125 free physics calculators

Equations of motion

Jerk

s = u*t + 1/2*a0*t^2 + 1/6*j*t^3

What the displacement formula becomes when the acceleration itself changes - solve s = ut + 1/2 a0 t^2 + 1/6 j t^3 four ways and see what SUVAT drops.

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Projectile motion

Projectile Motion

R = v0²·sin(2θ) / g

Range, peak height and flight time for any launch speed, angle and gravity.

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Free fall

Free Fall

h = ½·g·t², v = g·t

Distance, speed and time for an object dropped under gravity.

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Equations of motion

SUVAT

v² = u² + 2as

The five constant-acceleration equations — give any three, get the rest.

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Vectors

Vector Addition

R = sqrt(Rx² + Ry²), φ = atan2(Ry, Rx)

Add two vectors from magnitudes & angles or x/y components to get the resultant.

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Velocity

Velocity

v = d / t

Average velocity is distance travelled divided by the time taken.

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Acceleration

Acceleration

a = (v − u) / t

The rate at which velocity changes over time.

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Circular motion

Tangential Acceleration

a = sqrt((v^2/r)^2 + a_t^2)

What a = v^2/r becomes the moment the speed changes - solve the total acceleration on a bend four ways and read the angle it leans off the radius.

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Energy

Work Energy Theorem

F*d = 1/2*m*v2^2 - 1/2*m*v1^2

The net work done on a body equals its change in kinetic energy - solve for any one of the five quantities.

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Energy

Conservation of Energy

1/2*m*v1^2 + m*g*h1 = 1/2*m*v2^2 + m*g*h2 + E_loss

The energy balance between two points on any path, with or without a friction loss.

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Rotational motion

Angular Momentum

L = I*w = m*v*r

Angular momentum of a rotating body or of one mass on a circle.

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Newton's laws

Newton's Second Law

F = m·a

Force, mass and acceleration — the heart of dynamics.

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Elasticity

Hooke's Law

F = −k·x

Spring force, stiffness, extension and stored elastic energy.

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Rotation

Torque

τ = r·F·sinθ

Turning effect of a force about a pivot.

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Circular motion

Circular Motion

a_c = v² / r

Centripetal acceleration and force for motion in a circle.

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Energy

Kinetic Energy

KE = ½·m·v²

Energy of motion — note the v² dependence.

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Energy

Gravitational Potential Energy

PE = m·g·h

Stored energy of a raised mass, on any world.

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Energy

Work and Power

W = F·d, P = W / t

Work done by a force and how fast it is delivered.

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Momentum

Momentum

p = m·v

Mass in motion, impulse and the change in momentum.

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Momentum

Center of Mass

x_cm = (m1·x1 + m2·x2 + m3·x3) / (m1 + m2 + m3)

Weighted-average balance point of two or three point masses.

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Friction

Friction

F = μ·N

Friction force from the coefficient of friction and the normal force.

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Forces

Inclined Plane

a = g(sinθ - μ·cosθ)

Full force set on a ramp — mg sinθ, mg cosθ, N, friction and acceleration — from mass, angle and friction.

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Rotation

Angular Velocity

ω = θ / t

How fast an angle is swept out — plus rpm to rad/s.

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Momentum

Collision

m1u1 + m2u2 = m1v1 + m2v2

Final velocities in elastic and inelastic collisions.

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Circular motion

Centripetal Force

F = m·v² / r

Inward force keeping an object moving in a circle.

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Elasticity

Spring Constant

k = F / x

Spring stiffness from force and extension, plus elastic energy.

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Energy

Power

P = W / t

Rate of doing work or transferring energy.

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Rotation

Moment of Inertia

I = k·m·r²

Rotational inertia for spheres, cylinders, rods and hoops.

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Momentum

Impulse

J = F·Δt

Impulse from force and time, equal to change in momentum.

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Circular motion

Banked Curve

tanθ = v² / (r·g)

Ideal banking angle or speed for a curve.

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Simple machines

Mechanical Advantage

MA = F_load / F_effort

The force-for-distance trade of a lever, pulley or ramp — MA, IMA and efficiency.

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Elasticity

Young's Modulus

E = (F·L0) / (A·ΔL)

Modulus of elasticity from stress and strain — E = stress/strain.

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Power

Power-to-Weight Ratio

PWR = P / m

Power divided by mass — the key performance ratio for engines and vehicles.

