IB Physics HLTool 3 — MathematicsPractical Skillschecking equations with units~12 min read
Dimensional Analysis
Here’s a quiet superpower most students underuse: you can check whether an equation is even possible just by looking at its units. If the units on the left don’t match the units on the right, the equation is wrong — no matter how neat the algebra looked. This is dimensional analysis, and it catches mistakes before they cost you marks.
📚 What you need to know
Dimensional analysis checks the homogeneity of an equation using SI base units
The units on both sides of a correct equation must be the same
Break every quantity down into base units (kg, m, s, A, K, mol)
If the units don’t match, the equation needs adjusting
You can also use it to find the units of an unknown constant
A pure number or ratio (like an angle or a constant) is dimensionless
The core idea
Every term in a valid physics equation must reduce to the same combination of base units. This has to be true — you can’t add metres to seconds any more than you can add apples to hours. So if you break both sides down and they don’t match, something is broken.
⚛ Checking an equation’s homogeneity
Write down the units of each quantity in the equation.
Reduce each to SI base units (kg, m, s, A…).
Compare the left-hand side with the right-hand side.
If they match, the equation is homogeneous (dimensionally valid).
If they don’t, the equation is wrong and needs adjusting.
Each term of v = u + at reduces to m s⁻¹, so the equation is dimensionally consistent.
WE 1
Show that the equation v = u + at is homogeneous.
Step 1 — units of each termv and u are speeds: m s−1Step 2 — the at termat = (m s−2) × (s) = m s−1Step 3 — compare
All three terms are m s−1Homogeneous — the equation is dimensionally validThe seconds in acceleration and time cancel to leave a speed. When every term shares the same units, the equation passes the check.
Spotting a mistake
The real value shows up when an equation is wrong. Suppose you mis-remembered kinetic energy as E = mv (no square, no half). Energy has units kg m2 s−2, but mv gives kg m s−1. They don’t match — so the equation must be wrong, and you’d know instantly to look again.
This is the single best habit to build for calculation questions. After you rearrange an equation, run the units through it in your head. If the left comes out in one combination and the right in another, you’ve slipped up somewhere — and you’ve caught it in ten seconds, before touching the numbers. Examiners love awarding the marks you’d otherwise have thrown away.
Finding the units of an unknown
Dimensional analysis works in reverse too. If you know an equation but not the units of a constant inside it, just rearrange for that constant and reduce everything to base units.
WE 2
A drag force is modelled as F = kv, where v is speed. Find the SI base units of the constant k.
Step 1 — rearrange for kk = F / vStep 2 — substitute base unitsk = (kg m s−2) / (m s−1) = kg s−1k has units kg s−1The metres cancel and the powers of seconds subtract (−2 − (−1) = −1). No memorising needed — just careful bookkeeping of the base units.
WE 3
The period of a pendulum is T = 2π√(L/g). Show the right-hand side has units of seconds, so it correctly gives a time.
Step 1 — inside the rootL/g = m / (m s−2) = s2Step 2 — take the square root
√(s2) = sStep 3 — the 2π is dimensionless
so the whole right side has units of secondsRHS = s, matching the period T — consistentThe metres cancel inside the root, leaving s²; the root turns that into s. Constants like 2π carry no units, so they never affect the check.
💡 Top tips
Reduce every term to base units before comparing.
Pure numbers and ratios (angles, 2π, constants) are dimensionless.
Powers of the same base unit add and subtract when you multiply/divide.
Use it to check rearranged equations and to find unknown units.
Matching units doesn’t prove an equation is right — but mismatched units prove it’s wrong.
⚠ Common mistakes
Forgetting to reduce derived units (N, J) to base units first
Mishandling powers when a quantity is squared or rooted
Treating a dimensionless constant as if it had units
Assuming matching units means the equation is definitely correct
Dropping a unit partway through the reduction
Quick recap: Dimensional analysis checks that both sides of an equation reduce to the same base units. If they match, the equation is homogeneous; if they don’t, it’s wrong. Reduce every quantity to kg, m, s (etc.), remember constants are dimensionless, and use the method to check equations or find the units of an unknown.
Checking units keeps your equations honest. The next essential skill keeps your measurements honest — because no reading is ever exact, and physics demands you state how much it could be off by. That’s the world of Handling Uncertainties, coming up next.
Unit-checking not clicking yet?
Book a free meeting and we’ll practise reducing equations to base units and using dimensional analysis to catch errors — a fast way to protect marks across every calculation topic.
