All formulas (75)

Mechanics

8
Mechanics★★★★★

Newton's Second Law

F=maF = ma

Force equals mass times acceleration: heavier objects need more push.

Mechanics★★★★★

Hooke's Law

F=−kxF = -kx

A spring pushes back proportionally to how far you stretch it.

Mechanics★★★★★

Newton's Law of Universal Gravitation

F=GMmr2F = \frac{GMm}{r^2}

Every mass attracts every other mass; force drops with the square of distance.

Mechanics★★★★★

Kinetic Energy

KE=12mv2KE = \frac{1}{2}mv^2

Energy of motion: doubles with mass, quadruples with speed.

Mechanics★★★★★

Centripetal Acceleration

ac=v2ra_c = \frac{v^2}{r}

Moving in a circle requires constant inward acceleration; faster or tighter = more.

Mechanics★★★★★

Projectile Range

R=v2sin⁡(2θ)gR = \frac{v^2 \sin(2\theta)}{g}

Launch angle of 45° maximizes range; steeper or flatter angles fall shorter.

Mechanics★★★★★

Work-Energy Theorem

Wnet=ΔKE=12mvf2−12mvi2W_{\text{net}} = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2

Net work on an object equals the change in its kinetic energy.

Mechanics★★★★★

Simple Harmonic Motion Period

T=2πmkT = 2\pi\sqrt{\frac{m}{k}}

Heavier masses oscillate slower; stiffer springs oscillate faster.

Electromagnetism

11
Electromagnetism★★★★★

Coulomb's Law

F=ke∣q1q2∣r2F = k_e \frac{|q_1 q_2|}{r^2}

Electric force between two charges falls off as the square of the distance between them.

Electromagnetism★★★★★

Electric Field of a Point Charge

E⃗=14πε0qr2r^\vec{E} = \frac{1}{4\pi\varepsilon_0} \frac{q}{r^2} \hat{r}

Each charge creates a field that tells other charges how much force they would feel.

Electromagnetism★★★★★

Gauss's Law

∮E⃗⋅dA⃗=Qencε0\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}

The total electric flux through a closed surface equals the enclosed charge divided by ε₀.

Electromagnetism★★★★★

Electric Potential (Point Charge)

V=14πε0qrV = \frac{1}{4\pi\varepsilon_0} \frac{q}{r}

Potential is the energy per unit charge — it falls off as 1/r, not 1/r².

Electromagnetism★★★★★

Parallel Plate Capacitor

C=ε0AdC = \varepsilon_0 \frac{A}{d}

Bigger plates and smaller gaps store more charge per volt.

Electromagnetism★★★★★

Ohm's Law

V=IRV = IR

Voltage is the push, resistance is the friction, current is how much flows.

Electromagnetism★★★★★

Biot–Savart Law

dB⃗=μ04πI dl⃗×r^r2d\vec{B} = \frac{\mu_0}{4\pi} \frac{I \, d\vec{l} \times \hat{r}}{r^2}

Each bit of current creates a magnetic field perpendicular to both the current and distance.

Electromagnetism★★★★★

Ampère's Law (with Maxwell's correction)

∮B⃗⋅dl⃗=μ0(Ienc+ε0dΦEdt)\oint \vec{B} \cdot d\vec{l} = \mu_0 \left( I_{\text{enc}} + \varepsilon_0 \frac{d\Phi_E}{dt} \right)

Magnetic field loops around currents; total circulation equals enclosed current times μ₀.

Electromagnetism★★★★★

Faraday's Law of Induction

E=−dΦBdt\mathcal{E} = -\frac{d\Phi_B}{dt}

A changing magnetic flux through a loop induces a voltage that opposes the change.

Electromagnetism★★★★★

Lorentz Force Law

F⃗=q(E⃗+v⃗×B⃗)\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})

Electric fields push charges; magnetic fields deflect moving charges sideways.

Electromagnetism★★★★★

Gauss's Law for Magnetism

∮B⃗⋅dA⃗=0\oint \vec{B} \cdot d\vec{A} = 0

Magnetic field lines always close on themselves — no isolated north or south poles.