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Rotation

Coriolis Force

F = 2·m·ω·v·sinφ

The deflecting force felt in a rotating frame such as the spinning Earth.

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Rotation

Nm to ft-lb Converter

1 N·m = 0.73756 ft·lb

Convert torque between newton-metres and foot-pounds, both ways.

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Units and prefixes

SI Prefix Converter

result = v · 10^(a - b)

Convert a value between any two SI prefixes, quetta to quecto, or straight to the coherent SI unit.

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Gravitation

Escape Velocity

v_esc = sqrt(2·G·M / r)

Minimum speed to break free of a body’s gravity.

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Gravitation

Weight on Other Planets

W = m · g

How much you would weigh on the Moon, Mars and every planet.

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Universal gravitation

Gravitational Force

F = G·m1·m2 / r²

Gravitational attraction between two masses.

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Orbits

Kepler's Third Law

T² = 4π²·a³ / (G·M)

Links an orbit period to its size — weigh stars and design orbits.

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Pipe flow

Reynolds Number

Re = ρ·v·D / μ

Reynolds number from density, velocity, pipe diameter and viscosity, with the laminar or turbulent regime.

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Pressure

Atmospheric Pressure

P = P0 · (1 - L·h/T0)^5.2559

Air pressure at any altitude to 20 km from the standard atmosphere, or altitude from a measured pressure.

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Fluids

Density

ρ = m / V

Mass packed into a given volume — solves for density, mass or volume.

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Fluids

Buoyancy

F_b = ρ·V·g

Archimedes’ upward force on a submerged object.

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Pascal's law

Pascal's Law

F1/A1 = F2/A2

Hydraulic-press force multiplication — F1/A1 = F2/A2, with pressure and mechanical advantage.

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Fluid flow

Bernoulli Equation

P + ½ρv² + ρgh = constant

Pressure, speed and height trade-off along a streamline.

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Surface tension

Surface Tension

γ = F / L

Force per unit length, plus bubble pressure and capillary rise.

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Pressure

Pressure

P = F / A

Pressure as force over area, plus fluid pressure with depth.

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Drag

Terminal Velocity

v_t = sqrt(2mg / ρAC_d)

Maximum fall speed when drag balances gravity.

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Drag

Drag Force

F = ½·ρ·v²·C_d·A

Aerodynamic drag (air resistance) on a moving object.

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Hydraulics

Hydraulic Cylinder Force

F = P·A, A = π·D²/4

Push force of a hydraulic or pneumatic cylinder from pressure and bore.

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Viscosity

Viscosity Converter

μ = ν · ρ

Convert between kinematic (cSt) and dynamic (cP) viscosity via density.

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Pipe flow

Friction Factor

laminar f = 64/Re; turbulent: Colebrook

Darcy friction factor for pipe flow — laminar 64/Re and turbulent Colebrook.

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Open-channel flow

Manning's Equation

Q = (1/n)·A·R^(2/3)·S^(1/2)

Open-channel flow rate from roughness, area, hydraulic radius and slope.

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Oscillations

Pendulum Period

T = 2π·sqrt(L / g)

Swing time of a simple pendulum — independent of mass.

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Oscillations

Simple Harmonic Motion

ω = 2πf

Angular frequency, period and peak speed of an oscillator.

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Acoustics

Helmholtz Resonator

f = (c / 2π)·sqrt(A / (V·L))

The resonant frequency of an air cavity with a neck — a bottle, a port, a muffler.

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Electromagnetic spectrum

Ultraviolet Radiation

E = h·c / λ

Converts a UV wavelength in nanometres into a photon energy and names the UVA, UVB or UVC band.

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Waves

Wave Speed

v = f·λ

The fundamental wave relationship.

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Waves

Doppler Effect

f’ = f·(v ± v_o) / (v ∓ v_s)

Frequency shift from a moving source or observer.

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Diffraction

Diffraction Grating

d·sinθ = n·λ

Bright-fringe angles from a grating; highest visible order.

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Sound

Decibel

L = 10·log10(I / I0)

Sound level in decibels from intensity.

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Sound

Speed of Sound Calculator

v = sqrt(gamma R T / M)

How fast sound travels through a gas, from its temperature, adiabatic index and molar mass.

