# Kinematics/Basic Kinematics
## Translational

Relative Motion: ${\vec {r}}_{A,C}={\vec {r}}_{A,B}+{\vec {r}}_{B,C}$
| Variable | Unit | Context |
| ${\vec {r}}$ | (m) | Position |
| _i,j | unitless | i wrt j |
Distance
* ${\vec {r}}_{sys}={\frac {\sum m_{i}{\vec {x}}_{i}}{\sum m_{i}}}$ Variable Unit Context m (kg) Mass (Inertia) ${\vec {r}}$ (m) Position _sys unitless Of System _i unitless wrt i
* ${\vec {r}}_{sys}={\frac {1}{\sum m_{i}}}\int {\vec {x}}dm$ Variable Unit Context m (kg) Mass (Inertia) ${\vec {r}}$ (m) Position _sys unitless Of System _i unitless wrt i
Velocity
* ${\vec {v}}={\frac {d{\vec {r}}}{dt}}$ Variable Unit Context ${\vec {r}}$ (m) Position t (s) Time ${\vec {v}}$ $(ms^{-1})$ Velocity i unitless Dummy Variable
* ${\vec {v}}_{sys}={\frac {\sum m_{i}v_{i}}{\sum m_{i}}}$ Variable Unit Context m (kg) Mass (Inertia) _sys unitless Of System _i unitless wrt i ${\vec {v}}$ $(ms^{-1})$ Velocity
* ${\vec {v}}_{sys}={\frac {1}{\sum m_{i}}}\int {\vec {v}}dm$ Variable Unit Context m (kg) Mass (Inertia) _sys unitless Of System _i unitless wrt i ${\vec {v}}$ $(ms^{-1})$ Velocity
Acceleration
* ${\vec {a}}={\frac {d{\vec {v}}}{dt}}$ Variable Unit Context t (s) Time ${\vec {v}}$ $(ms^{-1})$ Velocity ${\vec {a}}$ $(ms^{-2})$ Acceleration i unitless Dummy Variable
* ${\vec {a}}_{sys}={\frac {\sum m_{i}{\vec {a}}_{i}}{\sum m_{i}}}$ Variable Unit Context m (kg) Mass (Inertia) _sys unitless Of System _i unitless wrt i ${\vec {a}}$ $(ms^{-2})$ Acceleration
* ${\vec {a}}_{sys}={\frac {1}{\sum m_{i}}}\int {\vec {a}}dm$ Variable Unit Context m (kg) Mass (Inertia) _sys unitless Of System _i unitless wrt i ${\vec {a}}$ $(ms^{-2})$ Acceleration
Constant Linear Acceleration
* ${\vec {v}}={\vec {v}}_{0}+{\vec {a}}t$ Variable Unit Context t (s) Time ${\vec {v}}$ $(ms^{-1})$ Velocity ${\vec {a}}$ $(ms^{-2})$ Acceleration _0 unitless Initial
* ${\vec {r}}={\vec {r}}_{0}+{\vec {v}}_{0}t+{\frac {{\vec {a}}t^{2}}{2}}$ Variable Unit Context ${\vec {r}}$ (m) Position t (s) Time ${\vec {v}}$ $(ms^{-1})$ Velocity ${\vec {a}}$ $(ms^{-2})$ Acceleration _0 unitless Initial
* ${\vec {v}}^{2}-{\vec {v}}_{0}^{2}=2a({\vec {r}}-{\vec {r}}_{0})$ Variable Unit Context ${\vec {r}}$ (m) Position ${\vec {v}}$ $(ms^{-1})$ Velocity ${\vec {a}}$ $(ms^{-2})$ Acceleration _0 unitless Initial
* ${\vec {r}}-{\vec {r}}_{0}={\frac {({\vec {v}}-{\vec {v}}_{0})t}{2}}$ Variable Unit Context ${\vec {r}}$ (m) Position ${\vec {v}}$ $(ms^{-1})$ Velocity t (s) Time _0 unitless Initial
* Projectile Motion: ${\vec {r}}={\frac {{\vec {v}}_{0}^{2}\sin(2\theta _{0})}{g}}$ Polar Coordinates Variable Unit Context $\theta$ $(^{(R)})$ Angle ${\vec {r}}$ (m) Position ${\vec {v}}$ $(ms^{-1})$ Velocity _0 unitless Initial g = 9.80665 $(ms^{-}2)$ Near-Earth Gravitational Acceleration
Momentum
