Example problems

80 multi-step 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.

Chemistry (10)Civil, surveying & construction (1)Electricity & magnetism (4)Fluids, pumps & piping (12)Heat & thermodynamics (11)Modern physics & astronomy (2)Motion & mechanics (17)Water & wastewater treatment (6)Waves, sound & light (4)Other (13)

By course — for students following one code

Air distribution (1)
Rooftop unit supply duct check

Air handler (1)
Steam coil on an air handler: duty and condensate load

Cooling tower (4)
A year of inhibitor at design maximum · Biocide slug dose and when to re-dose · Chiller tonnage to daily makeup water · Cooling tower: the annual water and sewer bill

Cue sports (1)
The half-ball stun shot, measured

Dispersion screening (1)
How high the plume really goes

Flight mechanics (1)
Engine out: how far can you glide

Freight (1)
The freight quote, decoded

Gas absorption (1)
Sizing an absorption column

Heat exchangers (1)
Plate heat exchanger: duty, LMTD, area

Hydronics (2)
Heating main growth and the expansion loop · Water to 40% glycol: flow, speed and power

Irrigation scheduling (1)
When to irrigate next

MHF4U Grade 12 Advanced Functions (1)
Coffee cup cooling: time constant, drinking time, the cold hour

Nutrient planning (1)
From the yield check to the manure rate

Occupational noise (1)
Noise at the workstation

Piping (2)
Pipe friction from a clamp-on velocity reading · Water hammer when the valve slams

Power transmission (1)
From the belt to the key

Pump sizing (1)
Chilled-water pump: head worksheet to annual power cost

Pumps (1)
VFD turndown by the affinity laws

SCH3U Grade 11 Chemistry (5)
A gas syringe carried into the cold · Dilution of a brine stock to a mass percent · Molar mass from percent composition, then a standard solution · Stoichiometry and percent yield of a zinc reaction · Titration to molarity and mass of solute

SCH4U Grade 12 Chemistry (5)
Bomb calorimeter: glucose pellet to calorific value · Cell potential, the Nernst correction, and copper on the cathode · Hess's law to free energy to the equilibrium constant · Two kinetics runs to an activation energy and a half-life · Weak acid pH, percent ionisation, and the buffer that follows

Signal timing (1)
Timing one approach

SNC2D Grade 10 Science (3)
Converging lens: where the image lands and how tall it is · Melting snow for drinking water: sensible heat, then latent heat · Warm bath from hot and cold water: the mixing calculation

Snow shelters (1)
The quinzhee wall, rated

Source testing (1)
Stack test to fence line

SPH3U Grade 11 Physics (17)
Braking bus: stopping distance, stopping time, and the distance not travelled · Circuit energy cost: current to power to kilowatt-hours · Copper block in a water column: specific heat by mixtures · Crate on a ramp: normal force, friction, acceleration, net force · Dragging a load: work against friction, power, and the machine's efficiency · Energy and power of a falling mass · Guitar string: tension to wave speed to the fifth harmonic · Horizontal launch off a bluff: fall time, range, impact speed · Ice cube in hot water: melting heat and the energy balance · Immersion heater: water's specific heat measured electrically · Kettle on a balance: latent heat of vaporization from mass loss · Loading ramp: useful work, real work, and the efficiency · Projectile launched at an angle · Sound level at two distances: intensity, decibels, and the inverse-square law · Toboggan on the flat: energy in, friction out, distance and time · Tram between stations: accelerate, cruise, brake · Two lamps in parallel: total resistance, branch current, power, energy

SPH4C Grade 12 College Physics (1)
Rooftop tank to wash-down nozzle

SPH4C Grade 12 Physics (College) (2)
Single-pane window: film resistances and the overnight loss · Wheelbarrow as a lever: torque, advantage, and the lift

SPH4U Grade 12 Physics (12)
Atwood machine: acceleration, tension, and the first metre · Ballistic pendulum: momentum in, energy out · Banked track: from the survey stakes to the cornering force · Collision analysis: momentum kept, energy lost · Defibrillator capacitor: charge, capacitance, energy, power · Inside a solenoid: field, force on a wire, and an electron's circle · Low Earth orbit: speed, period, field strength and true weight · One wavelength, three experiments: grating, double slit, soap film · Photoelectric effect through to the electron's own wavelength · Slapshot: impulse, release speed, energy, power · Spring dart launcher: stiffness, stored energy, launch speed · Stopper on a string: period to speed to the tension

Sprayer calibration (1)
Sprayer calibration morning

Steam mains (1)
Sizing a steam main from the specific volume

Steam plant (2)
Boiler horsepower to a season of gas · Boiler horsepower to steam, blowdown and makeup

Steam systems (1)
Condensate flashing to the low-pressure header

Storm drainage (1)
The design storm through the culvert

Thermal processing (1)
The pasteurisation ladder: D, z and F

Thermodynamics (1)
Rankine cycle read off the steam tables

Well hydraulics (1)
A pump test that checks itself

Chemistry 10

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.

