Kinematics/Basic Kinematics

Translational

Relative Motion: ${\overrightarrow{r}}_{A,C}={\overrightarrow{r}}_{A,B}+{\overrightarrow{r}}_{B,C}$

Variable	Unit	Context
$\overrightarrow{r}$	(m)	Position
_i,j	unitless	i wrt j

Distance

• ${\overrightarrow{r}}_{sys}=\frac{\sum m_i{\overrightarrow{x}}_i}{\sum m_i}$

  Variable	Unit	Context
  m	(kg)	Mass (Inertia)
  $\overrightarrow{r}$	(m)	Position
  _sys	unitless	Of System
  _i	unitless	wrt i

• ${\overrightarrow{r}}_{sys}=\frac{1}{\sum m_i}\int \overrightarrow{x}dm$

  Variable	Unit	Context
  m	(kg)	Mass (Inertia)
  $\overrightarrow{r}$	(m)	Position
  _sys	unitless	Of System
  _i	unitless	wrt i

Velocity

• $\overrightarrow{v}=\frac{d\overrightarrow{r}}{dt}$

  Variable	Unit	Context
  $\overrightarrow{r}$	(m)	Position
  t	(s)	Time
  $\overrightarrow{v}$	$(m{s}^{-1})$	Velocity
  i	unitless	Dummy Variable

• ${\overrightarrow{v}}_{sys}=\frac{\sum m_i{v}_i}{\sum m_i}$

  Variable	Unit	Context
  m	(kg)	Mass (Inertia)
  _sys	unitless	Of System
  _i	unitless	wrt i
  $\overrightarrow{v}$	$(m{s}^{-1})$	Velocity

• ${\overrightarrow{v}}_{sys}=\frac{1}{\sum m_i}\int \overrightarrow{v}dm$

  Variable	Unit	Context
  m	(kg)	Mass (Inertia)
  _sys	unitless	Of System
  _i	unitless	wrt i
  $\overrightarrow{v}$	$(m{s}^{-1})$	Velocity

Acceleration

• $\overrightarrow{a}=\frac{d\overrightarrow{v}}{dt}$

  Variable	Unit	Context
  t	(s)	Time
  $\overrightarrow{v}$	$(m{s}^{-1})$	Velocity
  $\overrightarrow{a}$	$(m{s}^{-2})$	Acceleration
  i	unitless	Dummy Variable

• ${\overrightarrow{a}}_{sys}=\frac{\sum m_i{\overrightarrow{a}}_i}{\sum m_i}$

  Variable	Unit	Context
  m	(kg)	Mass (Inertia)
  _sys	unitless	Of System
  _i	unitless	wrt i
  $\overrightarrow{a}$	$(m{s}^{-2})$	Acceleration

• ${\overrightarrow{a}}_{sys}=\frac{1}{\sum m_i}\int \overrightarrow{a}dm$

  Variable	Unit	Context
  m	(kg)	Mass (Inertia)
  _sys	unitless	Of System
  _i	unitless	wrt i
  $\overrightarrow{a}$	$(m{s}^{-2})$	Acceleration

Constant Linear Acceleration

• $\overrightarrow{v}={\overrightarrow{v}}_0+\overrightarrow{a}t$

  Variable	Unit	Context
  t	(s)	Time
  $\overrightarrow{v}$	$(m{s}^{-1})$	Velocity
  $\overrightarrow{a}$	$(m{s}^{-2})$	Acceleration
  _0	unitless	Initial

• $\overrightarrow{r}={\overrightarrow{r}}_0+{\overrightarrow{v}}_{0}t+\frac{\overrightarrow{a}{t}^2}{2}$

  Variable	Unit	Context
  $\overrightarrow{r}$	(m)	Position
  t	(s)	Time
  $\overrightarrow{v}$	$(m{s}^{-1})$	Velocity
  $\overrightarrow{a}$	$(m{s}^{-2})$	Acceleration
  _0	unitless	Initial

• ${\overrightarrow{v}}^2-{\overrightarrow{v}}_0^2=2a(\overrightarrow{r}-{\overrightarrow{r}}_0)$

  Variable	Unit	Context
  $\overrightarrow{r}$	(m)	Position
  $\overrightarrow{v}$	$(m{s}^{-1})$	Velocity
  $\overrightarrow{a}$	$(m{s}^{-2})$	Acceleration
  _0	unitless	Initial

• $\overrightarrow{r}-{\overrightarrow{r}}_0=\frac{(\overrightarrow{v}-{\overrightarrow{v}}_0)t}{2}$

  Variable	Unit	Context
  $\overrightarrow{r}$	(m)	Position
  $\overrightarrow{v}$	$(m{s}^{-1})$	Velocity
  t	(s)	Time
  _0	unitless	Initial

• Projectile Motion: $\overrightarrow{r}=\frac{{\overrightarrow{v}}_0^2\sin (2{\theta }_0)}{g}$ Polar Coordinates