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Refraction

Snell's Law

n1·sinθ1 = n2·sinθ2

How light bends crossing between media; critical angle & TIR.

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Polarization

Malus's Law

I = I0·cos²θ

Transmitted intensity of polarized light through an analyzer.

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Geometric optics

Lens and Mirror

1/f = 1/do + 1/di

Thin-lens and mirror equation with magnification.

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Refraction

Refractive Index

n = c / v

Index of refraction from the speed of light in a medium.

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First law

Isothermal Process

W = n·R·T·ln(V2/V1)

Work and heat for a gas expanding or being squeezed at constant temperature - the area under the PV curve, with the entropy change beside it.

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Radiation

Blackbody Radiation

F = INT(lam1..lam2) M_lam dlam / (sigma T^4)

How much of a hot surface radiated power lands between two wavelengths - the band fraction of Planck law, with the Wien peak and the total exitance beside it.

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Conduction

Thermal Diffusivity

alpha = k / (rho x c)

How fast a temperature change travels through a material - solve alpha = k/(rho c) four ways and read the time to steady state.

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Gas laws

Combined Gas Law

P1·V1/T1 = P2·V2/T2

Pressure, volume and temperature all change at once and the amount of gas stays put - solve for whichever of the six quantities you are missing.

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Gas laws

Avogadro's Law

V1/n1 = V2/n2

Volume-amount ratio of a gas at constant pressure and temperature.

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Temperature

Absolute Zero

T(K) = T(°C) + 273.15

Temperature on all four scales, with the rms molecular speed, kT and the mean kinetic energy per molecule.

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Gas laws

Ideal Gas Law

P·V = n·R·T

The four state variables of an ideal gas.

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Gas laws

Charles's Law

V1/T1 = V2/T2

Volume-temperature ratio of a gas at constant pressure.

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Gas laws

Boyle's Law

P1·V1 = P2·V2

Pressure-volume product of a gas at constant temperature.

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Thermal

Thermal Expansion

ΔL = α·L0·ΔT

How materials grow when heated.

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Calorimetry

Specific Heat

Q = m·c·ΔT

Heat energy to change a material temperature, Q = mcΔT.

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Calorimetry

Latent Heat Calculator

Q = mL

Energy for a phase change, Q = mL.

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Heat engines

Carnot Efficiency

η = 1 − T_c / T_h

Maximum efficiency of a heat engine between two temperatures.

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Radiation

Stefan-Boltzmann

P = σ·A·ε·T^4

Power radiated by a black body, P = σAεT^4.

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Conduction

Thermal Conduction

Q/t = k·A·ΔT / d

Steady-state heat-conduction rate through a slab, Q/t = kAΔT/d.

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Gas laws

Gay-Lussac's Law

P1/T1 = P2/T2

Pressure-temperature ratio of a gas at constant volume.

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Radiation

Wien's Law

λ_max = b / T

Peak wavelength of black-body radiation, λmax = b/T.

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Gas laws

Gas Density

ρ = P·M / (R·T)

Density of an ideal gas from its pressure, molar mass and temperature.

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Thermochemistry

Enthalpy of Reaction

q = n · ΔH

Heat released or absorbed by a reaction from moles and molar enthalpy.

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Entropy

Entropy Change

ΔS = Q / T

Entropy change for reversible heat transfer at constant temperature.

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First law

First Law of Thermodynamics

ΔU = Q − W

Energy conservation for a thermodynamic system, ΔU = Q − W.

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Reaction kinetics

Activation Energy

Ea = R·ln(k2/k1) / (1/T1 − 1/T2)

Arrhenius activation energy from two rate constants at two temperatures.

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Heat

Water Heating Energy

q = m·c·ΔT, t = q / P

Energy and time to heat water, from q = mcΔT.

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Psychrometrics

Wet-Bulb Temperature

Tw = f(T, RH) (Stull 2011)

Wet-bulb temperature from air temperature and relative humidity.

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Magnetism

Magnetic Flux

Φ = B · A · cos(θ)

The magnetic flux through a flat loop in a uniform field, from the flux density, the area and the tilt.

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Magnetism

Right-Hand Rule

F = q v × B

Which way the magnetic force on a moving charge points, from the velocity and field vectors.

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Superconductivity

Superconductor Critical Field

Ic = 2 pi a (Bc0 (1 - (T/Tc)^2) - Bapp) / mu0

The current a superconducting wire carries with no resistance at a chosen temperature and applied field, with the critical field beside it.

