Search formulas
326 formulas found
SUVAT — final velocity
Final velocity from initial velocity, acceleration and time.
SUVAT — displacement
Displacement from initial velocity, acceleration and time.
SUVAT — v² = u² + 2as
Time-independent SUVAT relation.
SUVAT — s = ½(u+v)t
Displacement using average velocity (constant acceleration).
Newton's Second Law
Resultant force equals mass times acceleration.
Weight
Weight is the gravitational force on a mass.
Work done
Work done by a constant force at angle θ to displacement.
Kinetic Energy
Translational kinetic energy of a body.
Gravitational PE (near Earth)
Change in gravitational potential energy near the Earth's surface.
Power
Rate of energy transfer.
Efficiency
Efficiency as a percentage.
Linear Momentum
Momentum is mass times velocity.
Impulse
Impulse equals change in momentum.
Hooke's Law
Restoring force in a spring is proportional to extension within the elastic limit.
Elastic Potential Energy
Energy stored in a spring obeying Hooke's law.
Density
Mass per unit volume of a substance.
Pressure (force per area)
Pressure is the force applied normal to a surface per unit area.
Hydrostatic Pressure
Pressure due to a column of fluid of depth h and density ρ.
Moment of a Force
Turning effect of a force about a pivot, where d is the perpendicular distance from the pivot to the line of action of the force.
Centripetal Acceleration
Acceleration of a body in uniform circular motion, directed towards the centre.
Centripetal Force
Net inward force required for circular motion.
Newton's Law of Gravitation
Gravitational force between two point masses.
Gravitational field strength
Field strength at a distance r from a point mass M.
Orbital Speed
Speed of a body in circular orbit at radius r.
Kepler's Third Law
Orbital period squared is proportional to radius cubed.
Wave equation
Wave speed equals frequency times wavelength.
Period & Frequency
Period is the reciprocal of frequency.
Angular frequency
Angular frequency in radians per second.
SHM displacement
Displacement of a body in simple harmonic motion (starting at amplitude).
Period of a pendulum
Period of a simple pendulum (small angles).
Mass-spring period
Period of a mass on a spring.
Snell's Law
Refraction at the boundary between two media.
Double-slit fringe spacing
Fringe spacing in Young's double-slit experiment.
Ohm's Law
Voltage equals current times resistance.
Charge
Charge equals current times time.
Electrical Power
Power dissipated by a component.
Resistivity
Resistance of a conductor of uniform cross section.
EMF & Internal Resistance
EMF equals current times total circuit resistance.
Capacitance
Charge stored per unit voltage.
Energy in a capacitor
Energy stored in a capacitor.
Coulomb's Law
Electrostatic force between two point charges.
E-field in parallel plates
Uniform electric field between parallel plates.
Force on a current-carrying wire
Force on a length of wire in a magnetic field.
Force on a moving charge
Magnetic force on a charge moving perpendicular to B.
Faraday's Law
Induced EMF equals rate of change of flux linkage.
Transformer equation
Ideal transformer voltage ratio.
Specific Heat Capacity
Energy to change the temperature of a mass.
Specific Latent Heat
Energy required to change state.
Ideal Gas Equation
Equation of state for an ideal gas.
Radioactive decay
Number of undecayed nuclei after time t.
Half-life
Relationship between half-life and decay constant.
Mass–Energy
Rest mass-energy equivalence.
Photon energy
Energy of a photon.
Photoelectric equation
Einstein's photoelectric equation.
de Broglie wavelength
Matter-wave wavelength of a particle of momentum p.
Hubble's Law
Recession velocity of distant galaxies.
Stefan–Boltzmann Law
Luminosity of a black-body sphere.
Wien's Displacement Law
Peak wavelength of black-body emission.
Quadratic Formula
Solves the general quadratic ax² + bx + c = 0 for both roots when a ≠ 0.
Discriminant
Sign of Δ classifies the roots of ax² + bx + c = 0: Δ>0 two real, Δ=0 repeated, Δ<0 complex.
Completing the square
Rewrites a quadratic in vertex form (x + p)² + q to read off the turning point (−p, q).
