Physics formula sheet

Every equation behind a physics tool on this site, each linked to the tool that uses it. Many of those tools solve for whichever variable you choose, so you can check a rearrangement rather than trust it.

103 equations across 4 topics

Mechanics All Mechanics tools

Projectile Motion Simulator
Animate a trajectory and read off range, apex and flight time as you change launch speed and angle.

Galileo, Two New Sciences (1638)

$R = \frac{v_0^{2}\sin 2\theta}{g}$
Elastic and Inelastic Collision Simulator
Collide two bodies and watch momentum survive every time while kinetic energy only survives a perfect bounce.

Conservation of momentum with Newton’s law of restitution

$m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$
Simple Pendulum Simulator
Swing a pendulum, add damping, and watch the small-angle approximation fail at large amplitude.

Huygens, Horologium Oscillatorium (1673)

$T \approx 2\pi\sqrt{\frac{L}{g}}$
Orbit Simulator
One launch speed slider walks an orbit from circle through ellipse to escape.

Kepler’s laws (1609, 1619) with Newton’s law of gravitation (1687)

$T^{2} = \frac{a^{3}}{M}\quad(\text{AU, yr, } M_{\odot})$
Mass Spring Simulator
Simple harmonic motion with damping and a driving force, plus the resonance curve.

Hooke’s law (1678) with simple harmonic motion

$T = 2\pi\sqrt{\frac{m}{k}}$
Density Calculator
Density, mass or volume in g/cm³ and kg/m³, with lb/ft³ in the reference table.

Definition of density

$\rho = \frac{m}{V}$
Kinetic Energy Calculator
Kinetic energy, mass or speed from KE = ½mv², plus momentum.

Classical kinetic energy, from the work-energy theorem

$KE = \tfrac{1}{2}mv^{2}$
Force Calculator (F = ma)
Force, mass or acceleration from F = ma, with g-force and weight.

Newton, Principia (1687), second law of motion

$F = ma$
Pressure Calculator
Solve P = F/A in mixed units and read the answer in atm, bar and psi at once.

Definition of pressure

$P = \frac{F}{A}$
Bernoulli Equation Calculator
Pressure, speed or height at two points on a streamline from Bernoulli’s equation.

Daniel Bernoulli, Hydrodynamica (1738)

$P_1 + \tfrac{1}{2}\rho v_1^{2} + \rho g h_1 = P_2 + \tfrac{1}{2}\rho v_2^{2} + \rho g h_2$
Continuity Equation Calculator
Speed or area across a pipe from A₁v₁ = A₂v₂, with the flow rate Q = Av.

Conservation of mass for an incompressible flow

$A_1 v_1 = A_2 v_2 = Q$
Reynolds Number Calculator
Re = ρvD/μ for a pipe, with the flow regime and the kinematic viscosity.

Reynolds, Philosophical Transactions of the Royal Society (1883)

$\text{Re} = \frac{\rho v D}{\mu} = \frac{v D}{\nu}$
Hydrostatic Pressure Calculator
Absolute and gauge pressure at a depth from P = P₀ + ρgh.

Hydrostatic equilibrium, after Stevin (1586) and Pascal (1663)

$P = P_0 + \rho g h$
Work and Power Calculator
Work from force, distance and angle, plus the average power over a time.

Definitions of mechanical work and power

$W = Fd\cos\theta$
Efficiency Calculator
Percent efficiency, useful output or total input, plus the waste.

Definition of efficiency, with conservation of energy

$\text{efficiency} = \frac{\text{useful output}}{\text{total input}} \times 100\%$
Gravitational Potential Energy Calculator
PE = mgh solved any way round, with the impact speed for that drop.

Potential energy in a uniform gravitational field

$PE = mgh$
Hooke’s Law Calculator
Force, stiffness or extension from F = kx, plus the energy stored.

Hooke, De Potentia Restitutiva (1678)

$F = kx$
Torque Calculator
Torque from arm, force and angle, in N·m, lbf·ft and lbf·in.

Definition of the moment of a force

$\tau = rF\sin\theta$
Momentum Calculator
Momentum from mass and speed, plus the impulse and force to stop it.

