Kinematics of a point mass

$\displaystyle \vec{v}=\frac{\partial\vec{d}}{\partial t},\quad \vec{a}=\frac{\partial\vec{v}}{\partial t}=\frac{\partial^2\vec{d}}{\partial t^2},\quad \vec{d}=\int{\vec{v}\cdot\d{t}},\quad \vec{v}=\int{\vec{a}\cdot\d{t}};\quad$

Polar coordinate $\begin{cases}r=r(t)\\\theta=\theta(t)\end{cases}$:  $\dfrac{\partial\vec{e_r}}{\partial t}=\dot{\theta}\vec{e_\theta},\quad \dfrac{\partial\vec{e_\theta}}{\partial t}=-\dot{\theta}\vec{e_r};\quad$ $\vec{d}=r\vec{e_r},\quad \vec{v}=\dot{r}\vec{e_r}+r\dot{\theta}\vec{e_\theta},\quad \vec{a}=\left(\ddot{r}-r\dot{\theta}^2\right)\vec{e_r}+\left(2\dot{r}\dot{\theta}+r\ddot{\theta}\right)\vec{e_\theta};\quad$
Eigen coordinate:  $\vec{v}=v\vec{e_t},\quad \vec{a}=\dfrac{\partial v}{\partial t}\vec{e_t}+\dfrac{v^2}{R}\vec{e_n};\quad$ where $R$ is the curvature radius;

Motion related to a uniformly accelerating reference frame:  $\vec{v}=\vec{v_r}+\dot{\vec{r_0}},\quad \vec{a}=\vec{a_r}+\ddot{\vec{r_0}}=\vec{a_r}+\vec{a_0};\quad$
   where $\vec{v_r}$ and $\vec{a_r}$ are relative motion and acceleration, $\vec{v_0}$ and $\vec{a_0}$ are the velocity and acceleration of the reference frame;

* Motion related to a uniformly rotating reference frame:  $\vec{v}=\vec{v_r}+\vec{\omega}\times\vec{r},\quad \vec{a}=\vec{a_r}+\vec{\omega}\times\left(\vec{\omega}\times\vec{r}\right)+2\vec{\omega}\times\vec{v_r};$

Centrifugal acceleration:  $a=\dfrac{v^2}{r}=w^2r=4\pi^2rf^2=\dfrac{4\pi^2r}{T^2};$

Motion with constant acceleration

$\vec{v_1}=\vec{v_0}+\vec{a}\Delta t,\quad \Delta\vec{d}=\left(\dfrac{\vec{v_0}+\vec{v_1}}{2}\right)\Delta t,\quad \Delta\vec{d}=\vec{v_0}\Delta t+\dfrac{1}{2}\vec{a}\Delta t^2 =\vec{v_1}\Delta t-\dfrac{1}{2}\vec{a}\Delta t^2,\quad v_1^2=v_0^2+2\vec{a}\cdot\vec{d};\quad$

Projectile motion in 2D:  $\displaystyle \vec{d}(t)=\left(v_0\cos(\theta)t,\ v_0\sin(\theta)t-\frac{1}{2}gt^2\right),\quad t_1=\frac{2v_0\sin(\theta)}{g},\quad \vec{d_1}=\left(\frac{v_0^2\sin(2\theta)}{g},0\right),\quad \vec{d_{1/2}}=\left(\frac{v_0^2\sin(\theta)\cos(\theta)}{g},\frac{v_{0}^{2}\sin(\theta)^2}{2g}\right);\quad$

Rigid body

Fluid

Forces

Gravitational force:  $F=G\dfrac{Mm}{r^2},\quad \vec{F}=-G\dfrac{Mm}{r^2}\vec{e_r}=-G\dfrac{Mm\vec{r}}{|\vec{r}|^3};\quad$ where $G\approx6.674\times10^{-11}\ m^3kg^{-1}s^{-2}$;
Gravitational potential energy:  $E=-G\dfrac{Mm}{r};\quad$ Close to earth’s surface:  $E=mgh,\quad F=mg;$

Hooke’s law for springs:  $\vec{F}=-k\Delta\vec{x},\quad E=\dfrac{1}{2}(\Delta\vec{x})^2;$

Inertial force in uniformly accelerating reference frame:  $\vec{F}=-m\vec{a_0};$
Inertial force in uniformly rotating reference frame:  centrifugal force $\vec{F}=m\omega^2r\cdot\vec{e_r}$,  Coriolis force $\vec{F}=2m\cdot\vec{v}_{relative}\times\vec{\omega}$;

Momentum

$\vec{p}=m\vec{v},\quad \Delta\vec{p}=\vec{F}\Delta t;$

Work and kinetic energy

$W=\vec{F}\cdot\Delta\vec{d},\quad E_k=\dfrac{1}{2}mv^2,\quad W=\Delta E_k=\dfrac{1}{2}mv_1^2-\dfrac{1}{2}mv_0^2;\quad P=\dfrac{W}{t}=\dfrac{\Delta E}{\Delta t};$

