{t_2-t_1}$
Instantaneous acceleration	$\overrightarrow{a}(t)=\underset{t\to 0}{\text{lim}}\frac{\overrightarrow{v}(t+\text{Δ}t)-\overrightarrow{v}(t)}{\text{Δ}t}=\frac{d\overrightarrow{v}(t)}{dt}$
Instantaneous acceleration, component form	$\overrightarrow{a}(t)=\frac{d{v}_x(t)}{dt}\widehat{i}+\frac{d{v}_y(t)}{dt}\widehat{j}+\frac{d{v}_z(t)}{dt}\widehat{k}$
Instantaneous acceleration as second derivatives of position	$\overrightarrow{a}(t)=\frac{d^{2}x(t)}{d{t}^2}\widehat{i}+\frac{d^{2}y(t)}{d{t}^2}\widehat{j}+\frac{d^{2}z(t)}{d{t}^2}\widehat{k}$
Time of flight	$T_{\text{tof}}=\frac{2(v_0\text{sin}{\theta }_0)}{g}$
Trajectory	$y=(\text{tan}{\theta }_0)x-\left[\frac{g}{2{(v_0\text{cos}{\theta }_0)}^2}\right]x^2$
Range	$R=\frac{v_0^2\text{sin}2{\theta }_0}{g}$
Centripetal acceleration	$a_{\text{C}}=\frac{v^2}{r}$
Position vector, uniform circular motion	$\overrightarrow{r}(t)=A\text{cos}\omega t\widehat{i}+A\text{sin}\omega t\widehat{j}$
Velocity vector, uniform circular motion	$\overrightarrow{v}(t)=\frac{d\overrightarrow{r}(t)}{dt}=\text{−}A\omega \text{sin}\omega t\widehat{i}+A\omega \text{cos}\omega t\widehat{j}$
Acceleration vector, uniform circular motion	$\overrightarrow{a}(t)=\frac{d\overrightarrow{v}(t)}{dt}=\text{−}A{\omega }^2\text{cos}\omega t\widehat{i}-A{\omega }^2\text{sin}\omega t\widehat{j}$
Tangential acceleration	$a_{\text{T}}=\frac{d\left|\overrightarrow{v}\right|}{dt}$
Total acceleration	$\overrightarrow{a}={\overrightarrow{a}}_{\text{C}}+{\overrightarrow{a}}_{\text{T}}$
Position vector in frame S is the position vector in frame $S^{\prime }$ plus the vector from the origin of S to the origin of $S^{\prime }$	${\overrightarrow{r}}_{PS}={\overrightarrow{r}}_{P{S}^{\prime }}+{\overrightarrow{r}}_{S^{\prime }S}$
Relative velocity equation connecting two reference frames	${\overrightarrow{v}}_{PS}={\overrightarrow{v}}_{P{S}^{\prime }}+{\overrightarrow{v}}_{S^{\prime }S}$
Relative velocity equation connecting more than two reference frames	${\overrightarrow{v}}_{PC}={\overrightarrow{v}}_{PA}+{\overrightarrow{v}}_{AB}+{\overrightarrow{v}}_{BC}$
Relative acceleration equation	${\overrightarrow{a}}_{PS}={\overrightarrow{a}}_{P{S}^{\prime }}+{\overrightarrow{a}}_{S^{\prime }S}$