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Induction

Eddy Current Loss

P/V = pi^2 B^2 f^2 d^2 / (6 rho)

Power lost to eddy currents in a lamination, with the skin depth beside it so you can see when the formula still applies.

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Circuits

Watts to Amps

I = P / (V · PF)

The current a load draws from its watts, the supply voltage and the power factor, for DC, single-phase and three-phase supplies.

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Circuits

Ohm's Law

V = I·R, P = I·V

Voltage, current, resistance and power in a circuit.

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Circuits

Electric Current

I = Q/t

Current as the rate of charge flow, I = Q/t, with the electron count.

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Circuits

Potential Difference

V = W/Q

Potential difference as energy per unit charge, V = W/Q.

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Resistance

Resistivity

R = ρL / A

Wire resistance from material resistivity and geometry, R = ρL/A.

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Circuits

Kirchhoff's Law

Sum I = 0, Sum V = 0

Solves the two-loop circuit for all three branch currents.

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Circuits

RMS Voltage

Vrms = V0 / sqrt(2)

Convert between peak, peak-to-peak and RMS voltage, plus AC power.

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Electrostatics

Coulomb's Law

F = k·q1·q2 / r²

Electrostatic force between two point charges.

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Circuits

Resistor

Series: ΣR · Parallel: 1/Σ(1/R)

Total resistance of resistors wired in series or parallel.

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Energy use

Electricity Cost

Cost = kWh × rate

Running cost of an appliance from wattage, hours and rate.

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Electrostatics

Electric Field

E = k·Q / r²

Electric field strength around a point charge.

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Capacitors

Capacitance

C = Q / V

Capacitance from charge and voltage, plus stored energy.

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Magnetism

Magnetic Force

F = q·v·B·sinθ

Lorentz force on a moving charge in a magnetic field.

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Magnetism

Magnetic Field

B = μ0·n·I

Magnetic field inside a solenoid, B = μ0nI.

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Induction

Faraday's Law Calculator

ε = −N ΔΦ/Δt

Induced EMF from a changing magnetic flux, ε = −N ΔΦ/Δt.

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Circuits

RC Time Constant

τ = R·C

Charging time constant of an RC circuit, τ = RC.

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Circuits

Cutoff Frequency

fc = 1 / (2π·R·C)

The -3 dB corner frequency of an RC filter.

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Radioactive decay

Decay Chain

A2/A1 = L2/(L2-L1) x (1 - e^(-(L2-L1)t))

The daughter of a decay is radioactive too - work out its activity, the parent activity behind it, or how long ago the two were separated.

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Mass–energy equivalence

E = mc²

E = m·c²

Einstein’s mass–energy equivalence — the energy locked inside a mass.

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Radioactive decay

Half-Life

N = N0·(½)^(t/t½)

Time for a radioactive quantity to fall to half its value.

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Radioactive decay

Beta Decay Energy

Q = (Mp - Md) × 931.494 MeV

Decay energy Q from the parent and daughter atomic masses, for beta-minus, beta-plus and electron capture.

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Matter waves

De Broglie Wavelength

λ = h / (m·v)

Matter wavelength of a moving particle, λ = h/mv.

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Quantum

Photon Energy

E = h·f

Energy of one quantum of light from its frequency.

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Special relativity

Time Dilation

t = t0 / sqrt(1 − v²/c²)

Relativistic time dilation at high speed.

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Quantum

Photoelectric Effect

KE = h·f − φ

Maximum kinetic energy of photoelectrons, KE = hf − φ.

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Special relativity

Lorentz Factor

γ = 1 / sqrt(1 − v²/c²)

The relativistic factor γ that stretches time and contracts length.

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Quantum

Volt to eV

E = q·V (E[eV] = n·V)

Convert accelerating voltage to electronvolts and joules, E = qV.

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Quantum

Uncertainty Principle

Δx · Δp >= h-bar/2

The Heisenberg limit: Δp = h-bar/(2Δx), plus the energy–time form.

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Quantum

Bohr Model

E = -13.6 Z²/n² eV

Hydrogen-like energy levels, transition wavelengths and Bohr orbit radii.

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Quantum

Particle in a Box

E = n2 h2 / (8 m L2)

Energy levels of a particle trapped in a one-dimensional infinite well.

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