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.
For the first 380,000 years after the Big Bang, the universe was a hot, opaque plasma — photons and baryons (protons + electrons) were tightly coupled. As the universe expanded and cooled to ~3000 K, electrons combined with protons (recombination). The universe became transparent, and the photons last scattered. We observe this "last scattering surface" today as microwave radiation at T₀ = 2.725 K — redshifted by a factor of 1100.
The CMB temperature field is expanded in spherical harmonics Y_ℓm(θ,φ). The multipole ℓ describes the angular scale:
• ℓ=1: dipole — one hemisphere hot, one cold (our motion through space)
• ℓ=2: quadrupole — four patches, 90° scale
• ℓ=10: ~18° patches — galaxy supercluster scales
• ℓ=220: ~1° patches — the "sound horizon", largest possible acoustic feature
• ℓ=1000: ~0.2° patches — galaxy cluster scales, entering Silk damping
C_ℓ = ⟨|a_ℓm|²⟩ is the average power at multipole ℓ. The factor ℓ(ℓ+1)/2π gives the "flat" Sachs-Wolfe plateau at low ℓ a constant value, making the acoustic peaks clearly visible above it. D_ℓ is plotted in μK².
Before recombination, the baryon-photon fluid supported sound waves — pressure oscillations driven by gravity (dark matter clumps pulling baryons in) and radiation pressure (photons pushing back out). Modes that completed exactly 1, 2, 3... half-oscillations before recombination show up as peaks in D_ℓ:
• Peak 1 (ℓ≈220): compressed once — the fundamental "note" of the universe
• Peak 2 (ℓ≈540): compressed and rarefied — the octave. Weaker because baryons resist rarefaction
• Peak 3 (ℓ≈800): compressed again — ratio of peak 1 to peak 2 measures baryon density
• Peak 1 position (ℓ≈220): universe is spatially flat — Ω_total = 1.000 ± 0.002
• Peak 1/2 height ratio: baryon density Ω_b h² ≈ 0.0224 — only 4.9% of universe is ordinary matter
• Peak heights overall: dark matter Ω_c h² ≈ 0.120 — 26.4% is dark matter
• Damping tail slope: thickness of last-scattering surface, photon diffusion length
The remaining ~68.7% is dark energy — causing the universe to accelerate its expansion.
At small angular scales (ℓ ≳ 1000), photons random-walk between scattering events during recombination, diffusing across the structures they pass through. This "Silk damping" (or diffusion damping) exponentially suppresses the acoustic peaks. The damping scale encodes the duration of recombination and the photon mean free path — measuring the effective "blur" of the last-scattering surface.
The large-angle plateau (ℓ < 30) is the Sachs-Wolfe plateau from inflationary perturbations — nearly scale-invariant quantum fluctuations stretched to cosmic scales during inflation. The spectral index n_s ≈ 0.965 (slightly less than 1, "red-tilted") matches inflationary predictions. Measuring n_s ≠ 1 is one of the strongest observational confirmations of inflation.
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.
e⁺e⁻ → γ → μ⁺μ⁻: An electron and positron annihilate into a virtual photon, which materialises as a muon pair. This is the process used in colliders like LEP.
Compton Scattering: A photon bounces off an electron, exchanging energy and momentum. The wavelength shift is given by Δλ = (h/mₑc)(1−cosθ).
Beta Decay: A down quark in the neutron emits a W⁻ boson, flipping to an up quark (neutron → proton). The W then decays into an electron and antineutrino.
Quark Confinement: As two quarks separate, the colour field narrows into a flux tube. When its energy equals 2mq, it snaps and creates a new quark-antiquark pair — quarks can never be isolated.
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.
Introduced by biologist Robert May in 1976 as a model for population dynamics: x is the population as a fraction of its maximum, r is the growth rate. The (1−x) factor represents resource competition — the fuller the environment, the slower the growth. Despite its simplicity (one parameter, one variable), it encodes the full journey from order to chaos.
A fixed point satisfies x* = rx*(1−x*). The non-trivial solution x*=1−1/r is stable when |f'(x*)|=|r(1−2x*)|<1, which gives r<3. At r=3 the derivative equals −1 exactly, and the fixed point destabilises: the system bifurcates into a period-2 cycle.
Each bifurcation doubles the period: 1→2→4→8→16→... This cascade of doublings accumulates at r∞≈3.5699. Beyond this point the system is chaotic — its long-term behaviour is sensitively dependent on initial conditions. Mathematically, the Lyapunov exponent λ changes sign from negative (order) to positive (chaos) at r∞.
In the chaotic regime, two trajectories starting a distance ε apart diverge exponentially. After ~20 iterations with ε=10⁻¹⁰, they are completely uncorrelated. Try changing x₀ by 0.01 in the cobweb and observe: same map, wildly different trajectory after 50 steps. This is deterministic chaos — not random, but unpredictable in practice.
The ratio of successive bifurcation interval widths converges to δ. Feigenbaum (1978) discovered this constant is universal — identical for any smooth unimodal map (quadratic, sinusoidal, etc.). It has been observed experimentally in turbulence, dripping faucets, and electronic circuits. Like π or e, δ is a fundamental constant of mathematics, not of any particular physical system.
The chaotic region is not uniformly chaotic — periodic windows appear (period-3 at r≈3.82, period-5 at r≈3.74, etc.). By the Sharkovskii theorem, period 3 implies periods of every order exist. The period-3 window is visible as a bright gap in the bifurcation diagram — order emerging spontaneously from chaos before dissolving again.
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.
From LEO (~400 km), the SLS upper stage fires to boost the Orion capsule onto a trajectory that reaches the Moon's distance (~384,400 km). The minimum-energy path is a Hohmann-like ellipse; boosting slightly above this gives a hyperbolic encounter needed for a free-return.
The spacecraft is governed by gravity from both Earth and Moon simultaneously. There is no closed-form solution — the trajectory must be numerically integrated.
A free-return trajectory is intrinsically safe: if the engine fails, the Moon's gravity automatically sends the crew home. Apollo 13 used this property after its oxygen tank exploded. Artemis 2 will fly a "hybrid free-return" — a slight variant that allows a closer lunar approach than the pure free-return geometry.
Two burns are needed: one at Earth to enter the transfer ellipse, one at Mars to enter orbit. The total Δv is the minimum possible for an impulsive transfer between coplanar circular orbits.
Mars must be ahead of Earth at launch by the angle Mars covers during the ~259-day transit. The synodic period is how long until Earth and Mars return to the same relative alignment — about 780 days (26 months).
Earth–Mars distance varies from 54 million km (closest approach) to 401 million km. Every Mars mission (Curiosity, Perseverance, MAVEN, InSight) launched in narrow windows. A crewed mission would require ~6–9 months one-way, meaning astronauts would spend ~3 years total — requiring radiation shielding and closed-loop life support.
Every pure qubit state corresponds to a unique point on the Bloch sphere surface. The angles θ (colatitude) and φ (longitude) parameterise all possible states. The global phase is unobservable — so the sphere surface, not a 4D complex space, captures everything measurable.
Every single-qubit gate is a rotation of the Bloch sphere. X rotates π around the X-axis (|0⟩↔|1⟩). Z rotates π around Z (phase flip). H exchanges the X and Z axes, creating superposition. S and T are fractional Z rotations used in fault-tolerant quantum error correction.
Measuring in the Z-basis only reveals the Z-component of the Bloch vector. The X and Y components (encoded in phase φ) are invisible to a single Z-measurement — they require interference experiments. This is why phase gates like Z and T are "classically invisible" but crucial for quantum algorithms (Deutsch-Jozsa, Grover, Shor).