Newton, Principia (1687), definition of momentum

$p = mv$
Centripetal Force Calculator
Force and acceleration for a turn, with the result also in multiples of g.

Acceleration in uniform circular motion

$F = \frac{mv^{2}}{r}$
Moment of Inertia Calculator
Rotational inertia of standard shapes, with the parallel axis theorem for offset axes.

OpenStax University Physics Volume 1 (2016), figure 10.20 and section 10.5, the parallel axis theorem

$I = kmr^{2} + md^{2}$
Angular Momentum Calculator
L = Iω and its conservation, with the spin rate and kinetic energy after a change.

OpenStax University Physics Volume 1 (2016), sections 11.2 and 11.3

$L = I\omega,\quad I_1\omega_1 = I_2\omega_2$
Rotational Kinetic Energy Calculator
K = ½Iω² in rad/s or rpm, with the rolling split between spin and forward motion.

OpenStax University Physics Volume 1 (2016), equation 10.18, with rolling from section 11.1

$K = \frac{1}{2}I\omega^{2}$
Kinematics Calculator
Any three of u, v, a, s, t gives the other two, and it names the equation.

Constant-acceleration kinematics

$v = u + at,\quad s = ut + \tfrac{1}{2}at^{2}$
Gravitational Force Calculator
Mutual gravity between two masses, and why you never notice it.

Newton, Principia (1687), law of universal gravitation

$F = G\frac{m_1 m_2}{r^{2}}$
Inclined Plane Simulator
Tilt a slope, set the friction and a push, and see whether the block slides, how fast, and where its energy goes.

Amontons (1699) and Coulomb (1785), laws of friction

$a = g(\sin\theta - \mu_k\cos\theta), \quad \mu_s = \tan\theta_c$
Conservation of Energy Simulator
A cart on a track you reshape, with kinetic, potential and thermal energy bars whose total never changes.

Conservation of energy, counting friction’s work as heat (Joule, 1850)

$\tfrac{1}{2}mv^{2} + mgh + E_{\text{th}} = mgh_0$
Double Pendulum Simulator
Two linked arms, a ghost released a millionth of a degree away, and the moment the two part company.

Lyapunov (1892), measured on a double pendulum by Shinbrot, Grebogi, Wisdom and Yorke (1992)

$\delta(t) \approx \delta_0\,e^{\lambda t}$
Buoyancy Simulator
Lower a block into a liquid on a spring balance and watch the buoyant force share its weight.

Archimedes, On Floating Bodies, Book I

$F_b = \rho_f V_{\text{sub}}\, g$
Free Fall Calculator
Fall time and impact speed from any height, with and without air resistance.

NASA Glenn Research Center, the drag equation and terminal velocity

$t = \sqrt{\frac{2h}{g}}, \quad v = \sqrt{2gh}, \quad v_t = \sqrt{\frac{2mg}{\rho C_d A}}$
Time Dilation Calculator
The Lorentz factor, time dilation and length contraction from a speed, γ, two times or two lengths.

Einstein (1905), On the Electrodynamics of Moving Bodies

$\Delta t = \gamma\,\Delta t_0, \quad L = \frac{L_0}{\gamma}, \quad \gamma = \frac{1}{\sqrt{1 - v^{2}/c^{2}}}$
Motion Graphs Simulator
Run a cart through up to five stages and watch its x-t, v-t and a-t graphs draw, with live gradients and areas.

OpenStax University Physics Volume 1, chapter 3, Motion Along a Straight Line

$v = \frac{dx}{dt},\quad a = \frac{dv}{dt},\quad s = \tfrac{1}{2}(u + v)t$
PID Controller Simulator
Tune Kp, Ki and Kd on a cart or an oven, with actuator limits, anti-windup and a sampled controller.

Åström and Murray, Feedback Systems (2008)

$u(t) = K_p e(t) + K_i \int_0^t e(\tau)\,d\tau + K_d \frac{de}{dt}$
Free-Body Diagram Maker
Draw the forces on a block, a hanging weight, a mass in a lift or your own set to scale, and read off the net force.

