Units and dimensional analysis

TL;DR:

• Multiply by conversion factors that equal 1, such as 1000 m / 1 km, and let the units cancel like fractions.

• 72 km/h is 72 × 1000 / 3600 = 20 m/s.

• A formula must have the same units on both sides. Checking units catches many wrong formulas and slips.

A car's speedometer says 72 km/h and a physics problem asks for energy, which needs the speed in metres per second. Plug in 72 and the answer is wrong by a wide margin. Units are not decoration: they are part of the number, and they catch mistakes for you if you let them.

A number without a unit is half an answer

A measurement has a number and a unit. The SI system uses base units, such as the metre (m), kilogram (kg) and second (s), and builds others from them: newton (N = kg·m/s²), joule (J = N·m) and pascal (Pa = N/m²).

A dimension is the kind of quantity: length, mass or time, written L, M and T. Speed has dimensions L/T, and it can be measured in m/s, km/h or miles per hour. The dimension stays fixed while the unit changes.

Conversion chains

To change units, multiply by a fraction that equals 1. Since 1 km = 1000 m, the fraction 1000 m / 1 km equals 1, so multiplying by it changes the unit but not the quantity. Arrange each fraction so the unit you want to remove is on the opposite side to where it already stands, and it cancels.

Worked example: 72 km/h in metres per second

1. Write the start with units: 72 km/h.

2. Replace km by metres: 72 km/h × (1000 m / 1 km). The km cancels, leaving 72,000 m/h.

3. Replace hours by seconds: 72,000 m/h × (1 h / 3600 s). The h cancels, leaving 20 m/s.

In one line: 72 km/h × (1000 m / 1 km) × (1 h / 3600 s) = 72 × 1000 / 3600 m/s = 20 m/s. A handy shortcut follows: divide km/h by 3.6 to get m/s, because 1000 / 3600 = 1/3.6.

Check by going back: 20 m/s × 3.6 = 72 km/h. Also sanity check the size: 20 metres each second is a car crossing a basketball court in about a second and a half, which is a sensible city-road speed.

Now use it. A 1000 kg car at 20 m/s has kinetic energy ½mv² = ½ × 1000 × 20² = 200,000 J. Using 72 directly would give ½ × 1000 × 72² = 2,592,000, which is not in joules at all, and is about thirteen times too large (2,592,000 ÷ 200,000 ≈ 12.96).

Squared and cubed units

Areas and volumes are the usual trap. Since 1 cm = 0.01 m, we have 1 cm² = (0.01 m)² = 10⁻⁴ m², not 10⁻² m². A cubic centimetre is 10⁻⁶ m³. And 1 g/cm³ = 1000 kg/m³: the factor 1 g = 10⁻³ kg and 1 cm³ = 10⁻⁶ m³ give 10⁻³ / 10⁻⁶ = 10³.

Checking a formula with units

Both sides of a true equation must have the same units, and you can only add quantities that share units. This lets you test a formula without knowing the physics.

• Pendulum period. Is T = 2π√(L/g) plausible? √(m ÷ m/s²) = √(s²) = s. Yes, a time.

• Kinetic energy. ½mv² gives kg·(m/s)² = kg·m²/s², which is a joule. Good.

• A wrong guess. Suppose someone writes E = mv. That has units kg·m/s, which is momentum, not energy. The units say it is wrong.

• Distance with acceleration. d = vt + ½at². Each term is m/s × s = m and m/s² × s² = m. Consistent.

Units show that a formula is possible, but they cannot show that it is right. A formula can have the right units and still be off by a number such as 2 or π, so they catch a lot of mistakes but not all.

A famous unit mix-up: In 1999 NASA lost its Mars Climate Orbiter after one team's software produced results in pound-force seconds while another expected newton seconds. A plain unit mismatch can sink a spacecraft.

How to check your answer

• Cancel on paper. Write every unit and cancel them. What is left must be the unit you wanted.

• Round trip. Convert back and see if you reach the start.

• Size check. Ask whether the number is reasonable for the real thing.

• Prefixes. milli is 10⁻³, centi is 10⁻², kilo is 10³. Write them out as numbers.

Mistakes that cost marks

• Multiplying by the conversion factor upside down, so the unit does not cancel.

• Converting hours to seconds with 60 instead of 3600.

• Squaring a length factor wrongly for areas.

• Mixing units inside a formula, such as grams with metres per second in joules.

• Dropping units in the middle and putting them back at the end by guess.

Where you will use it next

Dimensional analysis is the first line of defence in every physics and chemistry course. It pairs with measurement uncertainty and significant figures, and it keeps big calculations such as orbits from going astray.

Frequently asked questions

Can I use miles or feet in a physics formula?

Only if every other quantity uses a matching system. The safest habit is to convert everything to SI before substituting.

Are angles dimensionless?

Radians are a ratio of two lengths, so they count as dimensionless. You should still write the unit so that you know which angle mode your calculator needs.