physicshub.github.io

# PhysicsHub

> Free, open-source interactive physics simulations with written theory for students, teachers and developers.

PhysicsHub hosts 17 browser-based physics simulations built on a custom, in-house physics engine, each paired with a written explainer. Physics is in SI units, Y-up. All content is free and released under the MIT licence.

## Articles
- [Class 12 Physics Complete Guide – All Chapters with Formulas & Examples](https://physicshub.github.io/blog/class-12-physics-complete-guide): Class 12 Physics (CBSE 2025–26) chapter by chapter: clear explanations, key formulas, derivations and exam strategy from electrostatics to semiconductors.
- [Why is the sky blue?](https://physicshub.github.io/blog/why-is-the-sky-blue): Air molecules scatter short-wavelength blue light about six times more than red — Rayleigh scattering. The same effect turns sunsets red and clouds white.
- [Why does metal feel colder than wood?](https://physicshub.github.io/blog/why-does-metal-feel-colder-than-wood): Metal and wood in the same room are at the same temperature. Metal feels colder because it conducts heat out of your skin much faster than wood does.
- [What is the difference between elastic and inelastic collisions?](https://physicshub.github.io/blog/elastic-inelastic-collisions): Momentum is conserved in every collision; kinetic energy only in elastic ones. See both cases with worked numbers and the restitution coefficient that links them.
- [What is the unit circle?](https://physicshub.github.io/blog/unit-circle-trigonometry): On a circle of radius 1, cosine and sine are the x and y coordinates of a point at angle θ. Tangent and the other functions are lengths too — and sin(ωθ+φ) is a wave.
- [Why do we see lightning before we hear thunder?](https://physicshub.github.io/blog/why-do-we-see-lightning-before-thunder): Light travels almost a million times faster than sound. Count the seconds between flash and thunder, divide by 3, and you have the distance in kilometres.
- [How do airplanes fly?](https://physicshub.github.io/blog/how-do-airplanes-fly): A wing pushes air downward and the air pushes the wing up. That lift grows with speed squared and balances the plane's weight in level flight.
- [How does an inclined plane work?](https://physicshub.github.io/blog/inclined-plane-forces): On a ramp, weight splits into mg sin θ along the slope and mg cos θ into it. Friction decides whether the block slides — the maths, with a simulation.
- [What is centripetal force?](https://physicshub.github.io/blog/circular-motion-centripetal-force): A body in circular motion always accelerates toward the centre, needing an inward force F = mv²/r. If that force vanishes, the body flies off straight.
- [Why do astronauts float in space?](https://physicshub.github.io/blog/why-do-astronauts-float-in-space): Gravity at the ISS is still almost 90% of what it is on the ground. Astronauts float because they and the station are in constant free fall around Earth.
- [Why is the double pendulum chaotic?](https://physicshub.github.io/blog/double-pendulum-chaos): Hang one pendulum from another and the motion becomes unpredictable: a start difference of one millionth of a radian grows to a completely different path within seconds.
- [What is physics? A visual introduction](https://physicshub.github.io/blog/what-is-physics): Physics is the science of matter, energy, space, time and the forces between them. A short visual guide to what it studies, how it works and how to start.
- [How does a bouncing ball work?](https://physicshub.github.io/blog/physics-bouncing-ball-comprehensive-educational-guide): A bouncing ball loses speed at each impact, keeping a fraction e (restitution). Every bounce reaches e² of the last height — the maths, with a simulation.
- [How do you code a bouncing ball?](https://physicshub.github.io/blog/coding-a-bouncing-ball-simulation): Gravity, a floor and a restitution factor make a bouncing ball — but a naive floor check changes the energy. Handle the impact time exactly, in JavaScript.
- [How does free fall work?](https://physicshub.github.io/blog/ball-free-fall-comprehensive-guide): In free fall every object accelerates at g ≈ 9.81 m/s², whatever its mass. Equations, gravity on other worlds, and how air resistance sets terminal velocity.
- [How do you code air resistance?](https://physicshub.github.io/blog/simulating-air-resistance-in-code): Add a drag term proportional to v² to a falling object and it reaches a terminal velocity. Code it in a few lines of JavaScript and check it against the exact solution.
