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Kinetic Energy

KE = \frac{1}{2}mv^2

Energy of a moving object.

Gravitational Potential Energy

PE = mgh

Energy stored due to height in a gravitational field.

Work Done

W = F d \cos\theta

Work done by a force over a distance.

Power

P = \frac{W}{t}

Rate of doing work.

Momentum

p = mv

Quantity of motion of an object.

Impulse

J = F\Delta t = \Delta p

Change in momentum from a force over time.

Kinematics: Velocity

v = u + at

Final velocity under constant acceleration.

Kinematics: Displacement

s = ut + \frac{1}{2}at^2

Displacement under constant acceleration.

Kinematics: v²

v^2 = u^2 + 2as

Velocity–displacement relation under constant acceleration.

Ohm's Law

V = IR

Voltage equals current times resistance.

Electrical Power

P = VI = I^2R

Power dissipated in an electrical component.

Mass–Energy Equivalence

E = mc^2

Energy equivalent of mass.

Wave Speed

v = f\lambda

Speed of a wave from frequency and wavelength.

Coulomb's Law

F = k\frac{q_1 q_2}{r^2}

Electrostatic force between two charges.

Snell's Law

n_1\sin\theta_1 = n_2\sin\theta_2

Refraction of light between two media.

Newton's Gravitation

F = G\frac{m_1 m_2}{r^2}

Attractive force between two masses.

Hooke's Law

F = -kx

Restoring force of a spring.

Simple Pendulum Period

T = 2\pi\sqrt{\frac{L}{g}}

Period of a simple pendulum.

Centripetal Force

F = \frac{mv^2}{r}

Force needed to keep an object in circular motion.

Density

\rho = \frac{m}{V}

Mass per unit volume.

Pressure

P = \frac{F}{A}

Force per unit area.

Heat Energy

Q = mc\Delta T

Heat needed to change temperature of a substance.

Doppler Effect

f' = f\frac{v \pm v_o}{v \mp v_s}

Observed frequency shift due to relative motion.

Speed

v = \frac{d}{t}

Speed equals distance divided by time.

Acceleration

a = \frac{\Delta v}{t}

Rate of change of velocity.

Weight

W = mg

The gravitational force on a mass.

Frequency and Period

f = \frac{1}{T}

Frequency is the inverse of the period.

Mechanical Advantage

MA = \frac{F_{out}}{F_{in}}

Force amplification of a simple machine.

Energy of a Photon

E = hf

Energy of a photon from its frequency.

Weight

W = mg

Gravitational force on a mass near Earth.

Hooke Law

F = -kx

Restoring force of a spring proportional to displacement.

Impulse

J = F\Delta t = \Delta p

Change in momentum from a force over time.

Torque

\tau = rF\sin\theta

Rotational effect of a force about a pivot.

Angular Momentum

L = I\omega

Rotational analog of linear momentum.

Rotational Kinetic Energy

KE = \frac{1}{2}I\omega^2

Kinetic energy of a rotating body.

Gravitational Potential Energy (General)

U = -\frac{GMm}{r}

Potential energy of two masses at distance r.

Orbital Velocity

v = \sqrt{\frac{GM}{r}}

Speed needed for a circular orbit.

Escape Velocity

v_e = \sqrt{\frac{2GM}{r}}

Minimum speed to escape a gravitational field.

Spring Oscillation Period

T = 2\pi\sqrt{\frac{m}{k}}

Period of a mass on a spring.

Friction Force

f = \mu N

Resistive force between surfaces.

Terminal Velocity

v_t = \sqrt{\frac{2mg}{\rho A C_d}}

Constant fall speed when drag balances gravity.

Projectile Range

R = \frac{v^2\sin 2\theta}{g}

Horizontal distance of a projectile on level ground.

Centripetal Force

F = \frac{mv^2}{r}

Inward force needed for circular motion.

Power from Work

P = \frac{W}{t}

Rate of doing work.

Elastic Potential Energy

U = \frac{1}{2}kx^2

Energy stored in a stretched or compressed spring.

Newton's First Law (Inertia)

\sum F = 0 \Rightarrow v = \text{const}

An object stays at rest or in uniform motion unless acted on by a net force.

Newton's Second Law

F = ma = \frac{dp}{dt}

Net force equals mass times acceleration (rate of change of momentum).

Newton's Third Law

F_{AB} = -F_{BA}

For every action there is an equal and opposite reaction.

Conservation of Momentum

\sum p_i = \sum p_f

Total momentum of an isolated system stays constant.

Conservation of Energy

E_{total} = \text{const}

Energy cannot be created or destroyed, only transformed.

Archimedes' Principle

F_b = \rho g V

Buoyant force equals the weight of fluid displaced.

Pascal's Law

P = \frac{F_1}{A_1} = \frac{F_2}{A_2}

Pressure applied to a confined fluid transmits equally in all directions.

Bernoulli's Principle

P + \tfrac{1}{2}\rho v^2 + \rho g h = \text{const}

Faster-moving fluid has lower pressure (energy conservation in flow).

Hooke's Law

F = -kx

The restoring force of a spring is proportional to its extension.

Law of Universal Gravitation

F = G\frac{m_1 m_2}{r^2}

Every mass attracts every other mass.

Kepler's First Law

r = \frac{a(1-e^2)}{1 + e\cos\theta}

Planets orbit the Sun in ellipses with the Sun at one focus.

Kepler's Second Law

\frac{dA}{dt} = \text{const}

A line from a planet to the Sun sweeps equal areas in equal times.

