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