# Kinematics/Basic Kinematics

## Translational

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

Distance

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

Velocity

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

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

Acceleration

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

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

Constant Linear Acceleration

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

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

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

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

Momentum

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

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

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

Newton's Laws

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

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

## Rotational

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

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

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

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

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

Constant Angular Acceleration

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

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

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

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

Force

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