Thermodynamics

8
Thermodynamics★★★★★

Ideal Gas Law

PV=nRTPV = nRT

Pressure times volume is proportional to temperature for a fixed amount of gas.

Thermodynamics★★★★★

First Law of Thermodynamics

ΔU=Q−W\Delta U = Q - W

Energy in (heat) minus energy out (work) equals the change stored inside.

Thermodynamics★★★★★

Fourier's Law of Heat Conduction

q=−kdTdxq = -k \frac{dT}{dx}

Heat flows from hot to cold, faster through better conductors and steeper gradients.

Thermodynamics★★★★★

Linear Thermal Expansion

ΔL=αL0ΔT\Delta L = \alpha L_0 \Delta T

Materials grow longer when heated — by an amount proportional to their length and temperature rise.

Thermodynamics★★★★★

Carnot Efficiency

η=1−TCTH\eta = 1 - \frac{T_C}{T_H}

No engine can beat the efficiency set by the ratio of its cold and hot reservoir temperatures.

Thermodynamics★★★★★

Stefan-Boltzmann Law

P=σAT4P = \sigma A T^4

Hot objects radiate energy as light — and the power skyrockets with temperature (fourth power!).

Thermodynamics★★★★★

Entropy Change

ΔS=∫δQrevT\Delta S = \int \frac{\delta Q_{\text{rev}}}{T}

Entropy measures how much energy has spread out — it always increases in the universe overall.

Thermodynamics★★★★★

Maxwell-Boltzmann Speed Distribution

f(v)=4πn(m2πkBT)3/2v2e−mv22kBTf(v) = 4\pi n \left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2 e^{-\frac{mv^2}{2k_BT}}

Gas molecules have a spread of speeds — most cluster near a peak, with a long tail of fast outliers.

Waves & Optics

8
Waves & Optics★★★★★

Snell's Law

n1sin⁡θ1=n2sin⁡θ2n_1 \sin\theta_1 = n_2 \sin\theta_2

Light bends toward the normal when entering a denser medium.

Waves & Optics★★★★★

Law of Reflection

θi=θr\theta_i = \theta_r

Light bounces off a mirror at the same angle it arrives.

Waves & Optics★★★★★

Wave Speed Equation

v=fλv = f\lambda

Wave speed equals how many wavelengths pass a point each second.

Waves & Optics★★★★★

Thin Lens Equation

1f=1do+1di\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}

Reciprocals of object and image distances always add up to the lens power.

Waves & Optics★★★★★

Young's Double Slit Interference

dsin⁡θ=mλd \sin\theta = m\lambda

Two overlapping wave sources create bright and dark bands.

Waves & Optics★★★★★

Single Slit Diffraction

asin⁡θ=mλa \sin\theta = m\lambda

A narrow slit spreads light into a pattern of bright and dark bands.

Waves & Optics★★★★★

Brewster's Angle

tan⁡θB=n2n1\tan\theta_B = \frac{n_2}{n_1}

At one special angle, reflected light becomes perfectly polarized.

Waves & Optics★★★★★

Doppler Effect

f′=fv+vov−vsf' = f \frac{v + v_o}{v - v_s}

Moving toward a wave source compresses waves; moving away stretches them.

Quantum

8
Quantum★★★★★

de Broglie Wavelength

λ=hp\lambda = \frac{h}{p}

Momentum and wavelength are inversely related through Planck's constant — big things have unmeasurably tiny wavelengths.

Quantum★★★★★

Photon Energy (Planck Relation)

E=hfE = hf

Light energy is quantized: each photon carries a fixed packet of energy set by its frequency.

Quantum★★★★★

Photoelectric Effect

Kmax=hf−ϕK_{max} = hf - \phi

A photon gives all its energy to one electron; the work function is the minimum escape cost.

Quantum★★★★★

Heisenberg Uncertainty Principle

Δx Δp≥ℏ2\Delta x \, \Delta p \geq \frac{\hbar}{2}

Position and momentum are conjugate — pinning one down spreads the other.

Quantum★★★★★

Bohr Hydrogen Energy Levels

En=−13.6 eVn2E_n = -\frac{13.6 \text{ eV}}{n^2}

Bound electrons can only sit on a quantized energy ladder; jumping down emits a photon.