Bomb calorimeter: glucose pellet to calorific value

4 stepsSCH4U Grade 12 Chemistry · ThermochemistryA glucose pellet burned in a bomb calorimeter: one temperature rise becomes the heat released, the calorific value per gram, and the molar enthalpy.

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.

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.

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.

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.

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.

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.

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.

Civil, surveying & construction 1

The design storm through the culvert

3 stepsStorm drainage · sizing check on an existing culvertA 20-minute design storm through a paved catchment: IDF intensity, Rational Method peak, and critical depth that decides whether the culvert chokes.

Electricity & magnetism 4

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.

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.

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.

Fluids, pumps & piping 12

Boiler horsepower to a season of gas

5 stepsSteam plant · boiler horsepower, steam rate and fuel costA 150 hp firetube boiler priced from the nameplate: heat output, steam rate, fuel input at 80% efficiency, and a heating season of gas in therms.

Chilled-water pump: head worksheet to annual power cost

5 stepsPump sizing · flow, head, power and operating costOne chilled-water pump from the head worksheet to the hydro bill: total dynamic head, water power, shaft power at 70%, and a year of kilowatt-hours.

Cooling tower: the annual water and sewer bill

7 stepsCooling tower · water balance priced at utility ratesThe tower bill nobody itemises: evaporation, blowdown at four cycles, drift, makeup, a year of volumes, and the net cost with the sewer credit.

Heating main growth and the expansion loop

4 stepsHydronics · thermal expansion of pipe and waterA 60 m steel heating main from cold fill to 90 °C: how far it grows, the loop leg that absorbs the growth, and the water the tank must accept.

Pipe friction from a clamp-on velocity reading

6 stepsPiping · Reynolds number, friction factor and system headA clamp-on meter reads 2.0 m/s in a 100 mm riser main: flow, Reynolds number, turbulent friction factor, the loss over 90 m, and the pump's head.

Rooftop tank to wash-down nozzle

4 stepsSPH4C Grade 12 College Physics · Hydraulic and Pneumatic SystemsA rooftop tank feeds a wash-down nozzle 12 m below: static pressure at the valve, jet speed, the flow the nozzle passes, and the jet's power.

Rooftop unit supply duct check

4 stepsAir distribution · duct velocity, equivalent round and velocity pressureOne rooftop unit supply duct: face velocity from CFM and sheet-metal size, the equivalent round diameter, velocity pressure, and air changes.

Sizing a steam main from the specific volume

4 stepsSteam mains · specific volume, velocity and pipe size5,000 lb/h through a 3-inch steam main: velocity from the specific volume, the bore the trade band allows, and why 15 psig needs a bigger pipe.

Steam coil on an air handler: duty and condensate load

3 stepsAir handler · steam coil duty and condensate loadA preheat coil lifting 8,000 cfm of winter air by 50 F°: the duty by the 1.08 rule, the steam it takes at 5 psig, and the coil in boiler horsepower.

VFD turndown by the affinity laws

5 stepsPumps · affinity laws and variable-speed savingsA VFD slows a condenser pump from 1,800 to 1,200 rpm: the new flow, head and power by the affinity laws, and what the cube law saves in a year.

Water hammer when the valve slams

4 stepsPiping · surge pressure and pipe pressure ratingA fast-closing valve on a 150 mm pump main: the flow velocity, the Joukowsky surge when it slams, that surge as head, and the pipe's Barlow rating.

Water to 40% glycol: flow, speed and power

4 stepsHydronics · glycol derate and the affinity lawsConverting a 120 gpm heating loop to 40% propylene glycol: the load it carries, the flow the glycol needs, and what the pump speed-up costs in power.

Heat & thermodynamics 11

Coffee cup cooling: time constant, drinking time, the cold hour

3 stepsMHF4U Grade 12 Advanced Functions · Exponential ModelsA cooling mug of coffee timed against Newton's law: one reading fixes the time constant, which then predicts drinking time and the cold hour ahead.

Condensate flashing to the low-pressure header

3 stepsSteam systems · flash recoveryCondensate let down from 150 to 15 psig: the flash percentage off the tables, the steam it releases, and what that is worth in boiler horsepower.

Copper block in a water column: specific heat by mixtures

4 stepsSPH3U Grade 11 Physics · Energy and SocietyThe classic method-of-mixtures lab: a boiled copper cube dropped into a measured water column, and two thermometer readings give its specific heat.

Ice cube in hot water: melting heat and the energy balance

4 stepsSPH3U Grade 11 Physics · Energy and SocietyAn ice cube melts in a cup of hot water: the heat required to melt it, the minimum hot water that could supply it, and the energy balance of the real mixture.