Kepler's Third Law

T^2 \propto a^3

The square of the orbital period is proportional to the cube of the semi-major axis.

Zeroth Law of Thermodynamics

A = B, \ B = C \Rightarrow A = C

Two systems in thermal equilibrium with a third are in equilibrium with each other.

First Law of Thermodynamics

\Delta U = Q - W

Energy conservation: change in internal energy equals heat added minus work done.

Second Law of Thermodynamics

\Delta S_{universe} \ge 0

The entropy of an isolated system never decreases.

Boyle's Law

P_1 V_1 = P_2 V_2

At constant temperature, pressure and volume of a gas are inversely proportional.

Charles's Law

\frac{V_1}{T_1} = \frac{V_2}{T_2}

At constant pressure, volume of a gas is proportional to absolute temperature.

Gay-Lussac's Law

\frac{P_1}{T_1} = \frac{P_2}{T_2}

At constant volume, pressure of a gas is proportional to absolute temperature.

Stefan–Boltzmann Law

j = \sigma T^4

Total energy radiated by a black body scales with the fourth power of temperature.

Ohm's Law

V = IR

Voltage across a resistor equals current times resistance.

Coulomb's Law

F = k\frac{q_1 q_2}{r^2}

Electrostatic force between two point charges.

Gauss's Law

\oint \vec{E} \cdot d\vec{A} = \frac{Q}{\varepsilon_0}

Electric flux through a closed surface is proportional to the enclosed charge.

Faraday's Law of Induction

\varepsilon = -\frac{d\Phi_B}{dt}

A changing magnetic flux induces an electromotive force.

Lenz's Law

\varepsilon \propto -\frac{d\Phi_B}{dt}

Induced current opposes the change in flux that produced it.

Ampère's Law

\oint \vec{B} \cdot d\vec{l} = \mu_0 I

A magnetic field circulates around an electric current.

Kirchhoff's Current Law

\sum I_{in} = \sum I_{out}

Current into a junction equals current out of it (charge conservation).

Kirchhoff's Voltage Law

\sum V = 0

The sum of voltages around any closed loop is zero.

Lens Maker's / Thin Lens

\frac{1}{f} = \frac{1}{v} - \frac{1}{u}

Relates focal length to object and image distances for a thin lens.

Snell's Law

n_1 \sin\theta_1 = n_2 \sin\theta_2

Light bends when passing between media of different refractive index.

Mass–Energy Equivalence

E = mc^2

Mass and energy are equivalent and interconvertible.

Planck–Einstein Relation

E = hf

The energy of a photon is proportional to its frequency.

Heisenberg Uncertainty Principle

\Delta x \, \Delta p \ge \frac{\hbar}{2}

Position and momentum cannot both be known with unlimited precision.

Newton First Law

\sum F = 0 \Rightarrow \text{constant } v

An object stays at rest or in uniform motion unless acted upon.

Newton Third Law

F_{AB} = -F_{BA}

Every action has an equal and opposite reaction.

Conservation of Energy

E_{total} = \text{constant}

Energy cannot be created or destroyed, only transformed.

Conservation of Momentum

\sum p_i = \sum p_f

Total momentum is conserved without external forces.

Conservation of Charge

\sum q = \text{constant}

Total electric charge in an isolated system is conserved.

First Law of Thermodynamics

\Delta U = Q - W

Change in internal energy equals heat added minus work done.

Second Law of Thermodynamics

\Delta S_{universe} \geq 0

Total entropy of an isolated system never decreases.

Hooke Law

F = -kx

Elastic restoring force is proportional to displacement.

Faraday Law of Induction

\varepsilon = -\frac{d\Phi_B}{dt}

A changing magnetic flux induces an EMF.

Lenz Law

\text{induced current opposes change}

Induced currents oppose the change producing them.

Archimedes Principle

F_b = \rho V g

Buoyant force equals the weight of displaced fluid.

Pascal Principle

\Delta P = \text{transmitted equally}

Pressure applied to a confined fluid transmits undiminished.

Mass-Energy Equivalence

E = mc^2

Energy contained in mass.

Time Dilation

\Delta t = \frac{\Delta t_0}{\sqrt{1 - v^2/c^2}}

Moving clocks run slow.

Length Contraction

L = L_0\sqrt{1 - v^2/c^2}

Moving objects shorten along motion.

Relativistic Momentum

p = \frac{mv}{\sqrt{1 - v^2/c^2}}

Momentum at speeds near light.

Lorentz Factor

\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}

Scaling factor in special relativity.

de Broglie Wavelength

\lambda = \frac{h}{p}

Wavelength associated with a particle.

Uncertainty Principle

\Delta x\,\Delta p \geq \frac{\hbar}{2}

Limit on knowing position and momentum together.

Planck Relation

E = hf

Quantised energy of radiation.

Photoelectric Effect

K_{max} = hf - \phi

Kinetic energy of ejected electrons.

Bohr Energy Levels

E_n = -\frac{13.6}{n^2}\ \text{eV}

Energy of the nth hydrogen orbit.

Radioactive Decay

N = N_0 e^{-\lambda t}

Number of nuclei remaining over time.

Half-Life

t_{1/2} = \frac{\ln 2}{\lambda}

Time for half a sample to decay.

Radioactive Activity

A = \lambda N

Decays per second in a sample.

Binding Energy

E_b = \Delta m\,c^2

Energy holding a nucleus together.

Wien's Displacement Law

\lambda_{max} = \frac{b}{T}

Peak emission wavelength of a hot body.

Compton Shift

\Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta)

Wavelength change of scattered photons.