Quantum★★★★★

Time-Dependent Schrödinger Equation

iℏ∂Ψ∂t=H^Ψi\hbar \frac{\partial \Psi}{\partial t} = \hat{H}\Psi

The Hamiltonian generates time evolution of the wavefunction in complex Hilbert space.

Quantum★★★★★

Particle in a 1D Box

En=n2π2ℏ22mL2E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}

Standing waves must fit inside the box; only integer half-wavelengths are allowed.

Quantum★★★★★

Dirac Equation

(iγμ∂μ−mc/ℏ)ψ=0(i\gamma^\mu \partial_\mu - mc/\hbar)\psi = 0

A first-order relativistic wave equation whose solutions naturally carry spin and antiparticles.

Relativity

8
Relativity★★★★★

Lorentz Factor

γ=11−v2/c2\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}

Quantifies how much time, length, and mass distort as velocity approaches c.

Relativity★★★★★

Time Dilation

Δt=γΔt0\Delta t = \gamma \Delta t_0

A moving clock ticks slower than a stationary one, as seen by an outside observer.

Relativity★★★★★

Length Contraction

L=L0/γL = L_0 / \gamma

Objects in motion appear shorter along their direction of travel.

Relativity★★★★★

Relativistic Momentum

p=γmvp = \gamma m v

Momentum diverges as velocity approaches c, preventing massive objects from reaching light speed.

Relativity★★★★★

Energy–Momentum Relation

E2=(pc)2+(mc2)2E^2 = (pc)^2 + (mc^2)^2

The full relativistic relation linking total energy, momentum, and rest mass.

Relativity★★★★★

Relativistic Velocity Addition

u=u′+v1+u′v/c2u = \frac{u' + v}{1 + u'v/c^2}

Velocities don't simply add at high speeds; the result never exceeds c.

Relativity★★★★★

Relativistic Doppler Effect

f=f01−β1+βf = f_0 \sqrt{\frac{1 - \beta}{1 + \beta}}

Receding sources redshift, approaching sources blueshift — with a relativistic gamma correction.

Relativity★★★★★

Schwarzschild Radius

rs=2GMc2r_s = \frac{2GM}{c^2}

The radius at which escape velocity equals c — the event horizon of a non-rotating black hole.

Modern Physics

8
Modern Physics★★★★★

Mass-Energy Equivalence

E=mc2E = mc^2

Mass is a highly concentrated form of energy; the speed of light squared is the conversion factor.

Modern Physics★★★★★

Lorentz Factor

γ=11−v2/c2\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}

As speed approaches c, time stretches and lengths contract by the factor γ.

Modern Physics★★★★★

Photoelectric Effect

Kmax=hν−ϕK_{max} = h\nu - \phi

Light comes in quanta of energy hν; only photons above the work-function threshold can free electrons.

Modern Physics★★★★★

De Broglie Wavelength

λ=hp\lambda = \frac{h}{p}

Every particle has a wavelength inversely proportional to its momentum — heavy/fast things have wavelengths so small they're unobservable.

Modern Physics★★★★★

Heisenberg Uncertainty Principle

Δx⋅Δp≥ℏ2\Delta x \cdot \Delta p \geq \frac{\hbar}{2}

Position and momentum cannot both be sharply defined; nature enforces a fundamental fuzziness at small scales.

Modern Physics★★★★★

Schrödinger Equation (Time-Independent)

−ℏ22m∇2ψ+Vψ=Eψ-\frac{\hbar^2}{2m}\nabla^2\psi + V\psi = E\psi

The total energy operator acting on the wavefunction returns the energy times the same wavefunction — an eigenvalue problem for reality.

Modern Physics★★★★★

Bohr Energy Levels (Hydrogen)

En=−13.6 eVn2E_n = -\frac{13.6\,\text{eV}}{n^2}

Electrons in hydrogen are stuck on a ladder of negative energies; the gaps determine the colors of light atoms emit.

Modern Physics★★★★★

Radioactive Decay Law

N(t)=N0e−λtN(t) = N_0 e^{-\lambda t}

Each nucleus has a fixed probability per unit time of decaying, producing a smooth exponential decline in the population.