* ${\vec {p}}=m{\vec {v}}$ Variable Unit Context ${\vec {p}}$ $(kgms^{-1})$ Momentum ${\vec {v}}$ $(ms^{-1})$ Velocity m (kg) Mass (Inertia)
* ${\vec {P}}=\sum {\vec {p}}$ Variable Unit Context ${\vec {p}}$ $(kgms^{-1})$ Momentum ${\vec {P}}$ $(kgms^{-1})$ Total Momentum
Mass: $M=\sum m_{i}$
| Variable | Unit | Context |
| m | (kg) | Mass (Inertia) |
| M | (kg) | Total Mass (Inertia) |
| i | unitless | Dummy Variable |
Density: $\rho ={\frac {m}{V}}$
| Variable | Unit | Context |
| V | $(m^{3})$ | Volume |
| m | (kg) | Mass (Inertia) |
| $\rho$ | $(kgm^{-3})$ | Density |
Newton's Laws
* First Law: ${\vec {F}}_{net}=0\implies {\vec {a}}=0$ Variable Unit Context ${\vec {F}}$ $(N)=(kgms^{-2})$ Force _net unitless Total ${\vec {a}}$ $(ms^{-2})$ Acceleration
* Second Law: ${\vec {F}}={\vec {a}}m$ Variable Unit Context ${\vec {a}}$ $(ms^{-2})$ Acceleration m (kg) Mass (Inertia) ${\vec {F}}$ $(N)=(kgms^{-2})$ Force
* Third Law: ${\vec {F}}_{A,B}=-{\vec {F}}_{B,A}$ Variable Unit Context ${\vec {F}}$ $(N)=(kgms^{-2})$ Force _i,j unitless i wrt j

## Rotational

[Ark](http://i.imgur.com/E18yHXv.png): ${\vec {s}}={\vec {\theta }}\times {\vec {r}}$
| Variable | Unit | Context |
| $\theta$ | $(^{(R)})$ | Angle |
| s | $(m^{(R)})$ | Ark |
| ${\vec {r}}$ | $(m)$ | Radius |
Uniform Circular Motion (Circles): |\vec{r} | is constant [Velocity](http://i.imgur.com/VyDaV6k.png)
* Angular Velocity: ${\vec {\omega }}={\frac {d\theta }{dt}}{\hat {\theta }}$ Variable Unit Context ${\vec {\omega }}$ $(^{(R)}s^{-1})$ Angular Velocity $\theta$ $(^{(R)})$ Angle t (s) Time
* Velocity Components: ${\vec {v}}={\vec {v}}_{\parallel }+{\vec {v}}_{\perp }$ Variable Unit Context ${\vec {v}}$ $(ms^{-1})$ Velocity ${\vec {v}}_{\parallel }$ $(ms^{-1})$ Parallel Velocity ${\vec {v}}_{\perp }$ $(ms^{-1})$ Perpendicular Velocity
* Uniform Circular Motion: ${\vec {v}}={\vec {\omega }}\times {\vec {r}}$ Variable Unit Context ${\vec {\omega }}$ $(^{(R)}s^{-1})$ Angular Velocity ${\vec {v}}$ $(ms^{-1})$ Velocity ${\vec {r}}$ $(m)$ Radius
[Acceleration](http://i.imgur.com/Gl0O3lS.png)
* Angular Acceleration: $\alpha _{a}={\frac {d{\vec {\omega }}}{dt}}$ Variable Unit Context ${\vec {\alpha }}_{a}$ $(^{(R)}s^{-2})$ Angular Acceleration ${\vec {\omega }}$ $(^{(R)}s^{-1})$ Angular Velocity t (s) Time
* Acceleration Components: ${\vec {a}}={\vec {a}}_{r}+{\vec {a}}_{T}$ Variable Unit Context ${\vec {a}}$ $(ms^{-2})$ Acceleration ${\vec {a}}_{r}$ $(ms^{-2})$ Radial Acceleration ${\vec {a}}_{T}$ $(ms^{-2})$ Tangential Acceleration
* Uniform Circular Motion
