Physics Formulary – Every Formula, Explained

55 formulas

Kinematics9

x = x_0 + v\,t

Uniform motion

Constant-velocity motion · Position update

An object moving at constant velocity covers equal distances in equal times, so its position grows linearly with time.

$x$ Positionm
$x_0$ Initial positionm
$v$ Velocitym/s
$t$ Times

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Variables

$x$	Position	m
$x_0$	Initial position	m
$v$	Velocity	m/s
$t$	Time	s

When it holds

Constant velocity: no acceleration
Applies to each component separately in 2-D or 3-D

Good to know

A simulation applies this rule once per time step, $\vec{x}_{t+\Delta t} = \vec{x}_t + \vec{v}\,\Delta t$ — during a short enough step, any motion looks uniform.

v = v_0 + a\,t

Velocity under constant acceleration

First equation of motion · SUVAT v = u + at

With a constant acceleration, velocity changes by the same amount every second: acceleration is the rate of change of velocity.

$v$ Final velocitym/s
$v_0$ Initial velocitym/s
$a$ Accelerationm/s²
$t$ Times

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Variables

$v$	Final velocity	m/s
$v_0$	Initial velocity	m/s
$a$	Acceleration	m/s²
$t$	Time	s

When it holds

Acceleration constant in magnitude and direction

Worked example

A car pulls away from rest with an acceleration of 3 m/s². How fast is it going after 4 s?

$v = 0 + 3 \times 4 = 12$ m/s, about 43 km/h.

x = x_0 + v_0 t + \tfrac{1}{2} a t^{2}

Displacement under constant acceleration

Second equation of motion · SUVAT s = ut + ½at²

Position under constant acceleration: the $v_0 t$ term is the distance you would cover anyway, the $\tfrac12 a t^2$ term is what the acceleration adds.

$x$ Positionm
$x_0$ Initial positionm
$v_0$ Initial velocitym/s
$a$ Accelerationm/s²
$t$ Times

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Variables

$x$	Position	m
$x_0$	Initial position	m
$v_0$	Initial velocity	m/s
$a$	Acceleration	m/s²
$t$	Time	s

When it holds

Acceleration constant in magnitude and direction

Common mistakes

Solving for $t$ means solving a quadratic: the calculator returns the first non-negative time the position is reached.

v = \sqrt{2 g h}

Speed after a free fall

Impact speed · Torricelli's equation from rest

An object dropped from rest reaches this speed after falling a height $h$ . Its mass does not appear: without air, everything falls alike.

$v$ Speedm/s
$g$ Gravitational accelerationm/s²
$h$ Height fallenm

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Variables

$v$	Speed	m/s
$g$	Gravitational acceleration	m/s²
$h$	Height fallen	m

When it holds

Starts from rest
No air resistance
$g$ constant over the fall

Good to know

It follows from energy conservation, $mgh = \tfrac12 m v^2$ , and the fall takes $t = \sqrt{2h/g}$ .

y = x\tan\theta - \dfrac{g\,x^{2}}{2\,v_0^{2}\cos^{2}\theta}

Projectile trajectory

Parabolic path · Equation of the trajectory

Eliminating time from the two equations of motion leaves height as a function of horizontal distance: a parabola.

$y$ Heightm
$x$ Horizontal distancem
$\theta$ Launch angle°
$v_0$ Launch speedm/s
$g$ Gravitational accelerationm/s²

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Variables

$y$	Height	m
$x$	Horizontal distance	m
$\theta$	Launch angle	°
$v_0$	Launch speed	m/s
$g$	Gravitational acceleration	m/s²

When it holds

No air resistance
Launched from the origin (add the launch height $h_0$ otherwise)

Good to know

Solving for $v_0$ answers the aiming question: how fast must I throw at this angle to pass through the point $(x, y)$ ?

R = \dfrac{v_0^{2}\sin 2\theta}{g}

Range of a projectile

Horizontal range

How far a projectile lands on level ground. For a given speed it peaks at 45°, where $\sin 2\theta = 1$ .

$R$ Rangem
$v_0$ Launch speedm/s
$\theta$ Launch angle°
$g$ Gravitational accelerationm/s²

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Variables

$R$	Range	m
$v_0$	Launch speed	m/s
$\theta$	Launch angle	°
$g$	Gravitational acceleration	m/s²

When it holds

Level ground: lands at launch height
No air resistance

Worked example

A ball is kicked at 20 m/s and 45°. How far does it go?

$R = 20^2 \cdot \sin 90° / 9.81 \approx 40.8$ m.

Common mistakes

Complementary angles ( $\theta$ and $90° - \theta$ ) give the same range. Solving for $\theta$ returns the lower one.

H = \dfrac{v_0^{2}\sin^{2}\theta}{2g}

Maximum height of a projectile

Apex height

Only the vertical part of the launch velocity, $v_0 \sin\theta$ , fights gravity — so only it sets how high the projectile climbs.

