Projectile motion decomposes velocity into two independent components. The horizontal component is constant (no force); the vertical component accelerates under gravity. They combine to produce a parabolic path.
sin(2θ) is maximised when 2θ = 90°, i.e. θ = 45°. With air drag, the optimal angle shifts lower because higher angles spend more time at altitude where drag accumulates more.
Artillery ballistics, sports science (football, javelin), orbital mechanics (re-entry trajectories), and video game physics engines all use this model as a foundation.
Set phase offset to 180° — the pattern inverts! This is how noise-cancelling headphones work.
Each source emits circular waves. At any point in space, the total displacement is the algebraic sum of both waves (superposition principle). Where they arrive in phase, amplitudes add; where out of phase, they cancel.
Wi-Fi antennas use phased arrays to steer signals. Active noise cancellation generates the anti-phase wave. Radio telescopes use interferometry to image black holes. Young's double-slit experiment first proved the wave nature of light (1801).
This is why your phone can compress audio, an MRI scanner can see your brain, and JPEG images work.
Any periodic function f(x) with period T can be written as an infinite sum of sines and cosines. The Fourier series converges to f(x) everywhere the function is continuous.
Only odd harmonics appear; amplitudes fall as 1/n. This is why a square wave sounds harsher than a sine — it contains all those higher harmonics.
MP3/AAC audio compression (drop imperceptible harmonics). JPEG image encoding (discrete cosine transform, a close relative). MRI reconstruction. Signal processing in every radio, phone, and scientific instrument.
Notice how they never cross — each point in space has only one field direction.
This geometry underpins how capacitors, transistors, and electric motors work.
The force between charges falls as the square of distance — the same inverse-square law as gravity, but ~10³⁶ times stronger.
The field at a point is the force per unit positive test charge. Field lines are tangent to this field vector at every point — which is why they never cross.
The total electric flux through any closed surface equals the enclosed charge divided by ε₀. This is one of Maxwell's four equations that describe all of classical electromagnetism.
Quantum mechanics describes every particle as a probability amplitude — a complex wave function ψ. The probability of detecting a particle at position x is |ψ(x)|². Particles don't have definite positions until measured.
d = slit separation, w = slit width, λ = de Broglie wavelength. The cos² term gives fringes; the sinc² envelope narrows them for wider slits.
When a which-slit detector is turned on, the particle must interact with it — this entangles the particle with the detector and collapses its wave function. The interference requires the particle to take both paths simultaneously. Knowing the path prevents that.
Every massive particle has a wavelength. For an electron, λ ≈ 1 Å — the size of an atom. This is why electron microscopes can see atoms but light microscopes can't.
At v=0.87c: γ≈2, time runs at half speed.
At v=0.99c: γ≈7, time runs 7× slower.
GPS satellites must correct for this or your maps drift by 11km/day.
γ equals 1 at rest and grows toward infinity as v → c. Time intervals measured by a moving clock are dilated by γ relative to a stationary observer.
A clock moving at 0.87c ticks at half speed (γ ≈ 2). This is not an illusion — muons created at the top of the atmosphere by cosmic rays survive long enough to reach the Earth's surface only because of time dilation.
Objects moving at relativistic speeds appear shorter along their direction of motion. The faster they go, the more contracted.
GPS satellites orbit at ~14,000 km/h (special relativity slows their clocks by 7 μs/day) and at high altitude (general relativity speeds them up by 45 μs/day). Net: +38 μs/day. Without correction, GPS would drift ~11 km per day.
σ (sigma) = Prandtl number, ρ (rho) = Rayleigh number, β (beta) = geometric factor. The classic parameters σ=10, ρ=28, β=8/3 produce the famous butterfly-shaped attractor.
The equations are completely deterministic — no randomness at all. Yet tiny differences in initial conditions grow exponentially (Lyapunov exponent λ ≈ 0.9 for classic parameters). This is why weather forecasting beyond ~10 days is a fundamental physical limit, not an engineering problem.