  Variable	Unit	Context
  $\theta$	${(}^{(R)})$	Angle
  $\overrightarrow{r}$	(m)	Position
  $\overrightarrow{v}$	$(m{s}^{-1})$	Velocity
  _0	unitless	Initial
  g = 9.80665	$(m{s}^{-}2)$	Near-Earth Gravitational Acceleration

Momentum

• $\overrightarrow{p}=m\overrightarrow{v}$

  Variable	Unit	Context
  $\overrightarrow{p}$	$(kgm{s}^{-1})$	Momentum
  $\overrightarrow{v}$	$(m{s}^{-1})$	Velocity
  m	(kg)	Mass (Inertia)

• $\overrightarrow{P}=\sum \overrightarrow{p}$

  Variable	Unit	Context
  $\overrightarrow{p}$	$(kgm{s}^{-1})$	Momentum
  $\overrightarrow{P}$	$(kgm{s}^{-1})$	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)
$\rho$	$(kg{m}^{-3})$	Density

Newton's Laws

• First Law: ${\overrightarrow{F}}_{net}=0⟹\overrightarrow{a}=0$

  Variable	Unit	Context
  $\overrightarrow{F}$	$(N)=(kgm{s}^{-2})$	Force
  _net	unitless	Total
  $\overrightarrow{a}$	$(m{s}^{-2})$	Acceleration

• Second Law: $\overrightarrow{F}=\overrightarrow{a}m$

  Variable	Unit	Context
  $\overrightarrow{a}$	$(m{s}^{-2})$	Acceleration
  m	(kg)	Mass (Inertia)
  $\overrightarrow{F}$	$(N)=(kgm{s}^{-2})$	Force

• Third Law: ${\overrightarrow{F}}_{A,B}=-{\overrightarrow{F}}_{B,A}$

  Variable	Unit	Context
  $\overrightarrow{F}$	$(N)=(kgm{s}^{-2})$	Force
  _i,j	unitless	i wrt j

Rotational

[Ark](http://i.imgur.com/E18yHXv.png): $\overrightarrow{s}=\overrightarrow{\theta }\times \overrightarrow{r}$

Variable	Unit	Context
$\theta$	${(}^{(R)})$	Angle
s	$(m^{(R)})$	Ark
$\overrightarrow{r}$	$(m)$	Radius

Uniform Circular Motion (Circles): |\vec{r} | is constant [Velocity](http://i.imgur.com/VyDaV6k.png)

• Angular Velocity: $\overrightarrow{\omega }=\frac{d\theta }{dt}\hat{\theta }$

  Variable	Unit	Context
  $\overrightarrow{\omega }$	${(}^{(R)}{s}^{-1})$	Angular Velocity
  $\theta$	${(}^{(R)})$	Angle
  t	(s)	Time

• Velocity Components: $\overrightarrow{v}={\overrightarrow{v}}_{\parallel }+{\overrightarrow{v}}_{\perp }$

  Variable	Unit	Context
  $\overrightarrow{v}$	$(m{s}^{-1})$	Velocity
  ${\overrightarrow{v}}_{\parallel }$	$(m{s}^{-1})$	Parallel Velocity
  ${\overrightarrow{v}}_{\perp }$	$(m{s}^{-1})$	Perpendicular Velocity

• Uniform Circular Motion: $\overrightarrow{v}=\overrightarrow{\omega }\times \overrightarrow{r}$

  Variable	Unit	Context
  $\overrightarrow{\omega }$	${(}^{(R)}{s}^{-1})$	Angular Velocity
  $\overrightarrow{v}$	$(m{s}^{-1})$	Velocity
  $\overrightarrow{r}$	$(m)$	Radius

[Acceleration](http://i.imgur.com/Gl0O3lS.png)

• Angular Acceleration: ${\alpha }_a=\frac{d\overrightarrow{\omega }}{dt}$

  Variable	Unit	Context
  ${\overrightarrow{\alpha }}_a$	${(}^{(R)}{s}^{-2})$	Angular Acceleration
  $\overrightarrow{\omega }$	${(}^{(R)}{s}^{-1})$	Angular Velocity
  t	(s)	Time

• Acceleration Components: $\overrightarrow{a}={\overrightarrow{a}}_r+{\overrightarrow{a}}_T$

  Variable	Unit	Context
  $\overrightarrow{a}$	$(m{s}^{-2})$	Acceleration
  ${\overrightarrow{a}}_r$	$(m{s}^{-2})$	Radial Acceleration
  ${\overrightarrow{a}}_T$	$(m{s}^{-2})$	Tangential Acceleration

• Uniform Circular Motion
  • Tangential Acceleration: ${\overrightarrow{a}}_T={\overrightarrow{\alpha }}_a\times \overrightarrow{r}$

    Variable	Unit	Context
    ${\overrightarrow{\alpha }}_a$	${(}^{(R)}{s}^{-2})$	Angular Acceleration
    ${\overrightarrow{a}}_T$	$(m{s}^{-2})$	Tangential Acceleration
    $\overrightarrow{r}$	$(m)$	Radius