$H$ Maximum heightm
$v_0$ Launch speedm/s
$\theta$ Launch angle°
$g$ Gravitational accelerationm/s²

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Variables

$H$	Maximum height	m
$v_0$	Launch speed	m/s
$\theta$	Launch angle	°
$g$	Gravitational acceleration	m/s²

When it holds

No air resistance
Measured from launch height

a_c = \dfrac{v^{2}}{r} = \omega^{2} r

Centripetal acceleration

Radial acceleration

Moving in a circle at constant speed is still accelerated motion: the velocity keeps turning, and that takes an acceleration pointing to the centre.

$a_c$ Centripetal accelerationm/s²
$v$ Speedm/s
$r$ Radiusm

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Variables

$a_c$	Centripetal acceleration	m/s²
$v$	Speed	m/s
$r$	Radius	m

When it holds

Uniform circular motion (constant speed)

Good to know

With angular speed $\omega = v / r$ the same acceleration reads $\omega^2 r$ .

T = \dfrac{2\pi r}{v} = \dfrac{2\pi}{\omega}

Period of circular motion

Orbital period

The time for one full lap: the circumference divided by the speed.

$T$ Periods
$r$ Radiusm
$v$ Speedm/s

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Variables

$T$	Period	s
$r$	Radius	m
$v$	Speed	m/s

When it holds

Uniform circular motion (constant speed)

Forces & dynamics13

\vec{F} = m\,\vec{a}

Newton's second law

F = ma · Fundamental law of dynamics

The net force on a body equals its mass times its acceleration. A force does not keep things moving — it changes how they move.

$F$ Net forceN
$m$ Masskg
$a$ Accelerationm/s²

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Variables

$F$	Net force	N
$m$	Mass	kg
$a$	Acceleration	m/s²

When it holds

Inertial (non-accelerating) frame of reference
Constant mass
Speeds far below the speed of light

Common mistakes

$F$ is the net force, the vector sum of every force on the body, not any single one of them.
Force and acceleration always point the same way; velocity need not.

History

Isaac Newton · 1687
Stated in the Principia as the second of three laws of motion.

W = m\,g

Weight

Force of gravity near the Earth

Weight is the gravitational force on a mass. Mass is how much stuff there is; weight is how hard gravity pulls on it.

$W$ WeightN
$m$ Masskg
$g$ Gravitational accelerationm/s²

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Variables

$W$	Weight	N
$m$	Mass	kg
$g$	Gravitational acceleration	m/s²

When it holds

Near a planet's surface, where $g$ is roughly constant

Common mistakes

Mass is measured in kilograms, weight in newtons. A 70 kg person weighs about 687 N on Earth and about 113 N on the Moon — with the same mass.

F = -k\,x

Hooke's law

Spring force · Restoring force · Elastic force

A spring pulls back with a force proportional to how far it is stretched or compressed, always towards its rest length.

$F$ Spring forceN
$k$ Spring constantN/m
$x$ Displacement from rest lengthm

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Variables

$F$	Spring force	N
$k$	Spring constant	N/m
$x$	Displacement from rest length	m

When it holds

Within the spring's elastic limit
Ideal, massless spring

Common mistakes

The minus sign is the physics: the force points opposite to the displacement. That is what makes the motion oscillate.

History

Robert Hooke · 1678
First hidden in the anagram ceiiinosssttuv (1676), then decoded: ut tensio, sic vis — as the extension, so the force.

F_c = \dfrac{m v^{2}}{r}

Centripetal force

Net radial force

The net force towards the centre that a body needs to follow a circle of radius $r$ at speed $v$ .

$F_c$ Centripetal forceN
$m$ Masskg
$v$ Speedm/s
$r$ Radiusm

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Variables

$F_c$	Centripetal force	N
$m$	Mass	kg
$v$	Speed	m/s
$r$	Radius	m

When it holds

Uniform circular motion (constant speed)

Common mistakes

Centripetal force is not a new kind of force. It is a job that tension, gravity, friction or a normal force does.

F_\parallel = m g \sin\theta

Weight along a slope

Parallel component of weight

On a slope, part of the weight pulls the body down along the surface. The steeper the slope, the bigger that part.

$F_\parallel$ Force along the slopeN
$m$ Masskg
$g$ Gravitational accelerationm/s²
$\theta$ Slope angle°

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Variables

$F_\parallel$	Force along the slope	N
$m$	Mass	kg
$g$	Gravitational acceleration	m/s²
$\theta$	Slope angle	°

When it holds

Flat, straight incline

N = m g \cos\theta

Normal force on a slope

Perpendicular component of weight

The surface pushes back on the body with a force that balances the part of the weight pressing into it.

