Quadratic Formula |
$ax^2+bx+c=0\qquad x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}$ |
Power Rule |
$x=t^n \quad\text{then}\quad \dot{x}=n t^{n-1}$ |
Velocity and Acceleration in Rectilinear Motion |
$v=\dfrac{dx}{dt}\qquad a=\dfrac{dv}{dt}\qquad a=\dfrac{d^2x}{dt^2}\qquad a=v\dfrac{dv}{dx}$ |
Uniformaly accelerated rectilinear motion |
$v=v_0+at$ $x=x_0+v_0t +\dfrac{1}{2}at^2$ $v^2=v_0^2+2a\left(x-x_0\right)$ |
Tangential and Normal Components |
$\vec{v}=v \hat{e_t}$ $\vec{a}=\dfrac{dv}{dt}\hat{e_t}+\dfrac{v^2}{\rho}\hat{e_n}$ |
Radial and Transverse Components |
$\vec{v}=\dot{r} \hat{e_r} + r\dot{\theta}\hat{e_\theta}$ $\vec{a}=\left(\ddot{r}-r\dot{\theta}^2\right)\hat{e_r} + \left(r\ddot{\theta}+2\dot{r}\dot{\theta}\right)\hat{e_\theta}$ |
Newton's second law |
$\sum F_x=ma_x \qquad \sum F_y=ma_y$ |
Newton's second law using tangential and normal components |
$\sum F_t=m\dfrac{dv}{dt} \qquad \sum F_n=m\dfrac{v^2}{\rho}$ |
Newton's second law using radial and transverse components |
$\sum F_r=m\left(\ddot{r}-r\dot{\theta}^2\right) \qquad \sum F_\theta=m\left(r\ddot{\theta}+2\dot{r}\dot{\theta}\right)$ |
Angular Momentum |
$\vert{\vec H}_0\vert=H_0=rmv\sin\phi$ |
Newton's Law of Universal Gravitation |
$F=G\dfrac{Mm}{r^2}\qquad G=\begin{cases}6.67\times 10^{-11}\frac{\text{N m}^2}{\text{kg}^2} \\3.44\times 10^{-8} \frac{\text{lb ft}^2}{\text{slug}^2}\end{cases}$ |
Mass and radius of the earth |
$M=\begin{cases}5.97\times 10^{24}~\text{kg}\\4.09\times 10^{23} \text{slug}\end{cases} \qquad R=\begin{cases}6.37\times 10^{6}~\text{m}\\20.9\times 10^{6}~\text{ft}\end{cases}$ |
Work of a Force |
$U_{1 \to 2}={\displaystyle \int_{A_1}^{A_2} {\vec F}\cdot d{\vec r}}$ |
Work of a Weight |
$U_{1\to 2}=Wy_1-W y_2$ |
Work of a Spring |
$U_{1\to 2}=\frac{1}{2}kx_1^2-\frac{1}{2}kx_2^2$ |
Work of the Gravitational Force |
$U_{1\to 2}=\dfrac{GMm}{r_2}-\dfrac{GMm}{r_1}$ |
Principle of Work and Energy |
$\text{KE}_1+U_{1\to 2}=\text{KE}_2\quad \text{where}\quad \text{KE}=\frac{1}{2}mv^2$ |
Power and Mechanical Efficiency |
${\textrm{Power}}=\dfrac{dU}{dt}={\vec F}\cdot {\vec v} \qquad\qquad \eta=\dfrac{\textrm{power output}}{\textrm{power input}}$ |
Conservative Forces and Potential Energy $\quad U_{1 \to 2}=V_1 - V_2$ |
$$ \begin{eqnarray} V_g&=&Wy && \textrm{Force of gravity}\\ V_G&=&-\dfrac{GMm}{r}\qquad &&\textrm{Gravitational Force}\\ V_e&=&\frac{1}{2}kx^2 &&\textrm{Elastic force by a Spring} \end{eqnarray} $$ |
Principle of work and energy (setting $U_{1\to 2}^{\text{NC}}=0$ results in the principle of energy conservation) |
$\text{KE}_1+\left(V_g\right)_1+\left(V_e\right)_1+U_{1\to 2}^{\text{NC}} = \text{KE}_2+\left(V_g\right)_2+\left(V_e\right)_2$ |
Impulse and Momentum |
$m{\vec v}_1 + \sum\vec{J}_{1\to 2}=m {\vec v}_1 \qquad \vec{J}_{1\to 2}={\displaystyle \int_{t_1}^{t_2} {\vec F}\,dt}$ |
Direct central impact |
$m_A v_A + m_Bv_B=m_A v_A^\prime + m_Bv_B^\prime\\ v_B^\prime - v_A^\prime=e\left(v_A - v_B\right)$ |
Oblique central impact |
$\left(v_B^\prime\right)_n - \left(v_A^\prime\right)_n=e\left[\left(v_A\right)_n - \left(v_B\right)_n\right]$ |
Coefficient of restitution |
$e=\begin{cases}0\quad\text{perfectly plastic}\\1\quad\text{perfectly elastic} \end{cases}$ |