The Lorenz attractor has fractal dimension ≈ 2.06. Trajectories are confined to a bounded region (the attractor) but never repeat — they trace out an infinitely complex fractal structure. This is characteristic of all chaotic systems.
The equations are nonlinear and coupled — θ₁ and θ₂ appear in each other's equations via trigonometric terms. This coupling is what produces the chaotic behaviour. The simulation uses RK4 integration for accuracy.
For most initial conditions, the Lyapunov exponent of the double pendulum is positive — meaning nearby trajectories diverge exponentially. The system is unpredictable beyond a few seconds of simulation time regardless of computational precision.
Slicing the 4-dimensional phase space (θ₁, θ₂, ω₁, ω₂) reveals the fractal structure of chaos. For small angles, the pendulum is regular; past a critical energy, it transitions suddenly to chaos — a phase transition visible in the Poincaré map.
Two coupled pendulums have exactly two normal modes — patterns of motion where every part oscillates at the same frequency. In the symmetric mode both pendulums swing together (spring unstretched); in the antisymmetric mode they swing in opposite directions (spring maximally stretched).
Normal modes are the foundation of molecular spectroscopy (CO₂ has 4 normal modes). Phonons in crystalline solids are quantised normal modes. In quantum field theory, every particle is an excitation of a normal mode of a quantum field.
Einstein showed that the mean square displacement of a Brownian particle grows linearly with time — a result that allowed Jean Perrin to measure Avogadro's number experimentally in 1908 and definitively prove that atoms exist.
d = dimensions, D = diffusion coefficient, k_B = Boltzmann constant, T = temperature, η = viscosity, r = particle radius. Higher temperature → faster diffusion.
The probability distribution of particle speeds in a gas. Its peak gives the most probable speed; its mean gives the average kinetic energy ½mv² = 3/2·k_BT. This links temperature directly to molecular motion.
Brownian motion models stock price fluctuations (Black-Scholes equation). It explains diffusion in biological cells. It's used to simulate protein folding. And it underpins the entire field of stochastic calculus.
The quantum harmonic oscillator describes any system with a restoring force proportional to displacement — from vibrating atoms to the Higgs field. Unlike the classical oscillator, energy is quantised: only discrete levels E_n = ℏω(n + ½) are allowed.
H₀=1, H₁=2x, H₂=4x²−2, H₃=8x³−12x. Each higher level has one extra node (zero crossing). The probability density |ψₙ|² has n+1 peaks — a quantum particle is most likely found where a classical particle moves slowest.
A coherent state is a superposition of eigenstates that mimics classical motion — a Gaussian wave packet that oscillates back and forth without spreading. This is the quantum state produced by a laser and the closest thing to a classical oscillator in quantum mechanics.
Molecular vibrations, phonons in crystals, laser photon modes, the Higgs mechanism, and quantum field theory — every quantum field is fundamentally a collection of harmonic oscillators. The ground state of each field gives the vacuum zero-point energy.
Poincaré proved in 1890 that there is no general closed-form solution to the three-body problem. The system is chaotic — small changes in initial conditions produce wildly different long-term trajectories. This was one of the first discoveries of deterministic chaos.
In 1993, Cris Moore discovered that three equal masses can follow a stable figure-8 path under gravity. It requires exquisitely precise initial conditions and is unstable under perturbations — but it exists. Hundreds of choreographic solutions have since been found.
Total energy E = KE + PE and total momentum p = Σmᵢvᵢ are conserved. Angular momentum L = Σmᵢ(rᵢ × vᵢ) is also conserved. These are monitored in real time — watch for numerical drift, which exposes the limits of any integration scheme.
Galaxy formation, spacecraft trajectory design (gravitational slingshots), binary star evolution, and planet formation all require N-body simulation. Modern simulations handle billions of particles using tree codes and GPU parallelisation.