Nuclear & Particle

10
Nuclear & Particle★★★★★

Mass–Energy Equivalence

E=mc2E = mc^2

Mass is a highly concentrated form of energy; converting even a tiny mass releases enormous energy.

Nuclear & Particle★★★★★

Radioactive Decay Law

N(t)=N0e−λtN(t) = N_0 e^{-\lambda t}

Undecayed nuclei fall exponentially; each has a fixed decay probability per unit time.

Nuclear & Particle★★★★★

Half-Life Relation

T1/2=ln⁡2λT_{1/2} = \frac{\ln 2}{\lambda}

Half-life is a fixed isotope property—independent of the sample size.

Nuclear & Particle★★★★★

Nuclear Binding Energy

B=Δm c2B = \Delta m \, c^2

The missing mass when nucleons bind into a nucleus appears as the energy holding them together.

Nuclear & Particle★★★★★

Q-Value of Nuclear Reaction

Q=(mi−mf)c2Q = (m_i - m_f) c^2

Lighter products mean the missing mass exits as kinetic energy and radiation.

Nuclear & Particle★★★★★

Relativistic Energy–Momentum Relation

E2=(pc)2+(mc2)2E^2 = (pc)^2 + (mc^2)^2

Energy and momentum combine to form a Lorentz invariant — the rest mass of the particle.

Nuclear & Particle★★★★★

de Broglie Wavelength (Relativistic)

λ=hp\lambda = \frac{h}{p}

A particle’s wavelength shrinks with momentum—more momentum probes smaller scales.

Nuclear & Particle★★★★★

Geiger–Nuttall Law

log⁡10T1/2=aZQ+b\log_{10} T_{1/2} = a \frac{Z}{\sqrt{Q}} + b

Alpha half-lives vary exponentially with decay energy via quantum tunneling.

Nuclear & Particle★★★★★

Semi-Empirical Mass Formula

B(A,Z)=aVA−aSA2/3−aCZ2A1/3−aA(A−2Z)2A+δ(A,Z)B(A,Z) = a_V A - a_S A^{2/3} - a_C \frac{Z^2}{A^{1/3}} - a_A \frac{(A-2Z)^2}{A} + \delta(A,Z)

Five competing terms—volume, surface, Coulomb, asymmetry, pairing—model the nuclear binding energy.

Nuclear & Particle★★★★★

Klein–Nishina Cross Section

dσdΩ=12α2rc2(ω′ω)2[ωω′+ω′ω−sin⁡2θ]\frac{d\sigma}{d\Omega} = \frac{1}{2} \alpha^2 r_c^2 \left(\frac{\omega'}{\omega}\right)^2 \left[\frac{\omega}{\omega'} + \frac{\omega'}{\omega} - \sin^2\theta\right]

QED correction to Thomson scattering: photon cross-section shrinks at high energy.

Fluid Mechanics

6
Fluid Mechanics★★★★★

Continuity Equation

A1v1=A2v2A_1 v_1 = A_2 v_2

What flows in must flow out — narrow pipes force faster flow.

Fluid Mechanics★★★★★

Bernoulli's Equation

P+12ρv2+ρgh=constP + \tfrac{1}{2}\rho v^2 + \rho g h = \text{const}

Pressure, kinetic, and potential energy per unit volume sum to a constant along a streamline.

Fluid Mechanics★★★★★

Hydrostatic Pressure

P=P0+ρghP = P_0 + \rho g h

Pressure grows linearly with depth because of the weight of fluid above.

Fluid Mechanics★★★★★

Archimedes' Principle

Fb=ρVgF_b = \rho V g

Buoyant force equals the weight of fluid displaced.

Fluid Mechanics★★★★★

Torricelli's Law

v=2ghv = \sqrt{2 g h}

Falling fluid trades height for speed, just like a dropped ball.

Fluid Mechanics★★★★★

Reynolds Number

Re=ρvLμRe = \frac{\rho v L}{\mu}

Ratio of inertial to viscous forces — high Re means inertia wins, turbulence reigns.