Seven base units for everything
The world agreed on one system — the International System of Units, or SI — so that a metre in Tokyo is exactly a metre in Paris. It rests on seven base units, and every other unit in physics is built from these.
The seven are: the metre (m, length), kilogram (kg, mass), second (s, time), ampere (A, electric current), kelvin (K, temperature), mole (mol, amount of substance), and candela (cd, luminous intensity). Almost all of mechanics — motion, forces, energy — needs only the first three: the metre, the kilogram, and the second.
Derived units: multiply and divide the base
Every remaining unit is a derived unit, built by multiplying and dividing base units. Speed is length over time, so its unit is metres per second; acceleration is speed-change over time, so its unit is metres per second squared.
Reading m/s as m·s⁻¹ makes the bookkeeping of units obvious.
Some combinations recur so often they earn their own name. Force is mass times acceleration — kg·m/s² — called the newton (N). Energy is force times distance — kg·m²/s² — the joule (J). Pressure is force per area — N/m² — the pascal (Pa).
A named unit is shorthand; expand it back to kg, m, s whenever an equation confuses you.
Prefixes and unit conversion
SI prefixes scale a unit by powers of ten: kilo (10³), milli (10⁻³), micro (10⁻⁶), nano (10⁻⁹), giga (10⁹). So 1 km = 10³ m, and 1 ms = 10⁻³ s. The prefix never changes what you are measuring, only the size of the yardstick.
The whole trick of unit conversion is to multiply by a fraction that equals 1. Because 1000 m and 1 km are the same length, the ratio (1000 m)/(1 km) equals 1 — and multiplying by 1 changes the number's clothing, never its value.
- Write the quantity with its units attached: 72\ \mathrm{km/h} = 72 \times \dfrac{\mathrm{km}}{\mathrm{h}}.
- Replace 1\ \mathrm{km} with 10^{3}\ \mathrm{m} — i.e. multiply by the ratio \tfrac{1000\,\mathrm{m}}{1\,\mathrm{km}} = 1.
- Replace 1\ \mathrm{h} with 3600\ \mathrm{s}.
- Do the arithmetic: \dfrac{72\times1000}{3600} = 20, so 72\ \mathrm{km/h} = 20\ \mathrm{m/s}.
One system, consistently applied
Before substituting numbers into any physics equation, put every quantity into base SI units — kilograms, metres, seconds. Mixing centimetres with metres, or grams with kilograms, is the single most common source of wrong answers in introductory physics.