Immersion heater: water's specific heat measured electrically

5 stepsSPH3U Grade 11 Physics · Energy and SocietyAn immersion heater warms a measured cup of water: volts times amps times seconds against the temperature rise gives water's specific heat capacity.

Kettle on a balance: latent heat of vaporization from mass loss

3 stepsSPH3U Grade 11 Physics · Energy and SocietyA kettle boils on a balance: the electricity metered in, divided by the mass that left as steam, measures water's latent heat of vaporization.

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.

Plate heat exchanger: duty, LMTD, area

3 stepsHeat exchangers · stream duty, LMTD and surface areaSizing a plate heat exchanger: duty from one stream's flow and ΔT, the log-mean temperature difference across it, and the area that duty demands.

Rankine cycle read off the steam tables

4 stepsThermodynamics · the Rankine cycle from the steam tablesFour state points off the tables for a 4 MPa steam cycle: the real exhaust from an 85 % turbine, its specific work, cycle efficiency and shaft power.

Single-pane window: film resistances and the overnight loss

4 stepsSPH4C Grade 12 Physics (College) · Energy TransformationsA single-glazed window on a winter night: the pane's tiny R-value, the air films that actually resist, the wattage, and the megajoules lost by dawn.

Warm bath from hot and cold water: the mixing calculation

3 stepsSNC2D Grade 10 Science · Climate ChangeMixing hot reservoir water with cold tap water to hit a target bath temperature: an energy balance gives the cold mass, then a heater-time cross-check.

Modern physics & astronomy 2

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.

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.

Motion & mechanics 17

Atwood machine: acceleration, tension, and the first metre

4 stepsSPH4U Grade 12 Physics · DynamicsTwo masses on a string over a pulley: the system's acceleration, the tension that must exceed one weight, and the speed and time over a 90 cm drop.

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.

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.

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.

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.

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.

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.

Horizontal launch off a bluff: fall time, range, impact speed

4 stepsSPH3U Grade 11 Physics · KinematicsA ball fired horizontally off a bluff: the fall time set by height alone, the distance out, the vertical speed gained, and the speed at the water.

Loading ramp: useful work, real work, and the efficiency

4 stepsSPH3U Grade 11 Physics · Energy and SocietyA crate hauled up a ramp instead of lifted: useful work, work actually paid, the ramp's efficiency, and the pull a frictionless ramp would need.

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.

Slapshot: impulse, release speed, energy, power

4 stepsSPH4U Grade 12 Physics · Energy and MomentumA stick meets a stationary puck: the impulse from force and contact time, the release speed it buys, the kinetic energy, and the power of the contact.

Spring dart launcher: stiffness, stored energy, launch speed

4 stepsSPH4U Grade 12 Physics · Energy and MomentumA toy dart launcher worked from spring to flight: stiffness from one measurement, the energy stored, the launch speed, and the height that buys.

Stopper on a string: period to speed to the tension

4 stepsSPH4U Grade 12 Physics · DynamicsA rubber stopper whirled on a cord: period to speed, speed to centripetal acceleration, then the cord tension — and how it dwarfs the weight.

Toboggan on the flat: energy in, friction out, distance and time

6 stepsSPH3U Grade 11 Physics · Energy and SocietyA toboggan coasts onto level snow: its kinetic energy, the friction force under it, the stopping distance the work-energy theorem fixes, and the time.

Tram between stations: accelerate, cruise, brake

5 stepsSPH3U Grade 11 Physics · KinematicsA tram pulls away, cruises, and brakes: each phase timed and measured with the one kinematic equation that fits it, no quadratics needed.

Wheelbarrow as a lever: torque, advantage, and the lift

4 stepsSPH4C Grade 12 Physics (College) · Mechanical SystemsA loaded wheelbarrow treated as a lever: the load's weight, its torque about the axle, the mechanical advantage, and the lift the handles demand.

Water & wastewater treatment 6

A pump test that checks itself

3 stepsWell hydraulics · Cooper–Jacob with its own auditTwelve hours into a pump test: the drawdown the old report predicts, the transmissivity two wells actually measure, and the method's validity check.

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.

Boiler horsepower to steam, blowdown and makeup

4 stepsSteam plant · steam rate, cycles, blowdown and makeupA 150 BHP firetube boiler at five cycles: steam rate off the nameplate, the cycles the lab samples imply, the blowdown, and the makeup to treat.

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.

The pasteurisation ladder: D, z and F

3 stepsThermal processing · D, z, F on an HTST lineThermal kill's three constants in order: a D-value from a lab run, a z-value from two temperatures, and the F-value verdict on a 0.3-degree drift.

Waves, sound & light 4

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.

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.

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 use it three ways: its frequency, the fringes at a double slit, and the film that reflects it best.

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.