Newton, Principia (1687), Corollaries I and II: composition and resolution of forces

$F_{\text{net}} = \sqrt{(\sum F_x)^2 + (\sum F_y)^2}$
Asteroid Impact Simulator
Size, speed, angle and make-up in; airburst or crater, fireball, blast and quake out.

Collins, Melosh and Marcus (2005), Earth Impact Effects Program

$D_{tc} = 1.161\left(\frac{\rho_i}{\rho_t}\right)^{1/3} L^{0.78} v_i^{0.44} g^{-0.22}\sin^{1/3}\theta$
Moon Phase on Any Date
The Moon on any date, as seen from where you were: phase, percent lit and a picture to save.

Meeus, Astronomical Algorithms (1998), chapters 47 to 49 and 53

$k = \frac{1 + \cos i}{2}$
Shear Force and Bending Moment Diagram Calculator
Loads you can drag onto a beam, with reactions, shear force, bending moment and deflection redrawn live.

Euler-Bernoulli beam theory, with Macaulay (1919)

$\frac{dV}{dx} = -w, \quad \frac{dM}{dx} = V, \quad EI\,\frac{d^{2}v}{dx^{2}} = M$
Moment of Inertia and Section Modulus Calculator
Ix, Iy, section modulus, centroid and radius of gyration for six beam sections, drawn to scale, in mm or inches.

Gere and Goodno, Mechanics of Materials, 9th edition (2018)

$I_x = \int_A y^{2}\,dA, \quad S_x = \frac{I_x}{c}, \quad r_x = \sqrt{\frac{I_x}{A}}$
Stopping Distance Simulator
Thinking plus braking distance on the road and on a speed-squared chart, by speed, surface, slope and reaction time.

Constant deceleration under tyre friction, v² = u² + 2as

$S = v\,t_r + \frac{v^{2}}{2g(\mu\cos\theta + \sin\theta)}$
Three-Body Problem Simulator
Three stars, a planet if you add one, and a ghost started 150 km away that shows how fast chaos pulls the two apart.

Newton (1687), with the special solutions of Lagrange (1772) and of Chenciner and Montgomery (2000)

$\frac{d^{2}\mathbf{r}_i}{dt^{2}} = \sum_{j \neq i} \frac{G m_j \left(\mathbf{r}_j - \mathbf{r}_i\right)}{\left|\mathbf{r}_j - \mathbf{r}_i\right|^{3}}$

Electricity All Electricity tools

RC Charge and Discharge Simulator
Charge a capacitor through a resistor and watch where the time constant comes from.

Exponential RC response, from Kirchhoff’s laws

$V_C(t) = V_0\left(1 - e^{-t/RC}\right)$
RLC Circuit Simulator
A series RLC transient, from ringing through critical damping to sluggish.

Damped LC oscillation, after Thomson (1853)

$L\frac{d^{2}q}{dt^{2}} + R\frac{dq}{dt} + \frac{q}{C} = V$
LED Resistor Calculator
Series resistor value, next standard part up and power for any LED.

Ohm’s law applied across the diode forward voltage

$R = \frac{V_s - V_f}{I}$
RC Time Constant Calculator
Time constant, cutoff frequency and settling time for any RC pair.

RC time constant

$\tau = RC$
Ohm’s Law Calculator
Voltage, current or resistance from V = IR, plus power dissipation.

Ohm, Die galvanische Kette (1827)

$V = IR$
Electricity Cost Calculator
Running cost of any appliance per day, month and year, from watts, hours and price.

US Department of Energy, Estimating Appliance and Home Electronic Energy Use

$\text{cost} = \frac{P\,t}{1000} \times \text{price}\quad(P\text{ in W, } t\text{ in h})$
Electrical Power Calculator
Watts, amps or volts from P = IV, with load resistance and energy in J and kWh.

Joule (1841), heating by a current, with P = IV from the SI definition of the volt

$P = IV = I^{2}R = \frac{V^{2}}{R}$
Solar Panel Output Calculator
Solar kWh per day, month and year from panel watts, peak sun hours and losses.

