Formula Library and Calculators

Physics (83)

Kinematics (motion)

• First equation of motion — v = u + at
• Second equation of motion — s = ut + ½at²
• Third equation of motion — v² = u² + 2as
• Distance from average velocity — s = ½(u + v)t
• Speed, distance and time — v = s / t
• Free fall: distance fallen from rest — h = ½gt²
• Free fall: speed after a time — v = gt
• Free fall: speed after falling a height — v² = 2gh
• Projectile: horizontal range — R = u² sin 2θ / g
• Projectile: maximum height — H = u² sin²θ / 2g
• Projectile: time of flight — T = 2u sinθ / g

Dynamics (force and momentum)

• Newton's second law — F = ma
• Weight — W = mg
• Linear momentum — p = mv
• Impulse and change in momentum — Ft = m(v − u)
• Force of friction — f = μN
• Conservation of momentum (objects stick together) — m₁u₁ + m₂u₂ = (m₁ + m₂)v
• Torque (moment of a force) — τ = F × d

Circular and rotational motion

• Centripetal force — F = mv² / r
• Centripetal acceleration — a = v² / r
• Linear speed and angular velocity — v = rω
• Angular velocity from frequency — ω = 2πf
• Frequency and time period — f = 1 / T

Gravitation

• Newton's law of gravitation — F = G m₁m₂ / r²
• Gravitational field at a planet's surface — g = GM / R²
• Orbital speed — v = √(GM / r)
• Escape velocity — v = √(2GM / R)
• Kepler's third law (circular orbit) — T² = 4π²r³ / GM

Work, energy and power

• Work done by a force — W = Fs cosθ
• Kinetic energy — KE = ½mv²
• Gravitational potential energy — PE = mgh
• Work-energy theorem — W = ½m(v² − u²)
• Power — P = W / t
• Power from force and speed — P = Fv
• Efficiency — efficiency = (useful output ÷ input) × 100%

Properties of matter

• Density — ρ = m / V
• Pressure — P = F / A
• Pressure in a liquid — P = ρgh
• Buoyant force (Archimedes' principle) — F = ρVg
• Hooke's law — F = kx
• Energy stored in a spring — U = ½kx²
• Young's modulus — Y = (F/A) ÷ (ΔL/L₀)
• Equation of continuity — A₁v₁ = A₂v₂

Heat and thermodynamics

• Heat energy (specific heat capacity) — Q = mcΔT
• Latent heat (change of state) — Q = mL
• Linear thermal expansion — ΔL = αL₀ΔT
• Efficiency of a Carnot (ideal) engine — efficiency = (1 − Tⱼ / Tₕ) × 100%
• First law of thermodynamics — ΔU = Q − W

Waves and oscillations

• Wave speed — v = fλ
• Time period of a mass on a spring — T = 2π√(m / k)
• Time period of a simple pendulum — T = 2π√(L / g)

Light (optics)

• Refractive index — n = c / v
• Snell's law of refraction — n₁ sinθ₁ = n₂ sinθ₂
• Lens / mirror formula — 1/f = 1/p + 1/q
• Magnification — M = q / p
• Power of a lens — P = 1 / f

Current electricity

• Ohm's law — V = IR
• Electrical power (from V and I) — P = VI
• Electrical power (from I and R) — P = I²R
• Electrical power (from V and R) — P = V² / R
• Electric charge — Q = It
• Resistance of a wire — R = ρL / A
• Resistors in series — R = R₁ + R₂ + R₃
• Two resistors in parallel — 1/R = 1/R₁ + 1/R₂
• Electrical energy — E = Pt

Electrostatics

• Coulomb's law — F = k q₁q₂ / r²
• Electric field strength (from force) — E = F / q
• Electric field of a point charge — E = kq / r²
• Potential difference from work — V = W / q
• Capacitance — C = Q / V
• Energy stored in a capacitor — U = ½CV²
• Parallel-plate capacitor — C = ε₀A / d

Magnetism and electromagnetic induction

• Force on a current-carrying wire — F = BIL sinθ
• Force on a moving charge — F = qvB sinθ
• Induced emf (Faraday's law) — ε = N ΔΦ / Δt
• emf of a moving conductor — ε = BLv
• Transformer — Vₛ / Vₚ = Nₛ / Nₚ

Modern physics

• Energy of a photon — E = hf
• Photon energy from wavelength — E = hc / λ
• Photoelectric effect — KEₘₐₓ = hf − φ
• de Broglie wavelength — λ = h / mv
• Mass-energy equivalence — E = mc²
• Radioactive decay (half-life) — N = N₀ (½)^(t / T½)

Chemistry (37)

Mole concept and stoichiometry

• Moles from mass — n = m / M
• Moles from number of particles — n = N / Nₐ
• Moles and volume of a gas at STP — n = V / Vₘ
• Percentage composition by mass — % = (mass of part ÷ mass of whole) × 100
• Percentage yield — % yield = (actual yield ÷ theoretical yield) × 100

Atomic structure

• Mass number — A = Z + N
• Average atomic mass (two isotopes) — A = (m₁a₁ + m₂a₂) ÷ 100
• Hydrogen spectrum (Rydberg formula) — 1/λ = Rᴴ (1/n₁² − 1/n₂²)
• Energy of a hydrogen level (Bohr model) — Eₙ = −2.18 × 10⁻¹⁸ J / n²