- [How does a pendulum work?](https://physicshub.github.io/blog/physics-of-pendulum-explained): A pendulum's period depends only on its length and gravity, not its mass: T = 2π√(L/g). The physics of the swing, energy and damping, with a simulation.
- [How do you simulate a pendulum in code?](https://physicshub.github.io/blog/simulating-a-pendulum-in-code): Turn the pendulum equation into a state you can step forward in time. Compare explicit Euler, semi-implicit Euler and RK4 with real energy-drift numbers, in JavaScript.
- [How does ray tracing work?](https://physicshub.github.io/blog/how-does-ray-tracing-work): Ray tracing follows straight-line light rays backwards from the camera through each pixel, then applies the laws of optics where they hit to decide the pixel's colour.
- [Kirchhoff's Circuit Laws Explained](https://physicshub.github.io/blog/kirchhoffs-circuit-laws-explained): Why current splits at a junction and why voltages around a loop always add to zero — with a fully worked two-loop network, sign conventions that actually survive contact with a problem, and the nodal method the simulation uses to solve any circuit you build.
- [Vectors: components, addition, dot and cross products](https://physicshub.github.io/blog/comprehensive-guide-to-vector-operations): A vector has magnitude and direction. Split it into components and addition, scaling, dot and cross products all become simple arithmetic.
- [How does a ball accelerate toward a target?](https://physicshub.github.io/blog/ball-uniformly-accelerated-motion): Acceleration is how fast velocity changes. Aim a constant acceleration at a target and the ball speeds up, overshoots and loops back — here is why.
- [How does a spring work?](https://physicshub.github.io/blog/spring-connection): A spring pushes back in proportion to how far you stretch it (Hooke's law) — which is why a mass on a spring oscillates. The physics, with a simulation.
- [How does projectile motion work?](https://physicshub.github.io/blog/projectile-parabolic-motion): Gravity pulls a projectile down while its sideways speed stays constant; together they trace a parabola. The equations, range and angle, with a launcher.
- [The Three-Body Problem: The Equation That Broke Physics](https://physicshub.github.io/blog/physics-behind-three-body-problem): Two bodies under gravity are perfectly predictable; three are chaotic. Why the three-body problem has no formula — and what the Netflix show gets right.
- [How Two Sliding Blocks Compute π — The Most Surprising Result in Physics](https://physicshub.github.io/blog/pi-from-block-collisions-explained): Two frictionless blocks and a wall, all collisions elastic: count the clacks and you get the digits of π. The physics, from first collision to proof.

## Simulations
- [Bouncing Ball](https://physicshub.github.io/simulations/BouncingBall): Simulation of the ball bouncing off the walls. A playful first look at motion, speed and collisions — suited for middle school and even younger explorers.
- [Vector Operations Calculator](https://physicshub.github.io/simulations/VectorsOperations): Interactive 2D vector calculator — visualize addition, subtraction, and dot products in real time. The standard high-school tool for students learning vector physics.
- [Ball Acceleration](https://physicshub.github.io/simulations/BallAcceleration): Simulate a ball accelerating toward your cursor in real time. Explore velocity and uniform acceleration, a core topic from middle-school physics onward — free online.
- [Ball Gravity](https://physicshub.github.io/simulations/BallGravity): Watch a ball fall and bounce under gravity and external forces in real time. Adjust mass, elasticity, and gravitational force to explore Newton's laws — free interactive simulation for high-school physics.
- [Spring Mass System Simulator](https://physicshub.github.io/simulations/SpringConnection): Explore Hooke's Law and Simple Harmonic Motion interactively. Adjust mass and spring constant to visualize oscillations and energy conservation — an extended high-school (A-Level) topic.
- [Simple Pendulum Simulation](https://physicshub.github.io/simulations/SimplePendulum): Simulate a pendulum and calculate its period and frequency in real time. Explore how length, gravity, and energy are related, in a standard high-school topic — free interactive physics tool.
- [Projectile & Parabolic Motion](https://physicshub.github.io/simulations/ParabolicMotion): Simulate projectile motion and predict trajectory, range, and max height instantly. A staple of high-school kinematics — interactive parabolic motion calculator, free online.
- [Inclined Plane](https://physicshub.github.io/simulations/InclinedPlane): Simulate a block sliding down an inclined plane with friction. Adjust angle, mass, and surface to explore forces and Newton's laws — a core high-school dynamics experiment.