$N$ Normal forceN
$m$ Masskg
$g$ Gravitational accelerationm/s²
$\theta$ Slope angle°

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Variables

$N$	Normal force	N
$m$	Mass	kg
$g$	Gravitational acceleration	m/s²
$\theta$	Slope angle	°

When it holds

Flat, straight incline
No other force pressing into or lifting off the surface

Common mistakes

The normal force equals $mg$ only on flat ground. On a slope it is smaller.

f_{s,\max} = \mu_s N

Maximum static friction

Limiting friction

Static friction holds a body still up to a limit set by the surfaces ( $\mu_s$ ) and how hard they are pressed together ( $N$ ).

$f_{s,\max}$ Maximum static frictionN
$\mu_s$ Coefficient of static friction
$N$ Normal forceN

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Variables

$f_{s,\max}$	Maximum static friction	N
$\mu_s$	Coefficient of static friction
$N$	Normal force	N

When it holds

Dry surfaces (Amontons–Coulomb model)

Common mistakes

This is a ceiling, not the actual friction. Static friction takes whatever value is needed to prevent sliding, up to $\mu_s N$ .

\tan\theta_{\text{repose}} = \mu_s

Angle of repose

Critical slope angle

The steepest slope a body can rest on without sliding. Surprisingly, it does not depend on the mass.

$\theta_{\text{repose}}$ Angle of repose°
$\mu_s$ Coefficient of static friction

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Variables

$\theta_{\text{repose}}$	Angle of repose	°
$\mu_s$	Coefficient of static friction

When it holds

Dry friction model
Rigid block on a flat incline

Good to know

It comes from setting $mg\sin\theta = \mu_s\,mg\cos\theta$ : the mass cancels.

F_{\text{drag}} = \tfrac{1}{2}\,C_d\,\rho\,A\,v^2

Drag equation

Air resistance · Quadratic drag

Air resistance grows with the square of speed: go twice as fast and the drag is four times bigger.

$F_{\text{drag}}$ Drag forceN
$C_d$ Drag coefficient
$\rho$ Fluid densitykg/m³
$A$ Cross-sectional aream²
$v$ Speedm/s

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Variables

$F_{\text{drag}}$	Drag force	N
$C_d$	Drag coefficient
$\rho$	Fluid density	kg/m³
$A$	Cross-sectional area	m²
$v$	Speed	m/s

When it holds

High Reynolds number (turbulent flow): ordinary objects moving through air or water
$C_d$ depends on shape; 0.47 is a sphere

v_t = \sqrt{\dfrac{2\,m\,g}{C_d\,\rho\,A}}

Terminal velocity

Terminal speed

A falling body stops speeding up when drag grows to match its weight. That top speed is the terminal velocity.

$v_t$ Terminal velocitym/s
$m$ Masskg
$g$ Gravitational accelerationm/s²
$C_d$ Drag coefficient
$\rho$ Fluid densitykg/m³
$A$ Cross-sectional aream²

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Variables

$v_t$	Terminal velocity	m/s
$m$	Mass	kg
$g$	Gravitational acceleration	m/s²
$C_d$	Drag coefficient
$\rho$	Fluid density	kg/m³
$A$	Cross-sectional area	m²

When it holds

Quadratic drag
Constant air density

Worked example

A skydiver of 80 kg falls belly-down ( $C_d \approx 1$ , $A \approx 0.7$ m²). What is their terminal velocity?

$v_t = \sqrt{2 \cdot 80 \cdot 9.81 / (1 \cdot 1.225 \cdot 0.7)} \approx 43$ m/s, about 150 km/h.

L = \tfrac{1}{2}\,\rho\, v^{2}\, S\, C_L

Lift equation

Aerodynamic lift

The upward force on a wing grows with air density, wing area and the square of airspeed. The lift coefficient holds the shape and the angle of attack.

$L$ LiftN
$\rho$ Air densitykg/m³
$v$ Airspeedm/s
$S$ Wing aream²
$C_L$ Lift coefficient

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Variables

$L$	Lift	N
$\rho$	Air density	kg/m³
$v$	Airspeed	m/s
$S$	Wing area	m²
$C_L$	Lift coefficient

When it holds

Below the stall angle
Subsonic flight

\mathcal{L} = T - V

Lagrangian

Lagrange function

Kinetic minus potential energy. Written in any convenient coordinates, it holds all the dynamics of a system — no forces or constraint reactions needed.

$\mathcal{L}$ LagrangianJ
$T$ Kinetic energyJ
$V$ Potential energyJ

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Variables

$\mathcal{L}$	Lagrangian	J
$T$	Kinetic energy	J
$V$	Potential energy	J

When it holds

Conservative forces (potentials exist)

History

Joseph-Louis Lagrange · 1788
Mécanique analytique.

\dfrac{d}{dt}\dfrac{\partial \mathcal{L}}{\partial \dot{q}_i} - \dfrac{\partial \mathcal{L}}{\partial q_i} = 0

Euler–Lagrange equation

Lagrange's equations

One equation of motion per coordinate $q_i$ , straight from the Lagrangian. It is how the double pendulum's equations are derived.