Each spin sᵢ interacts with its nearest neighbours. J > 0 makes parallel alignment energetically favourable (ferromagnet). The external field h biases spins to align with it.
Pick a random spin, compute the energy change ΔE if flipped. Accept the flip with probability min(1, e^(-ΔE/kT)). At low T, only energy-lowering flips are accepted. At high T, random flips are accepted freely.
Lars Onsager solved the 2D Ising model exactly in 1944 — one of the great achievements of theoretical physics. Near Tc, the correlation length diverges and the system shows scale-free behaviour (fractal domain patterns).
Above Tc: average magnetisation ⟨M⟩ = 0 (disordered). Below Tc: ⟨M⟩ ≠ 0 even with no external field — the system spontaneously picks a direction. This is the simplest example of spontaneous symmetry breaking, the same mechanism behind the Higgs field giving mass to particles.
Near Tc, M ∝ (Tc-T)^β with β = 1/8 in 2D. The same exponents appear in completely different systems (liquid-gas, superconductors, superfluids) — universality classes mean that wildly different physical systems have identical critical behaviour.
A particle moving through a field-free region still picks up a phase if a solenoid encloses magnetic flux Φ. The particle never enters the field — yet the interference pattern shifts.
In classical physics, only the fields E and B matter. In quantum mechanics, the vector potential A is physical — it affects particles even in regions where B=0. This demonstrates gauge fields have direct physical reality.
When Φ = Φ₀, the phase shift is exactly 2π — one full fringe shift. This flux quantization appears in superconductors and is the basis of SQUID magnetometers.
The geometry of spacetime around a non-rotating mass M. The Schwarzschild radius r_s = 2GM/c² defines the event horizon — a one-way membrane where escape velocity equals c.
Null geodesics (light paths) satisfy this nonlinear ODE. Without the r_s term it's simple Kepler — the GR correction causes photon deflection. Light grazing the Sun bends by 1.75 arcseconds.
Photons with impact parameter b < b_crit spiral into the black hole. At b = b_crit they orbit the photon sphere at r = 1.5r_s. This creates the "black hole shadow" — a dark disk surrounded by an Einstein ring.
Before recombination, photons and baryons formed a tightly-coupled fluid. Pressure waves (sound) oscillated in this fluid. At recombination (z≈1100), the pattern was frozen into the CMB. Each acoustic peak corresponds to a mode that was at a maximum or minimum at that moment.
The first peak at ℓ≈220 tells us the universe is flat (Ω_total≈1). Higher baryon density Ω_b boosts odd peaks (compressions) relative to even peaks (rarefactions). Dark matter Ω_c suppresses all peaks by increasing the gravitational potential at horizon entry.
At high ℓ (small scales), photon diffusion smooths out fluctuations — "Silk damping". This exponential suppression tells us the thickness of the last-scattering surface.
Four terms: gauge kinetic energy (photons, W, Z, gluons), fermion kinetic energy + interactions, Yukawa couplings (masses via Higgs), and the Higgs potential that drives symmetry breaking.
Matter comes in three identical generations of increasing mass: (u,d,e,νe), (c,s,μ,νμ), (t,b,τ,ντ). Why three? We don't know. The top quark at 173 GeV/c² is heavier than a gold atom.
The strong coupling α_s decreases at high energy (asymptotic freedom) — quarks inside a proton barely interact. But at low energy, α_s→1, and quarks are permanently confined (confinement). This was Nobel Prize 2004.
The first equation is Newton's second law for a fluid parcel. The second enforces incompressibility — what flows in must flow out. These equations are unsolved analytically in the general case; the Clay Millennium Prize offers $1M for a proof of smooth solutions.
Re < 1: creeping flow (honey). Re ~ 100: laminar with wake. Re > 1000: turbulent. The transition to chaos in fluids is one of the great unsolved problems of physics.