IEC 61724 performance ratio (Marion et al. 2005), with the PVWatts V5 default losses (Dobos 2014)

$E = \frac{P\,N\,\mathrm{PSH}}{1000}\left(1 - \frac{L}{100}\right)\frac{\eta}{100}\quad(P\text{ in W, }E\text{ in kWh/day})$
Transformer Calculator
Voltage or turns from Vp/Vs = Np/Ns, plus currents and power.

Faraday’s law of induction (1831), with conservation of energy

$\frac{V_p}{V_s} = \frac{N_p}{N_s}, \quad V_s I_s = \eta\, V_p I_p$
Capacitor Energy Calculator
Energy and charge stored in a capacitor from capacitance and voltage.

Energy stored in a charged capacitor

$E = \tfrac{1}{2}CV^{2}$
Wire Resistance Calculator
Conductor resistance from length and cross-section, plus the voltage drop.

Resistivity and Pouillet’s law

$R = \frac{\rho L}{A}$
Voltage Drop Calculator
Voltage drop and percent drop for copper or aluminum wire by gauge.

Ohm (1827) over the run, with AWG sizes and resistivities from NBS Handbooks 100 (1966) and 109 (1972)

$V_d = 2\,I R L\ \text{(single-phase, DC)},\quad V_d = \sqrt{3}\,I R L\ \text{(three-phase)}$
Resistor Network Calculator
Series and parallel networks, with the current in each resistor shown.

Kirchhoff’s circuit laws (1845)

$\frac{1}{R_p} = \sum_i \frac{1}{R_i}, \quad R_s = \sum_i R_i$
Coulomb’s Law Calculator
Force between two point charges, plus the field one of them produces.

Coulomb (1785)

$F = k\frac{q_1 q_2}{r^{2}}$
Solenoid Magnetic Field Simulator
The field through and around a real coil, and the halving at its ends.

Biot and Savart (1820), with Ampere (1823)

$B(z) = \frac{\mu_0 n I}{2}\left[\frac{\ell/2 + z}{\sqrt{R^2 + (\ell/2+z)^2}} + \frac{\ell/2 - z}{\sqrt{R^2 + (\ell/2-z)^2}}\right]$
Resistor Color Code Calculator
Resistor bands to ohms and back, with tolerance, temperature coefficient and the nearest E-series value.

IEC 60062:2016, Marking codes for resistors and capacitors

$R = (10a + b) \times 10^{n}, \quad R = (100a + 10b + c) \times 10^{n}$
Electric Field Simulator
Point charges you can drag, with field lines, equipotentials and a probe reading E and V.

Coulomb (1785), with the superposition principle

$E = \frac{k|q|}{r^{2}}, \quad V = \sum_i \frac{kq_i}{r_i}$
Electromagnetic Induction Simulator
A magnet moved through a coil: induced EMF, current and Lenz’s law direction.

Faraday (1831) and Lenz (1834), with the flux of a magnetic dipole

$\varepsilon = -N\frac{d\Phi}{dt}, \quad \Phi = \frac{\mu_0 m a^{2}}{2(a^{2} + z^{2})^{3/2}}$
Capacitors in Series and Parallel Calculator
Series, parallel and mixed capacitor networks, with the charge on each capacitor.

Kirchhoff’s circuit laws (1845) and conservation of charge

$\frac{1}{C_s} = \sum_i \frac{1}{C_i}, \quad C_p = \sum_i C_i$
Series and Parallel Circuit Simulator
Bulbs, switches and meters in series, parallel and bridge circuits, solved exactly.

Kirchhoff’s circuit laws (1845), by modified nodal analysis

$R_{\text{s}} = R_1 + R_2 + R_3, \quad \frac{1}{R_{\text{p}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}, \quad P = I^{2}R$
Voltage Divider Calculator
Two resistors in series: the output voltage, or the resistor that gives a target output, with loading shown.

Ohm (1827), with Kirchhoff’s voltage law (1845)

$V_{\text{out}} = V_{\text{in}}\frac{R_2}{R_1 + R_2}$
Charged Particle in a Magnetic Field Simulator
Circles, helices, a velocity selector and a mass spectrometer, from the exact Lorentz force.

