Example problems
One answer becomes the next input. Every step names the formula that produced it, so you can follow the reasoning or jump into any solver and change the numbers.
Curriculum
Teaching sequences that follow how a course actually builds a solution.
Energy and power of a falling mass
3 stepsSPH3U Grade 11 Physics · Energy and SocietyTrack one drop from height to impact: gravitational energy becomes kinetic energy, then average power over the time of the fall.
Projectile launched at an angle
4 stepsSPH3U Grade 11 Physics · KinematicsFollow one angled launch all the way through: hang time, then apex height, then the constant horizontal speed that sets the range.
Collision analysis: momentum kept, energy lost
4 stepsSPH4U Grade 12 Physics · Energy and MomentumTwo carts lock together on a level track: momentum before, energy before, the shared speed afterwards, and the kinetic energy that survives.
Titration to molarity and mass of solute
4 stepsSCH3U Grade 11 Chemistry · Solutions and SolubilityTurn a burette reading into an answer: moles of titrant delivered, then the unknown acid concentration, then the mass of solute in the flask.
Circuit energy cost: current to power to kilowatt-hours
4 stepsSPH3U Grade 11 Physics · Electricity and MagnetismFrom a supply voltage and two series coils to the current drawn, the power dissipated, and the energy a shop heater burns over an evening.
Converging lens: where the image lands and how tall it is
3 stepsSNC2D Grade 10 Science · Light and Geometric OpticsOne object on an optical bench: focal length and object distance give the image distance, then the magnification, then the image height.
Melting snow for drinking water: sensible heat, then latent heat
4 stepsSNC2D Grade 10 Science · Climate ChangeA winter camp stove warms snow to the melting point, then melts it: two heat calculations and the two very different times they take.
Braking bus: stopping distance, stopping time, and the distance not travelled
3 stepsSPH3U Grade 11 Physics · KinematicsTwo kinematic equations on one stop: the time-free equation gives the braking distance, the average-velocity equation gives the time.
Crate on a ramp: normal force, friction, acceleration, net force
6 stepsSPH3U Grade 11 Physics · ForcesOne free-body diagram taken all the way through: weight, the two components on the slope, friction, the acceleration, and the net force.
Dragging a load: work against friction, power, and the machine's efficiency
6 stepsSPH3U Grade 11 Physics · Energy and SocietyA winch drags a sled over frozen ground: friction force, the work it costs, the power delivered, and how much of the motor's energy was wasted.
Guitar string: tension to wave speed to the fifth harmonic
4 stepsSPH3U Grade 11 Physics · Waves and SoundOne string, four steps: tension and linear density set the wave speed, the length sets the fundamental, and the fifth harmonic follows.
Sound level at two distances: intensity, decibels, and the inverse-square law
4 stepsSPH3U Grade 11 Physics · Waves and SoundA meter reading near a rink speaker gives its acoustic power, its decibel level, and then the intensity and level five times farther away.
Two lamps in parallel: total resistance, branch current, power, energy
5 stepsSPH3U Grade 11 Physics · Electricity and MagnetismA 12 V trailer circuit worked end to end: the parallel total, the current it draws, the split into branches, and the energy one lamp burns.
Banked track: from the survey stakes to the cornering force
4 stepsSPH4U Grade 12 Physics · DynamicsTurn a track's measured rise and run into a bank angle, then the frictionless design speed, the centripetal acceleration, and the force on the car.
Ballistic pendulum: momentum in, energy out
4 stepsSPH4U Grade 12 Physics · Energy and MomentumFollow a projectile into a hanging block: the energy it arrives with, the shared speed after impact, the energy left, and the height it lifts.
Low Earth orbit: speed, period, field strength and true weight
4 stepsSPH4U Grade 12 Physics · Gravitational, Electric and Magnetic FieldsOne orbital radius sets everything else: the orbital speed, the period, the gravitational field strength up there, and the satellite's real weight.