- [Circular Motion](https://physicshub.github.io/simulations/CircularMotion): Simulate uniform circular motion in real time. Visualize velocity and centripetal acceleration vectors while adjusting radius and speed — an extended high-school to first-year university topic.
- [Three Body Problem](https://physicshub.github.io/simulations/ThreeBody): Simulate a classic physics problem of three bodies operating under the gravity of each other. A numerically advanced topic aimed at university level (and keen high-school students).
- [Horizontal Spring](https://physicshub.github.io/simulations/HorizontalSpring): A spring attached to the left wall with a draggable mass. Explore Hooke's Law and oscillation on a horizontal plane — approachable from middle school, standard in high school.
- [Double Pendulum](https://physicshub.github.io/simulations/DoublePendulum): Simulate a double pendulum and explore chaotic motion. Watch how small changes in initial conditions lead to completely different trajectories — a classic university-level example of chaos theory.
- [1D Collision simulation](https://physicshub.github.io/simulations/CollisionSimulation): Simulate 1D elastic collisions with velocity vectors, momentum, and kinetic energy visualization — a core momentum topic in high-school physics.
- [Pi From Collisions](https://physicshub.github.io/simulations/PiCollisions): Simulate two colliding blocks of different masses to derive pi. A beautiful conservation-law puzzle for university students (and gifted high-schoolers): adjust the mass of the larger block by a power of 100 to see further precision of pi.
- [Trigonometric Circle](https://physicshub.github.io/simulations/TrigonometricCircle): Interactive unit circle with a draggable angle θ. Visualize sin, cos, tan, cot, sec and csc as circle segments, watch the generalized wave A·sin(ωθ+φ) plot in real time, and trace the inverse functions arcsin, arccos, arctan and arccot — a bridge from middle-school geometry to high-school trig.
- [Ray Tracing](https://physicshub.github.io/simulations/RayTracing): Follow a single ray of light from a camera into a 3D scene and watch a real ray tracer colour each pixel. Drag the camera, sphere and light to see the ray–sphere quadratic, Lambert's cosine law, the inverse-square law, shadows and the law of reflection update live.
- [Kirchhoff's Circuit Laws](https://physicshub.github.io/simulations/KirchhoffCircuit): Build and solve DC circuits online: single loop, series, parallel and a two-loop network with two cells. Watch charge flow, drag a voltmeter and an ammeter onto the circuit, and see Kirchhoff's current and voltage laws balance to zero in real time.

## Simulation concepts
- [Bouncing Ball — concept](https://physicshub.github.io/simulations/BouncingBall): A ball moves in a straight line at constant speed until it meets a wall, then rebounds. It is the simplest possible model of motion and collision: velocity, reflection, and — once gravity and restitution are switched on — energy loss on each bounce. Watching one velocity component flip at every wall makes the idea of a vector concrete.
- [Vector Operations Calculator — concept](https://physicshub.github.io/simulations/VectorsOperations): Vectors are the language of physics: force, velocity, acceleration and displacement are all vectors. Drag two vectors in the plane and see their sum, difference and dot product update live, alongside each vector's magnitude and direction. It is a sandbox for building intuition before vectors appear in every later topic.
- [Ball Acceleration — concept](https://physicshub.github.io/simulations/BallAcceleration): A ball is pulled toward your cursor: the further away the cursor, the longer the ball accelerates and the faster it is moving when it arrives. This is a hands-on picture of Newton's second law — a constant force produces a constant acceleration, and acceleration is the rate at which velocity changes, not the velocity itself.
- [Ball Gravity — concept](https://physicshub.github.io/simulations/BallGravity): A ball falls under gravity, bounces off the floor and loses a little energy on every impact. Adjust the gravitational acceleration, the ball's mass, the wind, the friction and the restitution to see how each one changes the motion. It connects free fall, Newton's laws and the coefficient of restitution in a single scene.
- [Spring Mass System Simulator — concept](https://physicshub.github.io/simulations/SpringConnection): A mass hangs from a spring and oscillates. The spring pulls back with a force proportional to how far it is stretched (Hooke's law), and that linear restoring force is exactly what produces simple harmonic motion. Change the mass, the spring constant or the damping and watch the period and decay respond.