Collision

Elastic collision:  $\displaystyle \sum_i{m_i\vec{u_i}}=\sum_i{m_i\vec{v_i}},\quad \sum_i{\dfrac{1}{2}m_iu_i^2}=\sum_i{\dfrac{1}{2}m_iv_i^2};\quad$

Elastic collision in one dimension:  $\vec{v_1}=\dfrac{(m_1-m_2)\vec{u_1}+2m_2\vec{u_2}}{m_1+m_2},\quad \vec{v_2}=\dfrac{(m_2-m_1)\vec{u_2}+2m_1\vec{u_1}}{m_1+m_2};\quad$   When $m_1=m_2$:  $\vec{v_1}=\vec{u_2},\ \vec{v_2}=\vec{u_1};$

Elastic non-central collision:  $m_1\vec{v_1}=m_1\vec{u_1}+\Delta{p}\cdot\mathbf{\vec{n}},\quad m_2\vec{v_2}=m_2\vec{u_2}-\Delta{p}\cdot\mathbf{\vec{n}},\quad \Delta{p}=\dfrac{2m_1m_2\left(\vec{u_2}-\vec{u_1}\right)\cdot\mathbf{\vec{n}}}{m_1+m_2};$    where $\mathbf{\vec{n}}$ is the unit normal (direction of force);

Geometrical optics

Fermat’s principle: the path taken by a ray between two given points is the path that takes the least time.

Reflection:  $\theta_r=\theta_i,\quad \vec{r_r}=\vec{r_i}-\left(2\vec{r_i}\cdot\mathbf{\vec{n}}\right)\mathbf{\vec{n}};\quad$  where $\mathbf{\vec{n}}$ is the unit normal of reflection;

Refraction (Snell’s law):  $n_1\sin\theta_1=n_2\sin\theta_2;\quad$ * $\vec{r_r}=\eta\cdot\vec{r_i}-\left(\sqrt{k}+\eta\vec{r_i}\cdot\mathbf{\vec{n}}\right)\mathbf{\vec{n}},\;\; k=1-\eta^2\left(1-\left(\vec{r_i}\cdot\mathbf{\vec{n}}\right)^2\right),\;\; \eta=\dfrac{n_i}{n_r};\quad$
   where $n=\dfrac{c}{v}$ is the index of refraction, $\vec{r_i}$ and $\vec{r_r}$ are unit vectors, $\mathbf{\vec{n}}$ is the unit normal with $\vec{r_i}\cdot\mathbf{\vec{n}}\le0$;   A negative $k=\cos^2\left(\theta_r\right)$ indicates a total internal reflection;

Equations of ideal mirrors and lenses:  $\dfrac{1}{f}=\dfrac{1}{d_i}+\dfrac{1}{d_o},\quad \dfrac{-h_i}{h_o}=\dfrac{d_i}{d_o}=-m;\quad$   $\left(d_i,h_i\right) = \dfrac{f}{d_o-f}\left(d_o,-h_o\right);\quad$
  • For mirrors: $f$ is positive for concave mirrors, a negative $d_i$ means behind the mirror;
  • For lenses: $f$ is positive for converging lenses, a negative $d_i$ means on the same side of the lens;

Wave basics

$f=\dfrac{1}{T},\;T=\dfrac{1}{f};\quad v=\dfrac{\lambda}{T}=f\lambda;\quad$   $\omega=\dfrac{2\pi}{T}=\dfrac{2\pi v}{\lambda},\; k=\dfrac{2\pi}{\lambda}=\dfrac{\omega}{v},\quad u=A\cos\left(\omega t\mp k\left(x-x_0\right)+\varphi\right);\quad$

Wave equation:  $\dfrac{\partial^2u}{\partial t^2}=v^2\left(\dfrac{\partial^2u}{\partial x_1^2}+\dfrac{\partial^2u}{\partial x_2^2}+\cdots+\dfrac{\partial^2u}{\partial x_n^2}\right);\quad$ General solution to wave equation in one dimension:  $u=C_1(x-vt)+C_2(x+vt);$

Speed of light in vacuum: $c=299\,792\,458\;m/s$;  Speed of wave on a string: $v=\sqrt{\dfrac{T}{\mu}}$;  Approximate speed of sound in air: $v=331.4m/s+(0.606m/s/℃)\cdot T_℃$;

* Doppler effect:  $f_{obs}=\dfrac{v_{wave}+v_{observer}}{v_{wave}+v_{source}}f_0;\quad$ $v_{wave}$ is positive when the wave source is moving away from the observer and negative when moving toward the observer;