Heaviside (1889) and Lorentz (1895), the force on a moving charge

$F = qvB\sin\theta, \quad r = \frac{mv}{qB}, \quad T = \frac{2\pi m}{qB}, \quad v = \frac{E}{B}$
Oscilloscope Simulator
Two signal generators on a scope screen: set the volts/div, time/div and trigger, then read the trace off the grid.

Tektronix, XYZs of Oscilloscopes (primer)

$V_{\text{pp}} = n_{y} \times \text{V/div}, \quad T = n_{x} \times \text{s/div}, \quad f = \frac{1}{T}$
Bode Plot and Filter Visualizer
Gain and phase of RC and RLC filters on log axes, with the straight-line asymptotes, the −3 dB point and Q.

Horowitz and Hill, The Art of Electronics, 3rd edition (2015)

$G = -10\log_{10}\left[1 + \left(\frac{f}{f_c}\right)^{2}\right],\quad f_c = \frac{1}{2\pi RC}$
Magnetic Field Simulator
Bar magnets, wires and coils you can drag, with field lines, iron filings, compasses and a B probe.

Biot and Savart (1820), with Ampère (1823)

$B = \frac{\mu_0 I}{2\pi r}, \quad B_{\text{centre}} = \left(\tfrac{4}{5}\right)^{3/2}\frac{\mu_0 N I}{R}$

Waves & Optics All Waves & Optics tools

Standing Wave Simulator
Harmonics on a string, with the two travelling waves that superpose to make them.

Normal modes of a vibrating string, Mersenne (1636)

$f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}$
Ray Diagram Simulator
The three principal rays, with the image located where they cross.

Gaussian optics, thin-lens and mirror equations

$\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$
Wavelength and Frequency Calculator
Wavelength to frequency and back, for light, sound or any wave speed.

Wave relation between speed, frequency and wavelength

$v = \lambda f$
Photon Energy Calculator
Photon energy from wavelength, in eV, joules and kJ per mole.

Planck (1900) and Einstein (1905)

$E = \frac{hc}{\lambda}$
Snell’s Law Calculator
Solve n₁sinθ₁ = n₂sinθ₂ for any term, with the critical angle shown for the pair of media.

Snell (1621), published by Descartes (1637)

$n_1 \sin\theta_1 = n_2 \sin\theta_2$
Thin Lens Calculator
Solve the thin lens equation and get magnification with the image type spelled out.

Thin-lens equation, Gaussian optics

$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$
Decibel Calculator
Sound level from intensity, with distance falloff and safe exposure time.

Logarithmic level definition, the decibel

$L = 10\log_{10}\!\left(\frac{I}{I_0}\right)$
Double Slit Simulator
Interference fringes, and the single-slit envelope shaping them.

Young (1801), double-slit interference

$d\sin\theta = m\lambda$
Doppler Effect Simulator
Wavefronts from a moving source, and the shift they produce.

Doppler (1842)

$f' = f\,\frac{v \pm v_o}{v \mp v_s}$
Electromagnetic Spectrum Explorer
Radio waves to gamma rays on one scale, with wavelength, frequency and photon energy linked.

Wave relation c = λf, with Planck (1900) and Einstein (1905) for E = hf

$c = \lambda{}f, \quad E = hf = \frac{hc}{\lambda}$
Wave Interference Simulator
Two coherent sources, their nodal lines, and the path difference at any point.

Young (1807), interference of two sets of circular waves

$|r_1 - r_2| = m\lambda \text{ (constructive)}, \quad |r_1 - r_2| = \left(m + \tfrac{1}{2}\right)\lambda \text{ (destructive)}$
Refraction Simulator
Rays through a boundary, a glass block and a prism, with total internal reflection and white light split into colours.

Snell (1621) and Descartes (1637), with Fresnel (1823) for reflection

$n_1\sin\theta_1 = n_2\sin\theta_2, \quad \sin\theta_c = \frac{n_2}{n_1}$
Photoelectric Effect Simulator
Which light frees electrons from a metal, how fast they leave, and what stops them.