$\mathcal{L}$ LagrangianJ
$q_i$ Generalised coordinate (an angle, a position…)
$\dot{q}_i$ Its time derivative

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Variables

$\mathcal{L}$	Lagrangian	J
$q_i$	Generalised coordinate (an angle, a position…)
$\dot{q}_i$	Its time derivative

When it holds

Holonomic constraints
Conservative forces

Energy & momentum8

K = \tfrac{1}{2} m v^{2}

Kinetic energy

Energy of motion

The energy a body has because it moves. It grows with the square of speed: twice as fast carries four times the energy.

$K$ Kinetic energyJ
$m$ Masskg
$v$ Speedm/s

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Variables

$K$	Kinetic energy	J
$m$	Mass	kg
$v$	Speed	m/s

When it holds

Speeds far below the speed of light

Worked example

How much kinetic energy does a 1000 kg car have at 50 km/h (about 14 m/s)?

$K = \tfrac12 \cdot 1000 \cdot 14^2 = 98\,000$ J. At 100 km/h it is four times as much.

U = m g h

Gravitational potential energy

GPE · Potential energy near the Earth

The energy stored by lifting a mass against gravity. Only changes in $h$ matter, so you choose where $h = 0$ is.

$U$ Potential energyJ
$m$ Masskg
$g$ Gravitational accelerationm/s²
$h$ Height above the reference levelm

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Variables

$U$	Potential energy	J
$m$	Mass	kg
$g$	Gravitational acceleration	m/s²
$h$	Height above the reference level	m

When it holds

Near a planet's surface, where $g$ is roughly constant

U = \tfrac{1}{2} k x^{2}

Elastic potential energy

Energy stored in a spring

The energy stored in a stretched or compressed spring. At the turning points of an oscillation all the energy is here, so the total is $\tfrac12 k A^2$ .

$U$ Elastic potential energyJ
$k$ Spring constantN/m
$x$ Displacement from rest lengthm

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Variables

$U$	Elastic potential energy	J
$k$	Spring constant	N/m
$x$	Displacement from rest length	m

When it holds

Within the elastic limit (Hooke's law holds)

v = \sqrt{2 g L\,(1 - \cos\theta_0)}

Pendulum speed at the bottom

Maximum speed of a pendulum

Released from an angle $\theta_0$ , the bob drops a height $L(1-\cos\theta_0)$ and turns it all into speed at the lowest point.

$v$ Speed at the bottomm/s
$g$ Gravitational accelerationm/s²
$L$ Lengthm
$\theta_0$ Release angle°

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Variables

$v$	Speed at the bottom	m/s
$g$	Gravitational acceleration	m/s²
$L$	Length	m
$\theta_0$	Release angle	°

When it holds

No friction or air resistance
Rigid, massless rod or taut string

m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2

Conservation of momentum

Momentum balance in a collision

In any collision the total momentum before equals the total after — elastic, inelastic or sticky.

$m_1$ Mass of body 1kg
$u_1$ Velocity of body 1 beforem/s
$m_2$ Mass of body 2kg
$u_2$ Velocity of body 2 beforem/s
$v_1$ Velocity of body 1 afterm/s
$v_2$ Velocity of body 2 afterm/s

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Variables

$m_1$	Mass of body 1	kg
$u_1$	Velocity of body 1 before	m/s
$m_2$	Mass of body 2	kg
$u_2$	Velocity of body 2 before	m/s
$v_1$	Velocity of body 1 after	m/s
$v_2$	Velocity of body 2 after	m/s

When it holds

Isolated system: no net external force during the collision

Common mistakes

Velocities are signed: pick a positive direction and give anything moving the other way a minus sign.
Momentum is conserved even when kinetic energy is not.

v_1 = \dfrac{m_1 - m_2}{m_1 + m_2}\,u_1 + \dfrac{2 m_2}{m_1 + m_2}\,u_2,\qquad v_2 = \dfrac{2 m_1}{m_1 + m_2}\,u_1 + \dfrac{m_2 - m_1}{m_1 + m_2}\,u_2

Elastic collision in one dimension

Final velocities of an elastic collision

When both momentum and kinetic energy are conserved, the velocities after a head-on collision follow from the masses and the velocities before.

$m_1$ Mass of body 1kg
$m_2$ Mass of body 2kg
$u_1$ Velocity of body 1 beforem/s
$u_2$ Velocity of body 2 beforem/s
$v_1$ Velocity of body 1 afterm/s
$v_2$ Velocity of body 2 afterm/s

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Variables

$m_1$	Mass of body 1	kg
$m_2$	Mass of body 2	kg
$u_1$	Velocity of body 1 before	m/s
$u_2$	Velocity of body 2 before	m/s
$v_1$	Velocity of body 1 after	m/s
$v_2$	Velocity of body 2 after	m/s

When it holds

Perfectly elastic ( $e = 1$ )
Head-on, along one line

Good to know

Equal masses simply swap velocities — the Newton's cradle effect.

e = \dfrac{v_2 - v_1}{u_1 - u_2}

Coefficient of restitution

Restitution · Bounciness

How much of the approach speed survives a collision as separation speed: 1 for a perfectly elastic bounce, 0 when the bodies stick.