Logarithm product law
Multiplication inside a log becomes addition outside (base a > 0, a ≠ 1, x, y > 0).
Exponential growth/decay
Continuous exponential model: k > 0 grows, k < 0 decays. Used for population, radioactive decay, cooling.
Arithmetic sequence — nth term
n-th term of an arithmetic progression with first term a and common difference d.
Arithmetic series — sum of n terms
Sum of first n terms of an arithmetic progression. Equivalent form: Sₙ = n(a + l)/2 where l is the last term.
Geometric sequence — nth term
n-th term of a geometric progression with first term a and common ratio r.
Geometric series — sum of n terms
Sum of first n terms of a geometric progression.
Geometric series — sum to infinity
A geometric series converges to a/(1 − r) only when |r| < 1.
Pythagoras' Theorem
In a right-angled triangle, the square of the hypotenuse c equals the sum of the squares of the other two sides.
Sine Rule
Solves non-right triangles when an angle and its opposite side are paired with one further side or angle.
Cosine Rule
Generalises Pythagoras to any triangle — use when two sides + included angle, or all three sides, are known.
Area of triangle (sine)
Area of any triangle using two sides and the included angle.
Arc length (radians)
Arc length of a circle sector with radius r and angle θ in radians.
Sector area (radians)
Area of a circular sector with radius r and angle θ in radians.
Distance between two points
Euclidean distance between (x₁,y₁) and (x₂,y₂) in the plane.
Midpoint of a segment
Midpoint of the segment joining (x₁,y₁) and (x₂,y₂).
Gradient of a line
Slope between two points; undefined when x₁ = x₂ (vertical line).
Circle equation
Circle with centre (a, b) and radius r. Distance from any point on the circle to the centre equals r.
Vector magnitude (3D)
Magnitude (length) of a 3D vector with components (x, y, z).
Power rule (derivative)
Derivative of xⁿ for any real n. Foundation of polynomial differentiation.
Power rule (integral)
Indefinite integral of xⁿ. The +C constant is essential.
Percentage change
Relative change between two values, expressed as a percentage. Negative result indicates decrease.
Simple interest
Interest paid only on the original principal P at annual rate r (decimal) over t years.
Compound interest
Final amount A when principal P grows at rate r per period (decimal) for t periods, compounding once per period.
Arithmetic mean
Mean of n values whose total is Σx.
Variance (population)
Population variance via the computational formula. Square root gives the standard deviation σ.
Binomial probability
Probability of exactly k successes in n independent Bernoulli trials with success probability p.
Poisson probability
Probability of k events occurring in a fixed interval when events occur independently at average rate λ.
Conditional probability
Probability of A given B has occurred, when P(B) > 0.
Standardising (z-score)
Number of standard deviations a value x lies from the mean μ. Used to look up normal probabilities.
Modulus of a complex number
Distance of z = a + bi from the origin in the Argand diagram.
Argument of a complex number
Principal argument of z = a + bi, returned in (−π, π]. Use atan2 to handle every quadrant correctly.
De Moivre's Theorem
Raising a complex number in polar form to the power n: modulus is raised to the n-th power, argument multiplied by n.
Modulus of a product
Multiplying complex numbers multiplies their moduli and adds their arguments.
2×2 Determinant
Determinant of a 2×2 matrix. Equals the signed area-scale factor of the linear map.
2×2 Matrix inverse
Inverse of a non-singular 2×2 matrix; swap a and d, negate b and c, divide by the determinant.
3×3 Determinant (cofactor expansion)
Cofactor expansion along the first row of a 3×3 matrix.
Polar → Cartesian (x-coordinate)
Cartesian x of a point given polar (r, θ).
Polar → Cartesian (y-coordinate)
Cartesian y of a point given polar (r, θ).
Hyperbolic sine
Definition of sinh in terms of exponentials. Odd function: sinh(−x) = −sinh x.
Hyperbolic cosine
Definition of cosh. Even function with cosh x ≥ 1 for all real x.
Hyperbolic identity
Fundamental hyperbolic identity (note the MINUS sign — different from sin² + cos² = 1).