Defibrillator capacitor: charge, capacitance, energy, power
4 stepsSPH4U Grade 12 Physics · Gravitational, Electric and Magnetic FieldsCharge a defibrillator capacitor and then dump it: the charge delivered, the capacitance it implies, the energy stored, and the power of a 4 ms shock.
Inside a solenoid: field, force on a wire, and an electron's circle
4 stepsSPH4U Grade 12 Physics · Gravitational, Electric and Magnetic FieldsBuild a field from turns and current, weigh it with a probe wire, then fire an electron across it and find the radius of the circle it is bent into.
One wavelength, three experiments: grating, double slit, soap film
4 stepsSPH4U Grade 12 Physics · The Wave Nature of LightMeasure a laser's wavelength on a grating, then spend it three ways: its frequency, the fringes at a double slit, and the film that reflects it best.
Photoelectric effect through to the electron's own wavelength
4 stepsSPH4U Grade 12 Physics · Revolutions in Modern PhysicsUltraviolet light on a metal: the photon's energy, the kinetic energy the freed electron keeps, the speed that implies, and its de Broglie wavelength.
Stoichiometry and percent yield of a zinc reaction
4 stepsSCH3U Grade 11 Chemistry · Quantities in Chemical ReactionsOne weighed strip of zinc taken all the way through: moles of metal, litres of hydrogen, the mass of salt predicted, and the yield actually reached.
Molar mass from percent composition, then a standard solution
4 stepsSCH3U Grade 11 Chemistry · Quantities in Chemical ReactionsAn elemental analysis becomes a molar mass, the molar mass becomes moles, and the moles become the concentration of the flask they were dissolved in.
Dilution of a brine stock to a mass percent
4 stepsSCH3U Grade 11 Chemistry · Solutions and SolubilityA pipetted portion of concentrated brine, diluted to the mark: the new concentration, the moles it holds, the mass of salt, and the percent by mass.
A gas syringe carried into the cold
4 stepsSCH3U Grade 11 Chemistry · Gases and Atmospheric ChemistryOne sealed sample of carbon dioxide under two sets of conditions: the new volume, the moles it always held, and the mass on the balance.
Hess's law to free energy to the equilibrium constant
3 stepsSCH4U Grade 12 Chemistry · Energy Changes and Rates of ReactionThree combustion enthalpies are summed into a reaction nobody can run in a calorimeter, then turned into a free energy and the constant it fixes.
Two kinetics runs to an activation energy and a half-life
5 stepsSCH4U Grade 12 Chemistry · Energy Changes and Rates of ReactionTwo timed decolourisation runs give two rate constants, the pair gives the activation energy, and that barrier predicts a half-life at 60 °C.
Weak acid pH, percent ionisation, and the buffer that follows
5 stepsSCH4U Grade 12 Chemistry · Chemical Systems and EquilibriumOne weak acid worked four ways: its pKa, the pH it reaches alone, the fraction actually ionised, and the pH once its conjugate base is added.
Cell potential, the Nernst correction, and copper on the cathode
4 stepsSCH4U Grade 12 Chemistry · ElectrochemistryA copper–zinc cell from tabulated half-cells, corrected for real concentrations, then the charge it passes and the metal that charge deposits.
Trade workflows
The order a job gets done in — each figure feeding the decision after it.
Chiller tonnage to daily makeup water
7 stepsCooling tower · water balance and chemical treatmentOne 250 ton chiller through its tower: heat rejection, condenser flow, evaporation, blowdown, drift, and the makeup water the building buys each day.
A year of inhibitor at design maximum
6 stepsCooling tower · chemical consumption and contract sizingSize a year of scale inhibitor the way a contract should be written: on the blowdown and drift that carry chemical out, never on the makeup.
Biocide slug dose and when to re-dose
6 stepsCooling tower · biocide dosing and holding timeA one-shot biocide slug on a cooling tower: system volume, product mass for a target active ppm, and the holding time index that sets the next dose.