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The most frequently used formulas from physics, math and statistics - each with a preview and copy-ready LaTeX code. Click a formula to see the code and how to use it.

Mechanics

Distance (uniformly accelerated motion)

s=s0+v0t+12at2s = s_0 + v_0 t + \tfrac{1}{2}a t^2

Newton's second law

F⃗=ma⃗\vec{F} = m\vec{a}

Kinetic energy

Ek=12mv2E_k = \tfrac{1}{2}m v^2

Potential energy

Ep=mghE_p = m g h

Momentum

p⃗=mv⃗\vec{p} = m\vec{v}

Pendulum period

T=2πLgT = 2\pi\sqrt{\dfrac{L}{g}}

Gravitational acceleration

g=4π2LT2g = \dfrac{4\pi^2 L}{T^2}

Work

W=F⃗⋅s⃗=Fscos⁡αW = \vec{F}\cdot\vec{s} = F s\cos\alpha

Electricity

Ohm's law

U=R IU = R\,I

Electric power

P=UI=I2R=U2RP = U I = I^2 R = \dfrac{U^2}{R}

Resistance of a conductor

R=ρlSR = \rho\dfrac{l}{S}

Resistors in series

Rz=R1+R2+⋯+RnR_z = R_1 + R_2 + \dots + R_n

Resistors in parallel

1Rz=∑i1Ri\dfrac{1}{R_z} = \sum_{i} \dfrac{1}{R_i}

Capacitance

C=QUC = \dfrac{Q}{U}

Coulomb's law

F=14πε0q1q2r2F = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1 q_2}{r^2}

Thermodynamics

Ideal gas law

pV=nRTpV = nRT

First law of thermodynamics

ΔU=Q−W\Delta U = Q - W

Heat

Q=mcΔTQ = m c \Delta T

Gibbs free energy

ΔG=ΔH−TΔS\Delta G = \Delta H - T\Delta S

Carnot efficiency

η=1−T2T1\eta = 1 - \dfrac{T_2}{T_1}

Waves and optics

Wave speed

v=λfv = \lambda f

Thin lens equation

1f=1x+1y\dfrac{1}{f} = \dfrac{1}{x} + \dfrac{1}{y}

Snell's law

n1sin⁡θ1=n2sin⁡θ2n_1\sin\theta_1 = n_2\sin\theta_2

Photon energy

E=hν=hcλE = h\nu = \dfrac{hc}{\lambda}

De Broglie wavelength

λ=hmv\lambda = \dfrac{h}{mv}

Statistics and uncertainty

Mean value

xˉ=1n∑i=1nxi\bar{x} = \dfrac{1}{n}\sum_{i=1}^{n} x_i

Standard deviation

s=1n−1∑i=1n(xi−xˉ)2s = \sqrt{\dfrac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2}

Type A uncertainty

uA=snu_A = \dfrac{s}{\sqrt{n}}

Combined uncertainty

uc=uA2+uB2u_c = \sqrt{u_A^2 + u_B^2}

Uncertainty propagation

uc(y)=∑i(∂y∂xiuxi)2u_c(y) = \sqrt{\sum_{i}\left(\dfrac{\partial y}{\partial x_i}u_{x_i}\right)^2}

Expanded uncertainty

U=k uc(k=2)U = k\,u_c \quad (k=2)

Mathematics

Binomial square formulas

(a±b)2=a2±2ab+b2(a\pm b)^2 = a^2 \pm 2ab + b^2

Quadratic formula

x=−b±b2−4ac2ax = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Product rule

(fg)’=f’g+fg’(fg)’ = f’g + fg’

Definite integral

∫abf(x) dx\int_{a}^{b} f(x)\,\mathrm{d}x

2×2 matrix

(abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix}

System of equations

{a1x+b1y=c1a2x+b2y=c2\begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}

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