Einstein (1905), photoelectric equation

$K_{\text{max}} = eV_0 = hf - \phi, \quad f_0 = \frac{\phi}{h}$
De Broglie Wavelength Calculator
Matter wavelength, h/p, from speed, kinetic energy, voltage or momentum.

de Broglie (1924), matter waves

$\lambda = \frac{h}{p} = \frac{h}{mv}, \quad \lambda = \frac{h}{\sqrt{2meV}}$
Sampling and Aliasing Visualizer
A sine sampled at any rate, and the slower sine its samples describe once the rate falls below twice its frequency.

Oppenheim and Schafer, Discrete-Time Signal Processing, 3rd ed., ch. 4

$f_{\text{alias}} = \left| f - k f_s \right|$
Blackbody Radiation Simulator
Planck’s curve, Wien’s peak, σT⁴ and the colour of anything hot, from molten lava to a blue star.

Planck (1900), with Wien (1893), Stefan (1879) and Boltzmann (1884)

$B_\lambda(T) = \frac{2hc^{2}}{\lambda^{5}}\,\frac{1}{e^{hc/\lambda k T} - 1}, \quad \lambda_{\max} T = b, \quad M = \sigma T^{4}$
Live Spectrogram and Spectrum Analyzer
Sing, whistle or play a file and watch its frequencies scroll past, with the pitch and note named.

Koenig, Dunn and Lacy (1946), The Sound Spectrograph

$X_m(k) = \sum_{n=0}^{N-1} x[n + mH]\,w[n]\,e^{-2\pi i k n/N}$

Thermodynamics All Thermodynamics tools

Heat Engine and Carnot Cycle Simulator
Run a gas around a thermodynamic cycle and read the work off the enclosed area.

Carnot, Réflexions sur la puissance motrice du feu (1824)

$\eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H}$
Ideal Gas Law Calculator
Solve PV = nRT for any variable, with kelvin conversion handled for you.

Clapeyron (1834), combining the Boyle, Charles and Avogadro laws

$PV = nRT$
Calorimetry Calculator
Heat, mass, specific heat or ΔT from q = mcΔT, in joules or calories.

Specific heat capacity, after Joseph Black (c. 1760)

$q = mc\Delta T$
Heating Curve Calculator
Energy from ice to steam, counting every stage a single m c dT misses.

Specific and latent heat, after Joseph Black (c. 1760)

$Q = \sum m c \Delta T + \sum m L$
Latent Heat Calculator
Heat to melt or boil a mass, and why the thermometer stops moving.

Latent heat, Joseph Black (c. 1761)

$Q = mL$
Thermal Expansion Calculator
Length change from a temperature change, plus the stress if it cannot move.

Linear thermal expansion coefficient

$\Delta L = \alpha L_0 \Delta T$
Combined Gas Law Calculator
Relate pressure, volume and temperature between two states of a sealed gas.

Boyle (1662), Charles (1787) and Gay-Lussac (1802) combined

$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$
Kinetic Theory Gas Simulator
Atoms in a box, with the gas law measured rather than assumed.

Kinetic theory of gases, Maxwell (1860) and Boltzmann (1872)

$P A = N k T,\quad f(v) \propto v\,e^{-mv^{2}/2kT}$
Boyle’s Law Calculator
Squeeze or expand a gas at one temperature and get its new pressure or volume, plus the work done.

Boyle (1662), pressure and volume at constant temperature

$P_1V_1 = P_2V_2$
States of Matter Simulator
Atoms that freeze, melt and evaporate as you heat and cool them.

Lennard-Jones (1931), the 12-6 potential between atoms

$U(r) = 4\varepsilon\left[\left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6}\right]$
Charles’s Law Calculator
Warm or cool a gas at constant pressure and get its new volume or temperature, with kelvin handled.

Gay-Lussac (1802), crediting unpublished work by Charles (1787)

$\frac{V_1}{T_1} = \frac{V_2}{T_2}$
Gibbs Free Energy Calculator
ΔG from ΔH, T and ΔS, or any one of them, with the spontaneity verdict, the crossover temperature and K.

