Science calculators

Physics, chemistry, biology and ecology: first-principles formulas for the lab, the field and the classroom.

Physics

Mechanics, electricity, energy and waves: first-principles formulas with every constant stated.

Ohm’s lawVoltage, current, resistance and power from any two.Resistors in series & parallelCombined resistance of up to four resistors.Voltage dividerOutput voltage across the lower resistor of a pair.Electricity running costWhat an appliance costs to run over time.Force, mass & accelerationNewton’s second law, with weight under gravity.Kinetic & potential energyEnergy of a moving or raised mass.Speed, distance & timeJourney arithmetic in metric and imperial.Acceleration & stopping distanceUniform acceleration, braking distance and reaction gap.Projectile motionRange, peak height and flight time of a launched object.Density, mass & volumeAny one of the three from the other two.Pressure from force or depthPressure under a load, and water pressure at depth.Wave speed, frequency & wavelengthRelates the three properties of any wave.Work, power & efficiencyEnergy transferred, the rate of transfer and what is lost.Momentum & impulseMomentum of a mass and the force needed to change it.Ideal gas lawPressure, volume, temperature and moles of a gas.Torque & rotational powerTurning force, and the power it delivers at a given speed.Centripetal force & circular motionForce, acceleration and period of an object moving in a circle at constant speed.Hooke's law & spring energyRestoring force and stored energy of a stretched or compressed spring.Pendulum periodSwing period of a simple pendulum for small angles.Elastic & inelastic collisionFinal velocities of two colliding masses, in a straight line.Capacitor charge & energyStored charge and energy of a capacitor at a given voltage.Inductor stored energyMagnetic energy stored in an inductor carrying current.RC circuit charge & dischargeHow a capacitor charges or discharges through a resistor over time.Carnot efficiencyMaximum possible efficiency of a heat engine between two temperatures.Linear thermal expansionHow much a material lengthens or shortens with a temperature change.Heat energy & specific heatHeat needed to change the temperature of a mass of material.Snell's law of refractionAngle of refraction as light passes between two media.Thin lens equationImage distance and magnification for a lens of known focal length.Doppler effect (sound)Observed frequency of a sound source moving relative to a listener.Sound intensity level (decibels)Decibel level of a sound from its intensity, referenced to the hearing threshold.Escape velocityMinimum speed to escape a body’s gravity from a given distance, ignoring drag.Circular orbital velocitySpeed and period needed to hold a circular orbit around a mass.Kepler's third lawOrbital period of a body from its orbit size and the mass it orbits.Buoyancy (Archimedes’ principle)Upward force on an object from the fluid it displaces, and whether it floats.Terminal velocityMaximum falling speed once drag balances gravity.Angular momentum (point mass)Rotational momentum of a point mass moving on a circular path.

Chemistry

Concentrations, pH, dilutions and reaction yields: the maths behind the lab bench.

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Molarity from massConcentration of a solution made from a known mass of solute.Solution dilution (C₁V₁ = C₂V₂)Stock volume and diluent needed to hit a target concentration.pH from hydrogen ion concentrationConversion between [H⁺] and pH, with pOH and [OH⁻] at 25 °C.Henderson–Hasselbalch (buffer pH)Buffer pH from the acid dissociation constant and the base/acid ratio.Percent yieldHow much product a reaction actually gave versus the theoretical maximum.Mass percent concentrationSolute mass as a percentage of total solution mass, plus ppm.Radioactive decay & half-lifeAmount of a substance remaining after a given elapsed time.Beer–Lambert law (absorbance)Absorbance and transmittance of a solution from its concentration.Osmotic pressure (van’t Hoff)Pressure needed to stop osmosis across a semipermeable membrane.

Biology

Population genetics, enzyme kinetics, growth curves and lab concentration formulas.

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Hardy–Weinberg equilibriumExpected genotype frequencies in a non-evolving population from allele frequency.DNA/RNA concentration from A₂₆₀Nucleic acid concentration from a spectrophotometer absorbance reading.Log reduction (microbial kill)How many orders of magnitude a treatment reduces a microbial count.Michaelis–Menten kineticsEnzyme reaction rate at a given substrate concentration.Bacterial/cell growth (doubling time)Population size after exponential growth at a fixed doubling time.Q₁₀ temperature coefficientHow much a biological or chemical rate changes per 10 °C rise.Plant water potentialTotal water potential from solute and pressure potential in a plant cell.

Ecology

Population and diversity formulas for the field: carrying capacity, species diversity and predator-prey dynamics.

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Logistic population growthPopulation size over time under a fixed carrying capacity (Verhulst model).Shannon diversity indexBiodiversity of a community from species counts.Lotka–Volterra predator–prey ratesInstantaneous population growth rates for a classic predator–prey pair.Kaya identity (CO₂ emissions)Global CO₂ emissions decomposed into population, wealth, energy and carbon factors.