# Kinematics/Basic Kinematics

## Translational

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

Distance

• ${\displaystyle {\vec {r}}_{sys}={\frac {\sum m_{i}{\vec {x}}_{i}}{\sum m_{i}}}}$
• ${\displaystyle {\vec {r}}_{sys}={\frac {1}{\sum m_{i}}}\int {\vec {x}}dm}$

Velocity

• ${\displaystyle {\vec {v}}={\frac {d{\vec {r}}}{dt}}}$
• ${\displaystyle {\vec {v}}_{sys}={\frac {\sum m_{i}v_{i}}{\sum m_{i}}}}$

• ${\displaystyle {\vec {v}}_{sys}={\frac {1}{\sum m_{i}}}\int {\vec {v}}dm}$

Acceleration

• ${\displaystyle {\vec {a}}={\frac {d{\vec {v}}}{dt}}}$
• ${\displaystyle {\vec {a}}_{sys}={\frac {\sum m_{i}{\vec {a}}_{i}}{\sum m_{i}}}}$

• ${\displaystyle {\vec {a}}_{sys}={\frac {1}{\sum m_{i}}}\int {\vec {a}}dm}$

Constant Linear Acceleration

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

• ${\displaystyle {\vec {v}}^{2}-{\vec {v}}_{0}^{2}=2a({\vec {r}}-{\vec {r}}_{0})}$

• ${\displaystyle {\vec {r}}-{\vec {r}}_{0}={\frac {({\vec {v}}-{\vec {v}}_{0})t}{2}}}$

• Projectile Motion: ${\displaystyle {\vec {r}}={\frac {{\vec {v}}_{0}^{2}\sin(2\theta _{0})}{g}}}$  Polar Coordinates

Momentum

• ${\displaystyle {\vec {p}}=m{\vec {v}}}$
• ${\displaystyle {\vec {P}}=\sum {\vec {p}}}$

Mass: ${\displaystyle M=\sum m_{i}}$

Density: ${\displaystyle \rho ={\frac {m}{V}}}$

Newton's Laws

• First Law: ${\displaystyle {\vec {F}}_{net}=0\implies {\vec {a}}=0}$
• Second Law: ${\displaystyle {\vec {F}}={\vec {a}}m}$

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

## Rotational

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

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

• Angular Velocity: ${\displaystyle {\vec {\omega }}={\frac {d\theta }{dt}}{\hat {\theta }}}$
• Velocity Components: ${\displaystyle {\vec {v}}={\vec {v}}_{\parallel }+{\vec {v}}_{\perp }}$
• Uniform Circular Motion: ${\displaystyle {\vec {v}}={\vec {\omega }}\times {\vec {r}}}$

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

• Angular Acceleration: ${\displaystyle \alpha _{a}={\frac {d{\vec {\omega }}}{dt}}}$
• Acceleration Components: ${\displaystyle {\vec {a}}={\vec {a}}_{r}+{\vec {a}}_{T}}$
• Uniform Circular Motion
• Tangential Acceleration: ${\displaystyle {\vec {a}}_{T}={\vec {\alpha }}_{a}\times {\vec {r}}}$
• Radial Acceleration: ${\displaystyle {\vec {\alpha }}_{r}={\frac {{\vec {v}}^{2}}{r}}{\hat {r}}={\vec {\omega }}^{2}{\vec {r}}}$

Constant Angular Acceleration

• ${\displaystyle {\vec {\omega }}={\vec {\omega }}_{0}+{\vec {\alpha }}_{a}t}$
• ${\displaystyle \theta =\theta _{0}+{\vec {\omega }}_{0}t+{\frac {{\vec {\alpha }}_{a}t^{2}}{2}}}$
• ${\displaystyle {\vec {\omega }}^{2}-{\vec {\omega }}_{0}^{2}=2{\vec {\alpha }}_{a}(\theta -\theta _{0})}$
• ${\displaystyle \theta -\theta _{0}={\frac {({\vec {\omega }}-{\vec {\omega }}_{0})t}{2}}}$

Moment of Inertia (Angular Mass): ${\displaystyle I=\sum m_{i}r_{i}^{2}=\int _{m}r^{2}dm}$

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

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

Force

• Torque: ${\displaystyle \tau ={\vec {r}}\times {\vec {F}}}$
• Dipole Moment: ${\displaystyle {\vec {\tau }}={\vec {p}}\times {\vec {E}}}$
• Centripetal/Tangential Force: ${\displaystyle F_{c}=m\alpha _{T}}$
• Uniform Circular Motion: ${\displaystyle \tau =I\alpha _{a}}$