Sum of first n integers
Closed form for 1 + 2 + … + n.
Sum of first n squares
Closed form for 1² + 2² + … + n².
Moles from mass
Moles equals mass divided by relative molecular mass.
Concentration
Concentration in mol/dm³.
Ideal Gas Equation
Equation of state for an ideal gas (R = 8.314).
Percentage yield
Yield as a percentage of the theoretical maximum.
Atom economy
Percentage of reactant mass becoming desired product.
pH
pH from hydrogen ion concentration.
Ionic product of water
Kw = 1.0 × 10⁻¹⁴ at 25 °C.
Acid dissociation constant
Equilibrium constant for weak acid dissociation.
pKa
Negative log of Ka.
Henderson–Hasselbalch
pH of a buffer solution.
Heat transferred (calorimetry)
Energy released/absorbed by a solution.
Arrhenius equation
Temperature dependence of rate constant.
Gibbs Free Energy
Spontaneity of a reaction.
Microscope magnification
Magnification = image size ÷ actual size.
Cardiac output
Cardiac output = heart rate × stroke volume.
Hardy–Weinberg (alleles)
Allele frequencies sum to 1.
Hardy–Weinberg (genotypes)
Genotype frequencies under HWE.
Simpson's Diversity Index
Higher D = greater diversity (single species).
Chi-squared statistic
One term of the chi-squared sum.
SA:Vol of a sphere
Surface area to volume ratio of a sphere.
Price Elasticity of Demand
Responsiveness of quantity demanded to price.
Income Elasticity of Demand
Responsiveness of demand to income changes.
Cross Elasticity of Demand
Responsiveness of demand for A to price of B.
Price Elasticity of Supply
Responsiveness of supply to price changes.
GDP (expenditure approach)
Aggregate demand identity.
Keynesian Multiplier
Multiplier from marginal propensity to consume.
Inflation rate (from CPI)
Year-on-year inflation rate.
Break-even output
Units to sell to cover all fixed costs.
Contribution per unit
Selling price minus variable cost.
Margin of safety
Buffer between current sales and break-even.
Return on Capital Employed
Profitability of capital used.
Current Ratio
Short-term liquidity measure.
Acid-test ratio
Liquidity excluding inventory.
Gearing
Long-term debt as a share of capital.
Average Rate of Return
Investment appraisal — accounting return.
Straight-line depreciation
Depreciation per year.
Stream discharge
Discharge equals cross-section area times velocity.
Hydraulic radius
Cross-section area divided by wetted perimeter.
Population density
People per unit area.
Spearman's rank (for ties-free data)
Correlation between two ranked variables.
Binary → Decimal (8-bit)
Convert an 8-bit unsigned binary value to decimal.
File transfer time
Time to transfer a file at a given bandwidth.
Bitmap image file size
Pixels times bit depth (in bits).
Sound file size
Sample rate × bit depth × duration × channels.
Standard deviation (sample)
Sample standard deviation.
Chi-squared (one term)
One contribution to the chi-squared sum.
Distance between two points
Euclidean distance between two coordinates in 2D.
Midpoint x-coordinate
x-coordinate of the midpoint of a line segment.
Gradient between two points
Slope of the line through two points.
Arc length (radians)
Arc length of a circular sector with angle in radians.
Geometric series sum
Sum of the first n terms of a geometric sequence (r ≠ 1).
Binomial coefficient
Number of ways to choose k items from n.
Z-score
Standardised score: how many standard deviations x is from the mean.
Standard error of the mean
Standard error of a sample mean.
Binomial probability
Probability of exactly k successes in n independent Bernoulli trials.
Real GDP
Inflation-adjusted GDP using a price deflator (base 100).
Marginal propensity to consume
Fraction of additional income spent on consumption.
Sample mean
Arithmetic mean of a data set.
Chi-square component
Single-cell contribution to the chi-square statistic.
Spearman's rank correlation
Non-parametric correlation between two ranked variables.
Population density
People per unit area.
Natural increase rate
Natural population change per 1000 people.
Gross profit margin
Percentage of revenue retained after cost of sales.