- [Simple Pendulum Simulation — concept](https://physicshub.github.io/simulations/SimplePendulum): A bob on a string swings back and forth about its lowest point. For small swings the motion is simple harmonic and the period depends only on the string length and gravity — not on the mass, and only weakly on the amplitude, the property Galileo called isochronism. Increase the initial angle to see the small-angle approximation break down.
- [Projectile & Parabolic Motion — concept](https://physicshub.github.io/simulations/ParabolicMotion): Launch a projectile and watch its path. With gravity as the only force, horizontal and vertical motion are independent: the horizontal velocity stays constant while the vertical velocity changes at g, and the combination traces a parabola. Add drag or wind to see how the ideal trajectory deforms.
- [Inclined Plane — concept](https://physicshub.github.io/simulations/InclinedPlane): A block rests on a ramp. Gravity still points straight down, but the surface splits it into a component along the slope that drives sliding and a component into the slope that sets the normal force — and therefore the maximum friction. Change the angle to find the point where static friction can no longer hold the block.
- [Circular Motion — concept](https://physicshub.github.io/simulations/CircularMotion): An object moves around a circle at constant speed. Its velocity is constant in size but always changing direction, so it is always accelerating — toward the centre. That centripetal acceleration needs a real inward force; remove it and the object flies off along the tangent, not outward.
- [Three Body Problem — concept](https://physicshub.github.io/simulations/ThreeBody): Three masses attract each other through Newtonian gravity. Unlike the two-body problem, this system has no general closed-form solution: the motion is chaotic, and tiny changes in the starting positions or velocities lead to completely different futures. The simulation integrates the equations numerically with small sub-steps to stay accurate through close approaches.
- [Horizontal Spring — concept](https://physicshub.github.io/simulations/HorizontalSpring): A mass on a frictionless horizontal surface is tied to a wall by a spring. With gravity removed from the direction of motion, the spring force alone governs the oscillation — the cleanest possible demonstration of Hooke's law and simple harmonic motion. Drag the mass to set an amplitude and release it.
- [Double Pendulum — concept](https://physicshub.github.io/simulations/DoublePendulum): A second pendulum hangs from the end of the first. The system has just two degrees of freedom, yet its motion is famously chaotic: run it twice from almost-identical starts and the arms soon diverge. This simulation integrates the exact Lagrangian equations for the two angles rather than using a constraint solver, so the chaos is physical and not a numerical artefact.
- [1D Collision simulation — concept](https://physicshub.github.io/simulations/CollisionSimulation): Two bodies move along a line and collide. Momentum is always conserved in the collision; kinetic energy is conserved only if the collision is perfectly elastic. Set the masses, the initial velocities and the restitution, then read off the momentum and kinetic energy before and after to see which quantities survive.
- [Pi From Collisions — concept](https://physicshub.github.io/simulations/PiCollisions): Two blocks slide on a frictionless floor with a wall on one side. When the left block is heavier than the right by a factor of 100ᴺ, the total number of collisions equals the first N+1 digits of π. It is a startling link between elastic-collision conservation laws and geometry — the collisions trace out an angle on a circle in velocity space.
- [Trigonometric Circle — concept](https://physicshub.github.io/simulations/TrigonometricCircle): Drag the angle θ around the unit circle and watch every trigonometric function appear as a real length: sine and cosine as the coordinates of the point, tangent on the vertical line x = 1, and their reciprocals as related segments. A generalized sine wave A·sin(ωθ + φ) plots alongside, tying the circle to the graph.
- [Ray Tracing — concept](https://physicshub.github.io/simulations/RayTracing): A ray tracer makes an image by following rays of light backwards: from the camera, through each pixel, into the scene. At the first surface a ray hits, the laws of geometric optics decide the colour — how squarely the surface faces the light, how far away the light is, whether something casts a shadow, and what the surface reflects. Click any pixel to follow its ray and see every equation with its live numbers.
- [Kirchhoff's Circuit Laws — concept](https://physicshub.github.io/simulations/KirchhoffCircuit): Four DC circuits — a single loop, three resistors in series, three in parallel, and a two-loop network driven by two cells — solved live by Kirchhoff's laws. Charge carriers drift along each wire at a speed set by that branch's current, resistors glow with the power they dissipate, and the readout walks each loop term by term so you can watch the voltages add up to exactly zero.