For small angles sin θ ≈ θ, giving simple harmonic motion. For large angles the full nonlinear equation must be integrated numerically.
The correction terms grow rapidly near 180°, where the pendulum can balance upright indefinitely — an unstable equilibrium.
Underdamped (b² < 4km): oscillates with decaying amplitude. Critically damped (b² = 4km): returns fastest without oscillating. Overdamped (b² > 4km): creeps to equilibrium slowly.
The field of a magnetic dipole falls as 1/r³ — faster than a monopole (1/r²). This means magnetic effects become negligible quickly with distance.
Unlike electric fields, magnetic field lines always close on themselves. There is no magnetic equivalent of a lone charge. This is one of Maxwell's equations. Magnetic monopoles are predicted by some GUT theories but have never been observed.
f is focal length (positive for converging, negative for diverging). d_o is object distance, d_i is image distance. Negative d_i means a virtual image on the same side as the object.
Light bends toward the normal when entering a denser medium. Glass (n≈1.5) bends light enough to focus it. The critical angle for total internal reflection is θ_c = arcsin(n₂/n₁).
The intensity pattern is the square of the Fourier transform of the aperture. The sinc² envelope sets the locations of zeros at a·sinθ = mλ for integer m ≠ 0.
Two point sources can just be resolved when the central maximum of one falls on the first minimum of the other. The Hubble Space Telescope's 2.4m mirror gives θ_min ≈ 0.05 arcseconds at 500nm.
Pressure × Volume = amount × gas constant × temperature. This is an approximation that treats molecules as point particles with no interactions — excellent for dilute gases.
Most probable speed: v_p = √(2k_BT/m). Root-mean-square: v_rms = √(3k_BT/m). Mean: v̄ = √(8k_BT/πm). The tail of the distribution determines evaporation rates and chemical reaction rates.
c is the wave speed. For a drum membrane, c = √(T/σ) where T is tension and σ is surface density. The simulation uses an explicit finite-difference scheme with stability condition c·dt/dx < 1/√2.
Unlike a 1D string, the modes are not harmonically related — which is why drums sound less "musical" than strings. Chladni patterns visualize these modes with sand on vibrating plates.
Originally a model for population dynamics (r = growth rate, x = fraction of max population). Despite its simplicity, it displays the full route from order to chaos.
The ratio of successive bifurcation widths converges to δ. Remarkably, this constant is the same for any smooth unimodal map — it is a universal constant of chaos theory, analogous to π or e.
1st: Orbits are ellipses with the star at one focus. 2nd: Equal areas are swept in equal times (conservation of angular momentum). 3rd: T² ∝ a³.
Gives orbital speed at any point. At perihelion r is smallest so v is largest. At aphelion they swap. If v exceeds √(2GM/r) the object escapes — escape velocity.
The wave function ψ(x,t) contains all information about the particle. |ψ|² is the probability density. The simulation uses a split-operator method: propagate kinetic and potential parts alternately in Fourier space.
Tunneling probability falls exponentially with barrier width d and the square root of the energy deficit. This underpins tunnel diodes, STM microscopes, nuclear fusion in the Sun, and radioactive α-decay.
The magnetic force is always perpendicular to velocity — it does no work, only curves the trajectory. This is why magnetic fields can steer but not accelerate charged particles (particle accelerators need electric fields for the acceleration).
In crossed E and B fields the particle drifts perpendicular to both, at speed E/B — independent of the particle's charge or mass. This E×B drift is used in plasma confinement and Hall-effect thrusters.
Each mode has n half-wavelengths fitting the string length L. The modes are harmonically related (f_n = n·f₀) which is why strings produce musical tones. Drums have inharmonic modes, which is why they sound "thumpy".
T is tension, μ is linear mass density. Guitarists change pitch by fretting (changing L) or tuning (changing T). Thicker strings (larger μ) vibrate lower — hence bass strings are wound with metal wire.