Dilution equation
Conservation of moles when diluting a solution.
pH + pOH
Auto-ionisation of water at 25 °C.
Body mass index
Body mass index from mass and height.
Weight
Weight equals Mass times Grav. field strength.
Density
Density equals Mass times Volume.
Pressure
Pressure equals Force times Area.
Hydrostatic pressure
Pressure at depth h in a fluid of density ρ.
v² = u² + 2as
v² = u² + 2as
s = ut + ½at²
s = ut + ½at²
s = ½(u+v)t
s = ½(u+v)t
Momentum
Momentum equals Mass times Velocity.
Impulse-momentum
Impulse-momentum
Kinetic energy
Kinetic energy
Gravitational PE (near surface)
Gravitational PE (near surface)
Power
Power equals Energy times Time.
Work done
Work equals Force times Distance.
Efficiency
Efficiency (%) as a percentage.
SHM angular frequency
SHM angular frequency
SHM period (mass-spring)
SHM period (mass-spring)
SHM period (pendulum)
SHM period (pendulum)
Wave speed
Speed equals Frequency times Wavelength.
Snell's law
Snell's law
Thin lens equation
Thin lens equation
Doppler shift (sound, source moving)
Doppler shift (sound, source moving)
Diffraction grating
Diffraction grating
Centripetal acceleration
Centripetal acceleration
Centripetal force
Centripetal force
Newton's law of gravitation
Newton's law of gravitation
Escape velocity
Escape velocity
Kepler's third law
Kepler's third law
Photon energy
Photon energy
Photon energy (wavelength)
Photon energy (wavelength)
Photoelectric equation
Photoelectric equation
de Broglie wavelength
de Broglie wavelength
Mass-energy equivalence
Mass-energy equivalence
Radioactive decay
Radioactive decay
Half-life and decay constant
Half-life and decay constant
Charge
Charge equals Current times Time.
Ohm's law
P.D. equals Current times Resistance.
Electrical power
Power equals P.D. times Current.
Electrical power (I²R)
Electrical power (I²R)
Resistivity
Resistivity
Coulomb's law
Coulomb's law
Capacitor energy
Capacitor energy
Capacitor charge
Capacitor charge
Transformer ratio
Transformer ratio
Force on current carrier
Force on a current-carrying wire perpendicular to B.
Specific heat capacity
Specific heat capacity
Latent heat
Heat equals Mass times Specific latent heat.
Ideal gas (pV=nRT)
Ideal gas (pV=nRT)
Stefan-Boltzmann law
Stefan-Boltzmann law
Wien's displacement law
Wien's displacement law
Hubble's law
Hubble's law
Sine rule
Sine rule: each side of any triangle is proportional to the sine of its opposite angle — used to find unknown sides or angles in non-right triangles.
Cosine rule
Cosine rule
Triangle area
Triangle area
Arc length
Arc length
Sector area
Sector area
AP nth term
Sine rule relates each side of a triangle to the sine of its opposite angle, used to find unknown sides or angles in non-right triangles.
Arithmetic series sum
Sum of the first n terms of an arithmetic progression with first term a and common difference d.
GP nth term
GP nth term
GP sum (finite)
GP sum (finite)
GP sum to infinity
GP sum to infinity
Combinations nCr
Combinations nCr
Permutations nPr
Permutations nPr
Circle equation
Circle equation
Trapezium rule
Trapezium rule
Newton-Raphson step
Newton-Raphson step
Binomial coefficient
Binomial coefficient
Volume of cylinder
Volume of cylinder
Volume of sphere
Volume of sphere
Surface area of sphere
Surface area of sphere
Volume of cone
Volume of cone
Pythagoras
Pythagoras
Z-score (standard score)
Number of standard deviations a value x lies from the mean of its distribution.
Complex modulus
Complex modulus
Complex argument
Complex argument
de Moivre's theorem
de Moivre's theorem
Sum of first n integers
Sum of first n integers
Sum of squares
Sum of squares
Sum of cubes
Sum of cubes
2×2 determinant
2×2 determinant
Hyperbolic identity
Hyperbolic identity
Polar area
Polar area
Moles from mass
Moles equals Mass divided by Molar mass.
