Write "the speed is 30" and you have said almost nothing. Thirty what? Every physical quantity is a magnitude paired with a unit, and every unit, however exotic, is built from a handful of SI base units. That fact gives you a quiet superpower: you can check an equation just by checking its units.
A physical quantity has a magnitude and a unit. Seven SI base quantities, each with one base unit, generate every other unit by multiplication and division. Because both sides of a correct equation must reduce to the same base units, dimensional homogeneity is a fast first test of whether an equation can be right.
Reduce each side of an equation to SI base units and put them on a balance. If the powers of kg, m, s and A do not match, the equation is wrong, whatever the numbers say. Pick an equation, predict, then weigh it.
Five base quantities carry almost all of AS mechanics and electricity. Everything else is derived from them.
| Base quantity | Base unit | Symbol |
|---|---|---|
| mass | kilogram | kg |
| length | metre | m |
| time | second | s |
| electric current | ampere | A |
| thermodynamic temperature | kelvin | K |
A derived unit is just a product of base units. The newton is kg m s⁻², the joule is kg m² s⁻², the watt is kg m² s⁻³. Prefixes then scale the size of a unit in steps of a thousand.
| Prefix | Symbol | Factor |
|---|---|---|
| tera | T | 10¹² |
| giga | G | 10⁹ |
| mega | M | 10⁶ |
| kilo | k | 10³ |
| centi | c | 10⁻² |
| milli | m | 10⁻³ |
| micro | µ | 10⁻⁶ |
| nano | n | 10⁻⁹ |
| pico | p | 10⁻¹² |
Four quick checks on quantities, base units, homogeneity and prefixes. Each correct answer earns XP and lights this skill on your star map.
A student writes "the resistance is 4.7". What is missing for this to be a complete physical quantity?
Which of these is an SI base unit?
In the simulator, an equation balances on the dimensional beam. What does that tell you?
A capacitor is labelled 2.2 nF. What is this in farads?
Do not treat a successful homogeneity check as proof. Units cannot detect a missing pure number such as the ½ in Eₖ = ½mv² or a factor of 2π, because pure numbers have no units. A homogeneous equation can still be wrong. The check only ever does one thing with certainty: if the base units do not match, the equation cannot be correct. Also watch the kilogram, the only base unit that already carries a prefix; the base unit of mass is the kilogram, not the gram.
Unlocks once the four checks above are done. Worth more XP, written to AS Paper 1 and 2 standard.
Pressure is force per unit area. Expressed in SI base units, the pascal is:
In the equation v = k ρa Pb, v is a speed, ρ is a density and P is a pressure, while k is a dimensionless constant. Using homogeneity, the values of a and b are:
A quantity is measured as 4.0 GW. Which is the same quantity?
Which statement about checking an equation by its units is correct?
This skill is now lit gold on your star map. Keep the chain going.