$e$ Coefficient of restitution
$u_1$ Velocity of body 1 beforem/s
$u_2$ Velocity of body 2 beforem/s
$v_1$ Velocity of body 1 afterm/s
$v_2$ Velocity of body 2 afterm/s

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Variables

$e$	Coefficient of restitution
$u_1$	Velocity of body 1 before	m/s
$u_2$	Velocity of body 2 before	m/s
$v_1$	Velocity of body 1 after	m/s
$v_2$	Velocity of body 2 after	m/s

When it holds

Head-on collision
$0 \le e \le 1$ for ordinary materials

Good to know

Against a fixed wall or floor it reduces to $e = |v_{\text{after}}| / |v_{\text{before}}|$ : each bounce reverses the velocity and scales it, $v \rightarrow -e\,v$ , and the rebound height by $e^2$ .

N_{\text{collisions}} = \lfloor \pi \cdot 10^{N} \rfloor

π from colliding blocks

Galperin's billiards · Block collision count

A block $100^N$ times heavier than a small one, sliding into it towards a wall, makes the two collide exactly as many times as the first $N+1$ digits of π.

$N_{\text{collisions}}$ Number of collisions
$N$ Mass ratio exponent: the big block is 100ᴺ times heavier

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Variables

$N_{\text{collisions}}$	Number of collisions
$N$	Mass ratio exponent: the big block is 100ᴺ times heavier

When it holds

Perfectly elastic collisions with each other and with the wall
No friction

Good to know

Both momentum and kinetic energy are conserved in every collision; drawn in the right coordinates, energy conservation is a circle and each collision a fixed rotation around it.

History

Gregory Galperin · 2003
Published as “Playing pool with π”.

Oscillations5

\omega = \sqrt{\dfrac{k}{m}}

Angular frequency of a mass on a spring

Natural frequency

How fast a mass on a spring oscillates: a stiffer spring speeds it up, a heavier mass slows it down.

$\omega$ Angular frequencyrad/s
$k$ Spring constantN/m
$m$ Masskg

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Variables

$\omega$	Angular frequency	rad/s
$k$	Spring constant	N/m
$m$	Mass	kg

When it holds

Ideal spring obeying Hooke's law
No damping

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

Period of a mass on a spring

Mass–spring period

The time for one full oscillation. The amplitude is missing on purpose: small or large swings take the same time.

$T$ Periods
$m$ Masskg
$k$ Spring constantN/m

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Variables

$T$	Period	s
$m$	Mass	kg
$k$	Spring constant	N/m

When it holds

Ideal spring obeying Hooke's law
No damping

Worked example

A 0.5 kg mass hangs from a spring with $k = 20$ N/m. What is its period?

$T = 2\pi\sqrt{0.5/20} \approx 0.99$ s — almost exactly one bounce per second.

Common mistakes

Hanging the spring vertically does not change the period: gravity only shifts the rest position.

x(t) = A\cos(\omega t + \varphi)

Simple harmonic motion

Position over time in SHM

A linear restoring force makes a body oscillate as a cosine: amplitude $A$ , angular frequency $\omega$ , phase $\varphi$ .

$x$ Displacementm
$A$ Amplitudem
$\omega$ Angular frequencyrad/s
$t$ Times
$\varphi$ Phaserad

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Variables

$x$	Displacement	m
$A$	Amplitude	m
$\omega$	Angular frequency	rad/s
$t$	Time	s
$\varphi$	Phase	rad

When it holds

Restoring force proportional to displacement
No damping

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

Period of a simple pendulum

Pendulum formula

For small swings a pendulum's period depends only on its length and on gravity — not on the mass, not on how far it swings.

$T$ Periods
$L$ Lengthm
$g$ Gravitational accelerationm/s²

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Variables

$T$	Period	s
$L$	Length	m
$g$	Gravitational acceleration	m/s²

When it holds

Small angles (below about 15°), where $\sin\theta \approx \theta$
Point mass on a massless string, no friction

Worked example

How long must a pendulum be to tick once per second (a period of 2 s)?

$L = g\,(T/2\pi)^2 = 9.81 \cdot (2/2\pi)^2 \approx 0.99$ m — the “seconds pendulum” of grandfather clocks.

Common mistakes

At large amplitudes the real period is longer: about 1.7% longer at 30°, 18% at 90°.

History

Galileo Galilei, Christiaan Huygens · 1602–1673
Galileo noticed that a pendulum's swings take equal times; Huygens derived the formula and built the first pendulum clock.