Moles from solution
Moles equals Concentration times Volume.
Percentage yield
Yield (%) as a percentage.
Atom economy
Atom economy (%) as a percentage.
Ideal gas (chem)
Ideal gas (chem)
Calorimetry
Calorimetry
Gibbs free energy
Gibbs free energy
Arrhenius equation
Temperature dependence of a reaction-rate constant via the activation energy Eₐ.
pH from hydrogen-ion concentration
Acidity expressed as the negative base-10 logarithm of the hydrogen-ion concentration.
pOH from hydroxide-ion concentration
Basicity expressed as the negative base-10 logarithm of the hydroxide-ion concentration.
Acid dissociation Ka
Acid dissociation Ka
Henderson-Hasselbalch
Henderson-Hasselbalch
Beer-Lambert law
Beer-Lambert law
Faraday's law (electrolysis)
Faraday's law (electrolysis)
Ionic product of water
Ionic product of water
Magnification
Magnification equals Image size times Actual size.
Surface area : volume (cube)
Surface area : volume (cube)
Hardy-Weinberg q²
Hardy-Weinberg q²
Hardy-Weinberg p+q=1
Hardy-Weinberg p+q=1
Lincoln index (mark-recapture)
Lincoln index (mark-recapture)
Simpson's diversity index
Simpson's diversity index
Cardiac output
Cardiac output
Body Mass Index (BMI)
Mass divided by the square of height — a screening measure of relative body weight (kg/m²).
Water potential
Water potential
Osmotic pressure
Osmotic pressure
Michaelis-Menten
Michaelis-Menten
Standard error
Standard error
Income elasticity (YED)
Income elasticity (YED)
Cross elasticity (XED)
Cross elasticity (XED)
Price elasticity of supply
Price elasticity of supply
Total revenue
Revenue equals Price times Quantity.
Inflation rate
Inflation rate
Unemployment rate
Unemployment rate
Marginal propensity to save
Marginal propensity to save
Real interest rate (Fisher)
Real interest rate (Fisher)
Money multiplier
Money multiplier
Quantity theory of money
Quantity theory of money
Gross profit margin
GPM (%) as a percentage.
Net profit margin
NPM (%) as a percentage.
ROCE
ROCE (%) as a percentage.
Current ratio
Current ratio equals Current assets divided by Current liabilities.
Acid test ratio
Acid test ratio
Break-even units
Break-even units
Contribution per unit
Contribution equals Price minus Variable cost.
Margin of safety (%)
MoS (%) as a percentage.
Payback period (even cashflow)
Payback period (even cashflow)
Average Rate of Return
Average Rate of Return
Labour turnover
LT (%) as a percentage.
Capacity utilisation
CU (%) as a percentage.
Market share
MS (%) as a percentage.
Straight-line depreciation
Straight-line depreciation
Gearing ratio
Gearing ratio
Two's complement range
Two's complement range
Bitmap image file size
Bitmap image file size
Uncompressed audio size
Uncompressed audio size
Network throughput
Network throughput
Usable hosts in subnet
Usable hosts in subnet
Key space size
Key space size
Hash table load factor
Hash table load factor
Crude birth rate
Crude birth rate
Crude death rate
Crude death rate
Natural increase rate
NIR (‰) equals CBR (‰) minus CDR (‰).
Doubling time (rule of 70)
Doubling time (rule of 70)
Dependency ratio
Dependency ratio
Nearest neighbour index
Nearest neighbour index
Bifurcation ratio
Bifurcation ratio
Dry adiabatic lapse rate
Dry adiabatic lapse rate
Arithmetic mean
Sum of all values divided by the number of values; the standard average for ratio data.
Range (descriptive statistic)
Spread of a data set: largest value minus smallest value.
Z-score (psy)
Z-score (psy)
IQ (ratio)
IQ (ratio)
IQ (deviation)
IQ (deviation)
Standard error (psy)
Standard error (psy)
Spearman-Brown reliability
Spearman-Brown reliability
Forgetting curve
Forgetting curve