Gases

• Boyle's law — P₁V₁ = P₂V₂
• Charles' law — V₁ / T₁ = V₂ / T₂
• Gay-Lussac's law — P₁ / T₁ = P₂ / T₂
• Combined gas law — P₁V₁ / T₁ = P₂V₂ / T₂
• Ideal gas equation — PV = nRT
• Density of a gas — ρ = PM / RT
• Graham's law of diffusion — rate₁ / rate₂ = √(M₂ / M₁)
• Dalton's law of partial pressures — Pₜ = P₁ + P₂ + P₃
• Partial pressure from mole fraction — Pₐ = xₐ × Pₜ
• Average kinetic energy of a gas molecule — KE = (3/2) kʙ T

Solutions and concentration

• Molarity — M = n / V
• Dilution — M₁V₁ = M₂V₂
• Molality — m = n / mass of solvent (kg)
• Mole fraction — xₐ = nₐ / (nₐ + nᵦ)
• Parts per million (ppm) — ppm = (mass of solute ÷ mass of solution) × 10⁶
• Boiling point elevation / freezing point depression — ΔT = K × m

Acids, bases and pH

• pH from hydrogen ion concentration — pH = −log₁₀[H⁺]
• pOH from hydroxide ion concentration — pOH = −log₁₀[OH⁻]
• pH and pOH — pH + pOH = 14
• Ionic product of water — [H⁺][OH⁻] = Kᵥᵥ = 1 × 10⁻¹⁴
• Weak acid: hydrogen ion concentration — [H⁺] = √(Kₐ × C)

Thermochemistry and energy

• Enthalpy change per mole — ΔH = q / n
• Gibbs free energy — ΔG = ΔH − TΔS

Electrochemistry

• Faraday's law of electrolysis — m = M I t / (n F)
• emf of an electrochemical cell — E°cell = E°cathode − E°anode

Reaction rates

• Average rate of reaction — rate = Δ[concentration] / Δt
• Half-life of a first-order reaction — t½ = 0.693 / k
• First-order decay of concentration — [A] = [A]₀ e^(−kt)

Mathematics (36)

Algebra

• Quadratic formula — x = (−b ± √(b² − 4ac)) / 2a
• Change of base for logarithms — logᵇ x = ln x / ln b
• Arithmetic mean (average) — mean = sum of values ÷ number of values
• Percentage — % = (part ÷ whole) × 100
• Percentage change — % change = ((new − old) ÷ old) × 100

Coordinate geometry

• Distance between two points — d = √((x₂ − x₁)² + (y₂ − y₁)²)
• Midpoint of a line segment — M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
• Gradient (slope) of a line — m = (y₂ − y₁) / (x₂ − x₁)
• Equation of a line (gradient-intercept form) — y = mx + c

Geometry and mensuration

• Pythagoras' theorem — c² = a² + b²
• Area of a rectangle — A = l × w
• Area of a triangle — A = ½ × base × height
• Heron's formula (area from three sides) — A = √(s(s−a)(s−b)(s−c)), s = (a+b+c)/2
• Area of a trapezium — A = ½(a + b) × h
• Area of a circle — A = πr²
• Circumference of a circle — C = 2πr
• Length of an arc — s = rθ (θ in radians)
• Area of a sector — A = ½r²θ (θ in radians)
• Volume of a sphere — V = (4/3)πr³
• Surface area of a sphere — A = 4πr²
• Volume of a cylinder — V = πr²h
• Total surface area of a cylinder — A = 2πr(r + h)
• Volume of a cone — V = ⅓πr²h

Trigonometry

• Sine of an angle (right triangle) — sin θ = opposite / hypotenuse
• Cosine of an angle (right triangle) — cos θ = adjacent / hypotenuse
• Tangent of an angle (right triangle) — tan θ = opposite / adjacent
• Sine rule — a / sin A = b / sin B
• Cosine rule — c² = a² + b² − 2ab cos C

Sequences and series

• Arithmetic sequence: nth term — aₙ = a + (n − 1)d
• Arithmetic series: sum of n terms — Sₙ = (n/2)[2a + (n − 1)d]
• Geometric sequence: nth term — aₙ = a rⁿ⁻¹
• Geometric series: sum of n terms — Sₙ = a(1 − rⁿ) / (1 − r)

Permutations and combinations

• Permutations — ⁿPᵣ = n! / (n − r)!
• Combinations — ⁿCᵣ = n! / (r!(n − r)!)

Percentage and finance

• Simple interest — I = P R T / 100
• Compound interest — A = P (1 + R/100)ᵀ

Biology (7)

Cells and microscopy

• Magnification of a microscope image — magnification = size of image ÷ actual size
• Surface area to volume ratio of a cube — SA : V = 6 / L

Genetics and populations

• Hardy-Weinberg equilibrium — p + q = 1 and p² + 2pq + q² = 1
• Population growth rate — growth rate (%) = ((births − deaths) ÷ population) × 100

Human physiology

• Cardiac output — cardiac output = heart rate × stroke volume
• Respiratory quotient — RQ = CO₂ produced ÷ O₂ used
• Body mass index (BMI) — BMI = mass ÷ height²