  • Radial Acceleration: ${\overrightarrow{\alpha }}_r=\frac{{\overrightarrow{v}}^2}{r}\hat{r}={\overrightarrow{\omega }}^2\overrightarrow{r}$

    Variable	Unit	Context
    ${\overrightarrow{a}}_r$	$(m{s}^{-2})$	Radial Acceleration
    $\overrightarrow{r}$	$(m)$	Radius
    $\overrightarrow{v}$	$(m{s}^{-1})$	Velocity
    $\overrightarrow{\omega }$	${(}^{(R)}{s}^{-1})$	Angular Velocity

Constant Angular Acceleration

• $\overrightarrow{\omega }={\overrightarrow{\omega }}_0+{\overrightarrow{\alpha }}_{a}t$

  Variable	Unit	Context
  t	(s)	Time
  $\overrightarrow{\omega }$	${(}^{(R)}{s}^{-1})$	Angular Velocity
  ${\overrightarrow{\alpha }}_a$	${(}^{(R)}{s}^{-2})$	Angular Acceleration
  _0	unitless	Initial

• $\theta ={\theta }_0+{\overrightarrow{\omega }}_{0}t+\frac{{\overrightarrow{\alpha }}_a{t}^2}{2}$

  Variable	Unit	Context
  $\theta$	${(}^{(R)})$	Angle
  t	(s)	Time
  $\overrightarrow{\omega }$	${(}^{(R)}{s}^{-1})$	Angular Velocity
  ${\overrightarrow{\alpha }}_a$	${(}^{(R)}{s}^{-2})$	Angular Acceleration
  _0	unitless	Initial

• ${\overrightarrow{\omega }}^2-{\overrightarrow{\omega }}_0^2=2{\overrightarrow{\alpha }}_a(\theta -{\theta }_0)$

  Variable	Unit	Context
  $\theta$	${(}^{(R)})$	Angle
  $\overrightarrow{\omega }$	${(}^{(R)}{s}^{-1})$	Angular Velocity
  ${\overrightarrow{\alpha }}_a$	${(}^{(R)}{s}^{-2})$	Angular Acceleration
  _0	unitless	Initial

• $\theta -{\theta }_0=\frac{(\overrightarrow{\omega }-{\overrightarrow{\omega }}_0)t}{2}$

  Variable	Unit	Context
  $\theta$	${(}^{(R)})$	Angle
  $\overrightarrow{\omega }$	${(}^{(R)}{s}^{-1})$	Angular Velocity
  t	(s)	Time
  _0	unitless	Initial

Moment of Inertia (Angular Mass): $I=\sum m_i{r}_i^2=\underset{m}{\int }{r}^{2}dm$

Variable	Unit	Context
I	$(kg{m}^2)$	Moment of Inertia (Angular Mass)
$\overrightarrow{r}$	$(m)$	Radius
m	(kg)	Mass (Inertia)
i	unitless	Dummy Variable

Angular Momentum: $\overrightarrow{L}=\overrightarrow{r}\times \overrightarrow{p}$

Variable	Unit	Context
$\overrightarrow{r}$	$(m)$	Radius
$\overrightarrow{L}$	$(Js)$	Angular Momentum
$\overrightarrow{p}$	$(kgm{s}^{-1})$	Momentum

Uniform Circular Motion: $\overrightarrow{L}=I\overrightarrow{\omega }$

Variable	Unit	Context
I	$(kg{m}^2)$	Moment of Inertia (Angular Mass)
$\overrightarrow{L}$	$(Js)$	Angular Momentum
$\overrightarrow{\omega }$	${(}^{(R)}{s}^{-1})$	Angular Velocity

Force

• Torque: $\tau =\overrightarrow{r}\times \overrightarrow{F}$

  Variable	Unit	Context
  $\tau$	$(J^{(R^{-1})})$	Torque
  $\overrightarrow{r}$	$(m)$	Radius
  $\overrightarrow{F}$	$(N)=(kgm{s}^{-2})$	Force

  • Dipole Moment: $\overrightarrow{\tau }=\overrightarrow{p}\times \overrightarrow{E}$

    Variable	Unit	Context
    $\overrightarrow{\tau }$	(N/m)	Torque on Rigid Dipole
    $\overrightarrow{p}$	(C m)	Electric Dipole Moment
    E	(N/C) = (V/m)	Electric Field

• Centripetal/Tangential Force: $F_c=m{\alpha }_T$

  Variable	Unit	Context
  $F_c$	${(}^{(R)}N)={(}^{(R)}kgm{s}^{-2})$	Tangential Force
  ${\overrightarrow{a}}_T$	$(m{s}^{-2})$	Tangential Acceleration
  m	(kg)	Mass (Inertia)

• Uniform Circular Motion: $\tau =I{\alpha }_a$

  Variable	Unit	Context
  $\tau$	$(J^{(R^{-1})})$	Torque
  I	$(kg{m}^2)$	Moment of Inertia (Angular Mass)
  ${\overrightarrow{\alpha }}_a$	${(}^{(R)}{s}^{-2})$	Angular Acceleration