## Formulary
- [Uniform motion](https://physicshub.github.io/formulas#uniform-motion): An object moving at constant velocity covers equal distances in equal times, so its position grows linearly with time. `x = x_0 + v\,t`
- [Velocity under constant acceleration](https://physicshub.github.io/formulas#velocity-constant-acceleration): With a constant acceleration, velocity changes by the same amount every second: acceleration is the rate of change of velocity. `v = v_0 + a\,t`
- [Displacement under constant acceleration](https://physicshub.github.io/formulas#displacement-constant-acceleration): 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 = x_0 + v_0 t + \tfrac{1}{2} a t^{2}`
- [Speed after a free fall](https://physicshub.github.io/formulas#free-fall-speed): An object dropped from rest reaches this speed after falling a height h. Its mass does not appear: without air, everything falls alike. `v = \sqrt{2 g h}`
- [Projectile trajectory](https://physicshub.github.io/formulas#projectile-trajectory): Eliminating time from the two equations of motion leaves height as a function of horizontal distance: a parabola. `y = x\tan\theta - \dfrac{g\,x^{2}}{2\,v_0^{2}\cos^{2}\theta}`
- [Range of a projectile](https://physicshub.github.io/formulas#projectile-range): How far a projectile lands on level ground. For a given speed it peaks at 45°, where \sin 2\theta = 1. `R = \dfrac{v_0^{2}\sin 2\theta}{g}`
- [Maximum height of a projectile](https://physicshub.github.io/formulas#projectile-max-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 = \dfrac{v_0^{2}\sin^{2}\theta}{2g}`
- [Centripetal acceleration](https://physicshub.github.io/formulas#centripetal-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 = \dfrac{v^{2}}{r} = \omega^{2} r`
- [Period of circular motion](https://physicshub.github.io/formulas#circular-period): The time for one full lap: the circumference divided by the speed. `T = \dfrac{2\pi r}{v} = \dfrac{2\pi}{\omega}`
- [Newton's second law](https://physicshub.github.io/formulas#newtons-second-law): The net force on a body equals its mass times its acceleration. A force does not keep things moving — it changes how they move. `\vec{F} = m\,\vec{a}`
- [Weight](https://physicshub.github.io/formulas#weight): Weight is the gravitational force on a mass. Mass is how much stuff there is; weight is how hard gravity pulls on it. `W = m\,g`
- [Hooke's law](https://physicshub.github.io/formulas#hookes-law): A spring pulls back with a force proportional to how far it is stretched or compressed, always towards its rest length. `F = -k\,x`
- [Centripetal force](https://physicshub.github.io/formulas#centripetal-force): The net force towards the centre that a body needs to follow a circle of radius r at speed v. `F_c = \dfrac{m v^{2}}{r}`
- [Weight along a slope](https://physicshub.github.io/formulas#incline-parallel-force): On a slope, part of the weight pulls the body down along the surface. The steeper the slope, the bigger that part. `F_\parallel = m g \sin\theta`
- [Normal force on a slope](https://physicshub.github.io/formulas#incline-normal-force): The surface pushes back on the body with a force that balances the part of the weight pressing into it. `N = m g \cos\theta`
- [Maximum static friction](https://physicshub.github.io/formulas#static-friction-max): 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} = \mu_s N`
- [Angle of repose](https://physicshub.github.io/formulas#angle-of-repose): The steepest slope a body can rest on without sliding. Surprisingly, it does not depend on the mass. `\tan\theta_{\text{repose}} = \mu_s`
- [Drag equation](https://physicshub.github.io/formulas#drag-equation): Air resistance grows with the square of speed: go twice as fast and the drag is four times bigger. `F_{\text{drag}} = \tfrac{1}{2}\,C_d\,\rho\,A\,v^2`
- [Terminal velocity](https://physicshub.github.io/formulas#terminal-velocity): A falling body stops speeding up when drag grows to match its weight. That top speed is the terminal velocity. `v_t = \sqrt{\dfrac{2\,m\,g}{C_d\,\rho\,A}}`
- [Lift equation](https://physicshub.github.io/formulas#lift-equation): 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 = \tfrac{1}{2}\,\rho\, v^{2}\, S\, C_L`