\ddot{\theta} + \dfrac{g}{L}\sin\theta = 0

Pendulum equation of motion

Nonlinear pendulum equation

The exact equation for a swinging pendulum. The $\sin\theta$ makes it nonlinear; replacing it with $\theta$ for small angles gives simple harmonic motion.

$\theta$ Angle from the verticalrad
$\ddot{\theta}$ Angular accelerationrad/s²
$g$ Gravitational accelerationm/s²
$L$ Lengthm

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Variables

$\theta$	Angle from the vertical	rad
$\ddot{\theta}$	Angular acceleration	rad/s²
$g$	Gravitational acceleration	m/s²
$L$	Length	m

When it holds

Point mass on a rigid, massless rod
No friction

Gravitation4

F = G\,\dfrac{m_1 m_2}{r^{2}}

Newton's law of universal gravitation

Inverse-square law of gravity

Every two masses attract each other along the line between them, with a force that falls off as the square of the distance.

$F$ Gravitational forceN
$m_1$ First masskg
$m_2$ Second masskg
$r$ Distance between the centresm

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$F$	Gravitational force	N
$m_1$	First mass	kg
$m_2$	Second mass	kg
$r$	Distance between the centres	m

$G$ = 6.674 × 10^-11N·m²/kg²

When it holds

Point masses or spherically symmetric bodies
Weak fields and slow speeds (otherwise general relativity)

Good to know

In vector form, the force on body $i$ from body $j$ is $\vec{F}_{ij} = -G\,\dfrac{m_i m_j}{|\vec{r}_{ij}|^{2}}\,\hat{r}_{ij}$ .

History

Isaac Newton · 1687
The same law explains a falling apple and the orbit of the Moon.

\vec{F}_i = \sum_{j \ne i} \vec{F}_{ij}

Net gravitational force

Superposition of forces · N-body force

With many bodies, each one feels the vector sum of the pulls from all the others. With three or more there is no general closed-form solution.

$\vec{F}_i$ Net force on body $i$ N
$\vec{F}_{ij}$ Force on body $i$ from body $j$ N

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Variables

$\vec{F}_i$	Net force on body $i$	N
$\vec{F}_{ij}$	Force on body $i$ from body $j$	N

When it holds

Newtonian gravity

v = \sqrt{\dfrac{G M}{r}}

Circular orbital speed

Orbital velocity

The speed that keeps a satellite on a circular orbit: gravity supplies exactly the centripetal force it needs.

$v$ Orbital speedm/s
$M$ Mass of the central bodykg
$r$ Orbit radius (from the centre)m

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Variables

$v$	Orbital speed	m/s
$M$	Mass of the central body	kg
$r$	Orbit radius (from the centre)	m

$G$ = 6.674 × 10^-11N·m²/kg²

When it holds

Circular orbit
Satellite much lighter than the central body

Good to know

It comes from setting $G M m / r^2 = m v^2 / r$ — the satellite's own mass cancels.

Worked example

The ISS orbits about 400 km up, so $r \approx 6771$ km. How fast does it go?

$v = \sqrt{6.674\times10^{-11} \cdot 5.972\times10^{24} / 6.771\times10^{6}} \approx 7.67$ km/s.

g(h) = g_0\left(\dfrac{R}{R+h}\right)^{2}

Gravity at altitude

Variation of g with height

Gravity weakens slowly with height. At the International Space Station it is still about 90% of its value on the ground.

$g(h)$ Gravitational acceleration at height hm/s²
$g_0$ Surface gravitational accelerationm/s²
$R$ Planet radiusm
$h$ Height above the surfacem

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Variables

$g(h)$	Gravitational acceleration at height h	m/s²
$g_0$	Surface gravitational acceleration	m/s²
$R$	Planet radius	m
$h$	Height above the surface	m

When it holds

Spherical planet
Above the surface

Common mistakes

Astronauts float because they are in free fall around the Earth, not because gravity is missing.

Fluids & heat1

\dot Q = -k\,A\,\dfrac{\Delta T}{d}

Fourier's law of heat conduction

Heat conduction · Thermal conduction

Heat flows through a material from hot to cold at a rate set by its conductivity, its area, and how steep the temperature drop is.

$\dot Q$ Heat flow rateW
$k$ Thermal conductivityW/(m·K)
$A$ Aream²
$\Delta T$ Temperature change across the layerK
$d$ Thicknessm

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Variables

$\dot Q$	Heat flow rate	W
$k$	Thermal conductivity	W/(m·K)
$A$	Area	m²
$\Delta T$	Temperature change across the layer	K
$d$	Thickness	m

When it holds

Steady state
Uniform slab of one material

Common mistakes

Metal feels colder than wood at the same temperature because its high $k$ draws heat out of your hand faster — not because it is colder.

History

Joseph Fourier · 1822
Théorie analytique de la chaleur, the book that also introduced Fourier series.

