Classical Mechanics

$$v=\frac{dx}{dt}$$

$$a=\frac{dv}{dt}=\frac{d^2x}{dt^2}$$

Galilei's Relations

$$v=v_0+a*t$$

$$x=x_0+v_0*t+\frac{a*t^2}{2}$$

$$v^2=v_0^2+2*a*x$$

Vertical throw

$$t_{climb}=\frac{v_0}{g}$$

$$h_{max}=\frac{v_0^2}{2*g}$$

Circular motion

$$\omega=\frac{d\alpha}{dt}$$ - angular speed

T - motion period

$$\nu$$ - motion frequency

$$\omega=\frac{2*\pi}{T}=2*\pi*\nu$$

$$T=\frac{1}{\nu}$$

v - instantaneous speed

R - curvature radius

$$v=\omega*R$$

$$a_{centripetal}=\omega^2*R$$

$$v=\omega*R$$

$$F_{centripetal}=m*\omega^2*R=\frac{m*v^2}{R}$$

Newton's third law

$$\vec{F}=m*\vec{a}=\frac{d\vec{p}}{dt}$$

$$\vec{p}=m*\vec{v}$$

Friction force $$F_f=\mu*N$$

Hooke's law

k - spring constant

E - Young's modules

$$\vec{F_e}=-k*\vec{\Delta l}=-\frac{E*S}{l_0}*\vec{\Delta l}$$

Dilation

$$l=l_0(1+\alpha *\Delta t)$$

$$F_d=E*S*\alpha *\Delta t$$

Position, speed, acceleration, impulse of the mass center of a system of mass points:

$$x_{mc}=\frac{\sum_{k=1}^n m_k*x_k}{\sum_{k=1}^n m_k}$$

Torque $$\vec{M}=\vec{b} \times \vec{F}$$

Translation equilibrium $$\sum_{k+1}{n} \vec{F_k}=0$$

Rotation equilibrium $$\sum_{k+1}{n} \vec{M_k}=0$$

W - mechanical work

$$E_c$$ - kinetic energy

$$E_p$$ - potential energy

P - power

$$W=F*d$$

$$E_c=\frac{m*v^2}{2}$$

$$E_p=m*g*h$$ (gravitational)

$$P=\frac{W}{\Delta t}$$

$$\eta=\frac{W_{out}}{W_{in}}=1-\frac{L_{friction}}{L_{total}}=\frac{\sin \alpha}{\sin \alpha + \mu \cos \alpha}$$

Heat emanated from a plastic collision

$$Q=\frac{m_r*v_r^2}{2}$$

$$m_r=\frac{m_1*m_2}{m_1+m_2}$$

$$v_r=\left\vert v_2-v_1 \right\vert$$

Elastic collision (1-dimensional)

$$v_1'=2\frac{m_1v_1+m_2v_2}{m_1+m_2}-v_1$$

$$v_2'=2\frac{m_1v_1+m_2v_2}{m_1+m_2}-v_2$$

Rockets

M - initial mass of the rocket

m - instantaneous mass of the rocket

v - the speed of the rocket

u - the speed of the ejected particles

$$m=Me^{-\frac{v}{u}}$$

Moment of inertia

$$I=\sum_{i=1}^nm_ir_i^2$$

$$r_i$$ - distance to the axis of rotation.

Rotational kinetic energy E_r=\frac{I\omega^2}{2}

$$F*r=I*\epsilon$$

r - distance to the axis of rotation

$$\epsilon$$ - angular acceleration

Capilary action

$$h=\frac{2*\sigma*\cos\theta}{\rho*g*r}$$

$$\sigma$$ - surface tension

$$\theta$$ - contact angle

r - radius of the tube

$$2*\sigma_0*\left( \frac{1}{R}-\frac{1}{r} \right)=\rho*g*h$$

Curvature radius of a trajectory

$$R=\frac{(\dot{x}^2+\dot{y}^2)^{\frac{3}{2}}}{\vert \dot{x}\ddot{y}-\ddot{x}\dot{y}\vert}$$

Table of Young's modulus for some materials