Gibbs (1876), On the Equilibrium of Heterogeneous Substances

$\Delta G = \Delta H - T\Delta S, \quad \Delta G^\circ = -RT\ln K$
Gay-Lussac’s Law Calculator
Heat or cool a gas in a sealed, rigid container and get its new pressure or temperature.

Amontons (1702) and Gay-Lussac (1802)

$\frac{P_1}{T_1} = \frac{P_2}{T_2}$
Psychrometric Chart Explorer
Drag a state on the psychrometric chart to read its dew point, wet bulb and enthalpy, then heat, cool or mix it.

ASHRAE Handbook Fundamentals (2017), chapter 1, with Hyland and Wexler (1983)

$W = 0.621945\,\frac{p_w}{p - p_w}, \quad h = 1.006\,t + W\,(2501 + 1.86\,t)$
Boiling Point at Altitude Simulator
Carry a pot up a mountain or seal a pressure cooker and watch where water boils.

Hyland and Wexler (1983) with the US Standard Atmosphere (1976)

$p_{ws}(T_b) = p, \quad p = 101.325\,(1 - 2.25577\times10^{-5}\,H)^{5.25588}\ \mathrm{kPa}$
Heat Index Chart Explorer
Drag a point across the heat index chart and read the feels-like temperature and its danger band.

Rothfusz (1990), NWS regression on Steadman (1979)

$\mathrm{HI} = c_1 + c_2 T + c_3 R + c_4 T R + c_5 T^{2} + c_6 R^{2} + c_7 T^{2} R + c_8 T R^{2} + c_9 T^{2} R^{2}\quad(T\text{ in °F, } R\text{ in \%})$
Wind Chill Calculator
Wind chill from air temperature and wind speed, with the frostbite risk level.

NWS and Environment Canada (2001), after Osczevski and Bluestein (2005)

$\mathrm{WCT} = 35.74 + 0.6215\,T - 35.75\,V^{0.16} + 0.4275\,T\,V^{0.16}\quad(\text{°F, mph})$
Humidex Calculator
Humidex from air temperature and dew point, with Environment Canada’s comfort level.

Masterton and Richardson (1979), Environment Canada

$H = T + 0.5555\,(e - 10), \quad e = 6.11\,\exp\!\left[5417.7530\left(\frac{1}{273.15} - \frac{1}{T_d}\right)\right]$
Air Conditioner BTU Calculator
Room air conditioner size in BTU/h from length and width, with tons and kW.

ENERGY STAR (US EPA), room air conditioner sizing chart

$Q = Q_{\text{chart}}(L \times W)\,\frac{100 + s}{100} + 600\,\max(n - 2,\ 0) + 4000\,k$
Dew Point Calculator
Dew point from temperature and relative humidity, or humidity from the dew point.

Hyland and Wexler (1983), ASHRAE Handbook Fundamentals (2017)

$p_{ws}(T_d) = \frac{\mathrm{RH}}{100}\,p_{ws}(T)$
R-Value and Heat Loss Calculator
Heat loss in Btu/h or W from R-value, area and ΔT, with layers in series, U-factor and RSI.

Fourier (1822), conduction; ISO 6946:2017, layers in series; NIST SP 811 (2008), appendix B

$Q = \frac{A\,\Delta T}{R_1 + R_2 + R_3}, \qquad U = \frac{1}{R}$

What is on the physics formula sheet?

Its 103 equations fall into four groups: mechanics (40), electricity (25), waves and optics (17), and thermodynamics (21). They run from Newton’s second law and the law of universal gravitation through Ohm’s and Coulomb’s laws to Snell’s law, the photoelectric equation, the ideal gas law and the Carnot efficiency.

Of the 103, 62 belong to calculators, which solve the equation for the numbers you enter. The other 41 belong to simulators (35) and visualizers (6), where you change the inputs and watch the equation at work.

Each row names its source: 74 cite a dated publication, from 1586 to 2018, and the other 29 name the definition, standard result or reference text the equation follows.

The constants these equations use, G, h, c and e among them, are in the physical constants table. The uncertainty a lab report asks for is on the math sheet: error propagation, percent error and the worst-fit line.

Other formula sheets

Chemistry formula sheet Biology formula sheet Math formula sheet Every equation by name