Electricity & magnetism2

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

Coulomb's law

Electrostatic force

Two charges push apart (same sign) or pull together (opposite signs) with a force that falls off as the square of the distance — the electric twin of Newton's gravity.

$F$ Electrostatic force (positive = repulsive)N
$q_1$ First chargeC
$q_2$ Second chargeC
$r$ Distancem

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Variables

$F$	Electrostatic force (positive = repulsive)	N
$q_1$	First charge	C
$q_2$	Second charge	C
$r$	Distance	m

$\tfrac{1}{4\pi\varepsilon_0}$ = 8.988 × 10⁹N·m²/C²

When it holds

Point charges at rest
In vacuum (in a material, divide by its relative permittivity)

History

Charles-Augustin de Coulomb · 1785
Measured with a torsion balance.

R = \dfrac{\rho L}{A}

Resistance of a wire

Resistivity formula

A longer wire resists current more, a thicker one less. The material enters through its resistivity $\rho$ .

$R$ ResistanceΩ
$\rho$ ResistivityΩ·m
$L$ Lengthm
$A$ Cross-sectional aream²

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Variables

$R$	Resistance	Ω
$\rho$	Resistivity	Ω·m
$L$	Length	m
$A$	Cross-sectional area	m²

When it holds

Uniform wire
Constant temperature

Light & optics6

n_1 \sin\theta_1 = n_2 \sin\theta_2

Snell's law

Law of refraction

Light bends when it crosses into a medium where it travels at a different speed. The refractive indices decide by how much.

$n_1$ Refractive index of the first medium
$\theta_1$ Angle of incidence°
$n_2$ Refractive index of the second medium
$\theta_2$ Angle of refraction°

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Variables

$n_1$	Refractive index of the first medium
$\theta_1$	Angle of incidence	°
$n_2$	Refractive index of the second medium
$\theta_2$	Angle of refraction	°

When it holds

Angles measured from the normal to the surface

Common mistakes

Going from a dense to a less dense medium, past the critical angle there is no refracted ray at all: total internal reflection.

History

Ibn Sahl, Willebrord Snellius · 984, 1621
First described by Ibn Sahl in Baghdad, rediscovered by Snellius six centuries later.

\hat{D}_r = \hat{D} - 2(\hat{D}\cdot\hat{N})\,\hat{N}

Law of reflection

Mirror reflection

A ray bounces off a mirror at the same angle it arrived ( $\theta_i = \theta_r$ ). In vector form: flip the part of the direction along the normal.

$\hat{D}$ Incoming direction (unit vector)
$\hat{N}$ Surface normal (unit vector)
$\hat{D}_r$ Reflected direction

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Variables

$\hat{D}$	Incoming direction (unit vector)
$\hat{N}$	Surface normal (unit vector)
$\hat{D}_r$	Reflected direction

When it holds

Smooth, mirror-like surface

E = \dfrac{I_0}{d^2}

Inverse-square law of light

Irradiance from a point source

Light from a point source spreads over a sphere, so its brightness falls with the square of the distance: twice as far, a quarter as bright.

$E$ Irradiance on the surfaceW/m²
$I_0$ Source intensityW/sr
$d$ Distance from the sourcem

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Variables

$E$	Irradiance on the surface	W/m²
$I_0$	Source intensity	W/sr
$d$	Distance from the source	m

When it holds

Point source
No absorption along the way

I_d = k_d\,E\,\max(0,\ \hat{N}\cdot\hat{L})

Lambert's cosine law

Diffuse shading

A matte surface looks brightest when it faces the light and darker as it turns away, following the cosine of the angle between normal and light.

$I_d$ Diffuse brightnessW/m²
$k_d$ Diffuse reflectance (albedo)
$E$ Incoming irradianceW/m²
$\theta$ Angle between normal $\hat{N}$ and light $\hat{L}$ °

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Variables

$I_d$	Diffuse brightness	W/m²
$k_d$	Diffuse reflectance (albedo)
$E$	Incoming irradiance	W/m²
$\theta$	Angle between normal $\hat{N}$ and light $\hat{L}$	°

When it holds

Ideal matte (Lambertian) surface

Good to know

For unit vectors, $\hat{N}\cdot\hat{L} = \cos\theta$ ; the $\max$ keeps surfaces facing away from the light black instead of negative.

History

Johann Heinrich Lambert · 1760
Photometria.

\vec{P}(t) = \vec{O} + t\,\hat{D}

Parametric ray

Ray equation

Every point on a ray: start at the origin $\vec{O}$ and walk a distance $t$ along the direction $\hat{D}$ . Ray tracing is finding the smallest $t$ that hits something.