- [Lagrangian](https://physicshub.github.io/formulas#lagrangian): 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} = T - V`
- [Euler–Lagrange equation](https://physicshub.github.io/formulas#euler-lagrange): One equation of motion per coordinate q_i, straight from the Lagrangian. It is how the double pendulum's equations are derived. `\dfrac{d}{dt}\dfrac{\partial \mathcal{L}}{\partial \dot{q}_i} - \dfrac{\partial \mathcal{L}}{\partial q_i} = 0`
- [Kinetic energy](https://physicshub.github.io/formulas#kinetic-energy): The energy a body has because it moves. It grows with the square of speed: twice as fast carries four times the energy. `K = \tfrac{1}{2} m v^{2}`
- [Gravitational potential energy](https://physicshub.github.io/formulas#gravitational-potential-energy): The energy stored by lifting a mass against gravity. Only changes in h matter, so you choose where h = 0 is. `U = m g h`
- [Elastic potential energy](https://physicshub.github.io/formulas#elastic-potential-energy): 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 = \tfrac{1}{2} k x^{2}`
- [Pendulum speed at the bottom](https://physicshub.github.io/formulas#pendulum-speed-bottom): 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 = \sqrt{2 g L\,(1 - \cos\theta_0)}`
- [Conservation of momentum](https://physicshub.github.io/formulas#momentum-conservation): In any collision the total momentum before equals the total after — elastic, inelastic or sticky. `m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2`
- [Elastic collision in one dimension](https://physicshub.github.io/formulas#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. `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`
- [Coefficient of restitution](https://physicshub.github.io/formulas#coefficient-of-restitution): How much of the approach speed survives a collision as separation speed: 1 for a perfectly elastic bounce, 0 when the bodies stick. `e = \dfrac{v_2 - v_1}{u_1 - u_2}`
- [π from colliding blocks](https://physicshub.github.io/formulas#pi-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}} = \lfloor \pi \cdot 10^{N} \rfloor`
- [Angular frequency of a mass on a spring](https://physicshub.github.io/formulas#spring-angular-frequency): How fast a mass on a spring oscillates: a stiffer spring speeds it up, a heavier mass slows it down. `\omega = \sqrt{\dfrac{k}{m}}`
- [Period of a mass on a spring](https://physicshub.github.io/formulas#spring-period): The time for one full oscillation. The amplitude is missing on purpose: small or large swings take the same time. `T = 2\pi\sqrt{\dfrac{m}{k}}`
- [Simple harmonic motion](https://physicshub.github.io/formulas#shm-position): A linear restoring force makes a body oscillate as a cosine: amplitude A, angular frequency \omega, phase \varphi. `x(t) = A\cos(\omega t + \varphi)`
- [Period of a simple pendulum](https://physicshub.github.io/formulas#pendulum-period): 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 = 2\pi\sqrt{\dfrac{L}{g}}`
- [Pendulum equation of motion](https://physicshub.github.io/formulas#pendulum-equation-of-motion): The exact equation for a swinging pendulum. The \sin\theta makes it nonlinear; replacing it with \theta for small angles gives simple harmonic motion. `\ddot{\theta} + \dfrac{g}{L}\sin\theta = 0`
- [Newton's law of universal gravitation](https://physicshub.github.io/formulas#newtons-law-of-gravitation): Every two masses attract each other along the line between them, with a force that falls off as the square of the distance. `F = G\,\dfrac{m_1 m_2}{r^{2}}`
- [Net gravitational force](https://physicshub.github.io/formulas#gravitational-superposition): 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 = \sum_{j \ne i} \vec{F}_{ij}`
- [Circular orbital speed](https://physicshub.github.io/formulas#orbital-speed): The speed that keeps a satellite on a circular orbit: gravity supplies exactly the centripetal force it needs. `v = \sqrt{\dfrac{G M}{r}}`
- [Gravity at altitude](https://physicshub.github.io/formulas#gravity-at-altitude): Gravity weakens slowly with height. At the International Space Station it is still about 90% of its value on the ground. `g(h) = g_0\left(\dfrac{R}{R+h}\right)^{2}`
- [Fourier's law of heat conduction](https://physicshub.github.io/formulas#heat-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 = -k\,A\,\dfrac{\Delta T}{d}`
