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