$\vec{P}(t)$ Point on the ray
$\vec{O}$ Ray origin (the camera)
$\hat{D}$ Direction (unit vector)
$t$ Distance along the raym

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Variables

$\vec{P}(t)$	Point on the ray
$\vec{O}$	Ray origin (the camera)
$\hat{D}$	Direction (unit vector)
$t$	Distance along the ray	m

When it holds

Straight-line propagation (geometric optics)

\Delta = b^2 - 4ac,\quad b = 2\,\hat{D}\cdot(\vec{O}-\vec{C}),\quad c = |\vec{O}-\vec{C}|^2 - r^2

Ray–sphere intersection

Ray–sphere discriminant

Substituting the ray into the sphere's equation gives a quadratic in $t$ . Its discriminant says whether the ray misses ( $\Delta < 0$ ), grazes, or hits the sphere.

$\Delta$ Discriminant
$\vec{C}$ Sphere centre
$r$ Sphere radiusm
$a$ $\hat{D}\cdot\hat{D}$ , equal to 1 for a unit direction

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Variables

$\Delta$	Discriminant
$\vec{C}$	Sphere centre
$r$	Sphere radius	m
$a$	$\hat{D}\cdot\hat{D}$ , equal to 1 for a unit direction

When it holds

Perfect sphere

Modern physics1

\lambda = \dfrac{h}{m v}

de Broglie wavelength

Matter wave

Every moving particle behaves like a wave whose wavelength shrinks as its momentum grows. For everyday objects it is far too small to notice.

$\lambda$ Wavelengthm
$m$ Masskg
$v$ Speedm/s

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Variables

$\lambda$	Wavelength	m
$m$	Mass	kg
$v$	Speed	m/s

$h$ = 6.626 × 10^-34J·s

When it holds

Non-relativistic speeds (otherwise use the relativistic momentum)

Worked example

What is the wavelength of an electron moving at $10^6$ m/s?

$\lambda = 6.63\times10^{-34} / (9.11\times10^{-31} \cdot 10^6) \approx 0.73$ nm — about the spacing of atoms in a crystal, which is why electrons diffract.

History

Louis de Broglie · 1924
Proposed in his PhD thesis; Nobel Prize 1929.

Mathematics6

|\vec{a}| = \sqrt{a_x^{2} + a_y^{2}}

Magnitude of a vector

Length of a vector · Norm

The length of a vector from its components — Pythagoras' theorem on the right triangle they form.

$|\vec{a}|$ Magnitude
$a_x$ x component
$a_y$ y component

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Variables

$\|\vec{a}\|$	Magnitude
$a_x$	x component
$a_y$	y component

When it holds

Cartesian components (add $a_z^2$ in 3-D)

Good to know

Its direction is $\theta = \arctan(a_y / a_x)$ , and back again: $a_x = |\vec{a}|\cos\theta$ , $a_y = |\vec{a}|\sin\theta$ .

\vec{a} + \vec{b} = (a_x + b_x,\; a_y + b_y)

Sum of two vectors

Vector addition · Tip-to-tail rule

To add vectors, add their components. Geometrically: place the tail of $\vec{b}$ on the tip of $\vec{a}$ .

$\vec{a}$ First vector
$\vec{b}$ Second vector

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Variables

$\vec{a}$	First vector
$\vec{b}$	Second vector

\vec{a}\cdot\vec{b} = a_x b_x + a_y b_y = |\vec{a}||\vec{b}|\cos\theta

Dot product

Scalar product · Inner product

Multiplies two vectors into a number that measures how much they point the same way: zero when they are perpendicular.

$\vec{a}\cdot\vec{b}$ Dot product
$|\vec{a}|$ Magnitude of a
$|\vec{b}|$ Magnitude of b
$\theta$ Angle between them°

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Variables

$\vec{a}\cdot\vec{b}$	Dot product
$\|\vec{a}\|$	Magnitude of a
$\|\vec{b}\|$	Magnitude of b
$\theta$	Angle between them	°

Good to know

In physics it gives work: $W = \vec{F}\cdot\vec{s}$ counts only the part of the force along the motion.

\sin^{2}\theta + \cos^{2}\theta = 1

Pythagorean identity

sin² + cos² = 1

On the unit circle, $\cos\theta$ and $\sin\theta$ are the legs of a right triangle with hypotenuse 1 — so this holds for every angle.

$\theta$ Any angle°

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$\theta$	Any angle	°

\tan\theta = \dfrac{\sin\theta}{\cos\theta}

Tangent as a ratio

Definition of tangent

The tangent is the slope of the line from the origin through the point on the unit circle. It blows up at 90°, where the cosine is zero.

$\theta$ Angle°

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Variables

$\theta$	Angle	°

y = A\sin(\omega\theta + \varphi)

Sine wave

Sinusoid · Generalised sine function

The shape of every oscillation and wave: $A$ stretches it vertically, $\omega$ squeezes it horizontally, $\varphi$ slides it sideways.

$y$ Value
$A$ Amplitude
$\omega$ Angular frequency
$\theta$ Angle°
$\varphi$ Phase°

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Variables

$y$	Value
$A$	Amplitude
$\omega$	Angular frequency
$\theta$	Angle	°
$\varphi$	Phase	°