** Tangential Acceleration: ${\vec {a}}_{T}={\vec {\alpha }}_{a}\times {\vec {r}}$ Variable Unit Context ${\vec {\alpha }}_{a}$ $(^{(R)}s^{-2})$ Angular Acceleration ${\vec {a}}_{T}$ $(ms^{-2})$ Tangential Acceleration ${\vec {r}}$ $(m)$ Radius
** Radial Acceleration: ${\vec {\alpha }}_{r}={\frac {{\vec {v}}^{2}}{r}}{\hat {r}}={\vec {\omega }}^{2}{\vec {r}}$ Variable Unit Context ${\vec {a}}_{r}$ $(ms^{-2})$ Radial Acceleration ${\vec {r}}$ $(m)$ Radius ${\vec {v}}$ $(ms^{-1})$ Velocity ${\vec {\omega }}$ $(^{(R)}s^{-1})$ Angular Velocity
Constant Angular Acceleration
* ${\vec {\omega }}={\vec {\omega }}_{0}+{\vec {\alpha }}_{a}t$ Variable Unit Context t (s) Time ${\vec {\omega }}$ $(^{(R)}s^{-1})$ Angular Velocity ${\vec {\alpha }}_{a}$ $(^{(R)}s^{-2})$ Angular Acceleration _0 unitless Initial
* $\theta =\theta _{0}+{\vec {\omega }}_{0}t+{\frac {{\vec {\alpha }}_{a}t^{2}}{2}}$ Variable Unit Context $\theta$ $(^{(R)})$ Angle t (s) Time ${\vec {\omega }}$ $(^{(R)}s^{-1})$ Angular Velocity ${\vec {\alpha }}_{a}$ $(^{(R)}s^{-2})$ Angular Acceleration _0 unitless Initial
* ${\vec {\omega }}^{2}-{\vec {\omega }}_{0}^{2}=2{\vec {\alpha }}_{a}(\theta -\theta _{0})$ Variable Unit Context $\theta$ $(^{(R)})$ Angle ${\vec {\omega }}$ $(^{(R)}s^{-1})$ Angular Velocity ${\vec {\alpha }}_{a}$ $(^{(R)}s^{-2})$ Angular Acceleration _0 unitless Initial
* $\theta -\theta _{0}={\frac {({\vec {\omega }}-{\vec {\omega }}_{0})t}{2}}$ Variable Unit Context $\theta$ $(^{(R)})$ Angle ${\vec {\omega }}$ $(^{(R)}s^{-1})$ Angular Velocity t (s) Time _0 unitless Initial
Moment of Inertia (Angular Mass): $I=\sum m_{i}r_{i}^{2}=\int _{m}r^{2}dm$
| Variable | Unit | Context |
| I | $(kgm^{2})$ | Moment of Inertia (Angular Mass) |
| ${\vec {r}}$ | $(m)$ | Radius |
| m | (kg) | Mass (Inertia) |
| i | unitless | Dummy Variable |
Angular Momentum: ${\vec {L}}={\vec {r}}\times {\vec {p}}$
| Variable | Unit | Context |
| ${\vec {r}}$ | $(m)$ | Radius |
| ${\vec {L}}$ | $(Js)$ | Angular Momentum |
| ${\vec {p}}$ | $(kgms^{-1})$ | Momentum |
Uniform Circular Motion: ${\vec {L}}=I{\vec {\omega }}$
| Variable | Unit | Context |
| I | $(kgm^{2})$ | Moment of Inertia (Angular Mass) |
| ${\vec {L}}$ | $(Js)$ | Angular Momentum |
| ${\vec {\omega }}$ | $(^{(R)}s^{-1})$ | Angular Velocity |
Force
* Torque: $\tau ={\vec {r}}\times {\vec {F}}$ Variable Unit Context $\tau$ $(J^{(R^{-1})})$ Torque ${\vec {r}}$ $(m)$ Radius ${\vec {F}}$ $(N)=(kgms^{-2})$ Force
** Dipole Moment: ${\vec {\tau }}={\vec {p}}\times {\vec {E}}$ Variable Unit Context ${\vec {\tau }}$ (N/m) Torque on Rigid Dipole ${\vec {p}}$ (C m) Electric Dipole Moment E (N/C) = (V/m) Electric Field
* Centripetal/Tangential Force: $F_{c}=m\alpha _{T}$ Variable Unit Context $F_{c}$ $(^{(R)}N)=(^{(R)}kgms^{-2})$ Tangential Force ${\vec {a}}_{T}$ $(ms^{-2})$ Tangential Acceleration m (kg) Mass (Inertia)
* Uniform Circular Motion: $\tau =I\alpha _{a}$ Variable Unit Context $\tau$ $(J^{(R^{-1})})$ Torque I $(kgm^{2})$ Moment of Inertia (Angular Mass) ${\vec {\alpha }}_{a}$ $(^{(R)}s^{-2})$ Angular Acceleration