Kinematics/Basic Kinematics

Translational
Relative Motion: $$ \vec{r}_{A,C} = \vec{r}_{A,B} + \vec{r}_{B,C} $$ Distance  $$ \vec{r}_{sys} = \frac{\sum m_{i} \vec{x}_{i}}{\sum m_{i}} $$  $$ \vec{r}_{sys} = \frac{1}{\sum m_{i}} \int \vec{x} dm $$   Velocity   $$ \vec{v} = \frac{d \vec{r}}{d t} $$ $$ \vec{v}_{sys} = \frac{\sum m_{i} v_{i}}{\sum m_{i}} $$ $$ \vec{v}_{sys} = \frac{1}{\sum m_{i}} \int \vec{v} dm $$ Acceleration  $$ \vec{a} = \frac{d \vec{v}}{dt} $$ $$ \vec{a}_{sys} = \frac{\sum m_{i} \vec{a}_{i}}{\sum m_{i}} $$ $$ \vec{a}_{sys} = \frac{1}{\sum m_{i}} \int \vec{a} dm $$ Constant Linear Acceleration  $$ \vec{v} = \vec{v}_{0} + \vec{a}t $$ $$ \vec{r} = \vec{r}_{0} + \vec{v}_{0} t + \frac{\vec{a}  t^{2} }{2} $$ $$ \vec{v}^{2} - \vec{v}_{0}^{2} = 2 a ( \vec{r} - \vec{r}_{0} ) $$ $$ \vec{r} -\vec{r}_0 = \frac{(\vec{v} - \vec{v}_{0}) t}{2} $$ Projectile Motion: $$ \vec{r}= \frac{\vec{v}_{0}^{2} \sin(2\theta_0)}{g}$$ Polar Coordinates 

Momentum  $$ \vec{p}=m \vec{v} $$ $$ \vec{P} = \sum \vec{p} $$ Mass: $$ M = \sum m_i $$ Density: $$ \rho = \frac{m}{V} $$ Newton's Laws <ul><li> First Law: $$ \vec{F}_{net} = 0 \implies \vec{a} = 0 $$ Second Law: $$ \vec{F} = \vec{a} m $$ Third Law: $$ \vec{F}_{A,B} = - \vec{F}_{B,A} $$ </li></ul>

Rotational
[Ark](http://i.imgur.com/E18yHXv.png): $$ \vec{s} = \vec{\theta} \times \vec{r} $$ Uniform Circular Motion (Circles): [Velocity](http://i.imgur.com/VyDaV6k.png) <ul><li>Angular Velocity: $$ \vec{\omega} = \frac{d\theta}{dt}\hat{\theta} $$ </li><li>Velocity Components: $$ \vec{v} = \vec{v}_{\parallel} + \vec{v}_{\perp} $$ </li><li>Uniform Circular Motion: $$ \vec{v} = \vec{\omega} \times \vec{r} $$ </li></ul> [Acceleration](http://i.imgur.com/Gl0O3lS.png) <ul><li>Angular Acceleration: $$ \alpha_{a} = \frac{d \vec{\omega}}{d t} $$ </li><li>Acceleration Components: $$ \vec{a} = \vec{a}_{r} + \vec{a}_{T} $$ </li><li>Uniform Circular Motion <ul><li>Tangential Acceleration: $$ \vec{a}_{T} =\vec{\alpha}_{a} \times \vec{r} $$ </li><li>Radial Acceleration: $$ \vec{\alpha}_{r} = \frac{\vec{v}^{2}}{r} \hat{r} = \vec{\omega}^{2} \vec{r} $$ </li></ul> </li></ul> Constant Angular Acceleration <ul><li>$$ \vec{\omega} = \vec{\omega}_{0} + \vec{\alpha}_a t $$ </li><li>$$ \theta = \theta_{0} + \vec{\omega}_{0} t + \frac{\vec{\alpha}_a  t^{2} }{2} $$ </li><li>$$ \vec{\omega}^{2} - \vec{\omega}_{0}^{2} = 2 \vec{\alpha}_a ( \theta - \theta_{0} ) $$ </li><li>$$ \theta-\theta_0 = \frac{(\vec{\omega} - \vec{\omega}_{0}) t}{2} $$ </li></ul> Moment of Inertia (Angular Mass): $$ I = \sum m_i r_i^2 = \int_m r^2 dm $$ Angular Momentum: $$ \vec{L} = \vec{r} \times \vec{p} $$ Uniform Circular Motion: $$ \vec{L} = I \vec{\omega} $$ Force <ul><li>Torque: $$ \tau = \vec{r} \times \vec{F} $$ <ul><li>Dipole Moment: $$ \vec{\tau} = \vec{p} \times \vec{E} $$ </li></ul> </li><li>Centripetal/Tangential Force: $$ F_{c} = m \alpha_{T} $$ </li><li>Uniform Circular Motion: $$ \tau = I \alpha_a $$ </li></ul>
 * \vec{r}
 * is constant