- [Coulomb's law](https://physicshub.github.io/formulas#coulombs-law): 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 = \dfrac{1}{4\pi\varepsilon_0}\,\dfrac{q_1 q_2}{r^{2}}`
- [Resistance of a wire](https://physicshub.github.io/formulas#wire-resistance): A longer wire resists current more, a thicker one less. The material enters through its resistivity \rho. `R = \dfrac{\rho L}{A}`
- [Ohm's law](https://physicshub.github.io/formulas#ohms-law): The voltage across a resistor is proportional to the current through it; the constant of proportionality is its resistance R. `V = IR`
- [Kirchhoff's current law](https://physicshub.github.io/formulas#kirchhoffs-current-law): The current flowing into any junction equals the current flowing out of it — charge is neither created nor stored at a node. `\sum_{\text{into node}} I = \sum_{\text{out of node}} I`
- [Kirchhoff's voltage law](https://physicshub.github.io/formulas#kirchhoffs-voltage-law): Going once round any closed loop, the rises in potential (across cells) and the drops (across resistances) add up to zero — energy conservation for a unit charge. `\sum_{\text{closed loop}} \Delta V = 0`
- [Terminal voltage of a real cell](https://physicshub.github.io/formulas#emf-terminal-voltage): A real cell loses part of its EMF \mathcal{E} across its own internal resistance r, so the voltage at its terminals drops as it delivers more current. `V = \mathcal{E} - Ir`
- [Power dissipated in a resistor](https://physicshub.github.io/formulas#joule-heating): A current through a resistance turns electrical energy into heat at a rate that grows with the square of the current. `P = I^{2} R`
- [Snell's law](https://physicshub.github.io/formulas#snells-law): Light bends when it crosses into a medium where it travels at a different speed. The refractive indices decide by how much. `n_1 \sin\theta_1 = n_2 \sin\theta_2`
- [Law of reflection](https://physicshub.github.io/formulas#law-of-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}_r = \hat{D} - 2(\hat{D}\cdot\hat{N})\,\hat{N}`
- [Inverse-square law of light](https://physicshub.github.io/formulas#inverse-square-law): 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 = \dfrac{I_0}{d^2}`
- [Lambert's cosine law](https://physicshub.github.io/formulas#lamberts-cosine-law): 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 = k_d\,E\,\max(0,\ \hat{N}\cdot\hat{L})`
- [Parametric ray](https://physicshub.github.io/formulas#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) = \vec{O} + t\,\hat{D}`
- [Ray–sphere intersection](https://physicshub.github.io/formulas#ray-sphere-intersection): 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 = b^2 - 4ac,\quad b = 2\,\hat{D}\cdot(\vec{O}-\vec{C}),\quad c = |\vec{O}-\vec{C}|^2 - r^2`
- [de Broglie wavelength](https://physicshub.github.io/formulas#de-broglie-wavelength): 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 = \dfrac{h}{m v}`
- [Magnitude of a vector](https://physicshub.github.io/formulas#vector-magnitude): The length of a vector from its components — Pythagoras' theorem on the right triangle they form. `|\vec{a}| = \sqrt{a_x^{2} + a_y^{2}}`
- [Sum of two vectors](https://physicshub.github.io/formulas#vector-sum): To add vectors, add their components. Geometrically: place the tail of \vec{b} on the tip of \vec{a}. `\vec{a} + \vec{b} = (a_x + b_x,\; a_y + b_y)`
- [Dot product](https://physicshub.github.io/formulas#dot-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} = a_x b_x + a_y b_y = |\vec{a}||\vec{b}|\cos\theta`
- [Pythagorean identity](https://physicshub.github.io/formulas#pythagorean-identity): 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. `\sin^{2}\theta + \cos^{2}\theta = 1`
- [Tangent as a ratio](https://physicshub.github.io/formulas#tangent-ratio): 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. `\tan\theta = \dfrac{\sin\theta}{\cos\theta}`
- [Sine wave](https://physicshub.github.io/formulas#sine-wave): The shape of every oscillation and wave: A stretches it vertically, \omega squeezes it horizontally, \varphi slides it sideways. `y = A\sin(\omega\theta + \varphi)`

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