Check a physics equation with base units
Test whether both sides and every added term in an equation have the same SI base-unit dimensions.
Section 1 · Physical quantities and units
AS Level · Cambridge International Physics 9702, 2025–2027 syllabus, Topic 1. This online sequence supports learning and practice; it does not replace supervised laboratory work, full timed assessment or human subject-teacher review.
Before you begin
- Read a number with its unit and use multiplication, division, powers and right-angled triangles.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the SI base-unit form of acceleration?
What you will learn
- Express a derived quantity in base units
- Check the homogeneity of an equation using SI base units
- Recognise what dimensional consistency can and cannot prove
Every term being added must match
An equation is dimensionally homogeneous when the units on both sides match and every term that is added or subtracted has the same dimensions. Convert named derived units to base units when needed: for example, N = kg m s⁻² and J = kg m² s⁻².
A number such as 1/2 is dimensionless. A trigonometric function's argument must also be dimensionless. Units cannot be added as if they were different kinds of quantities: a length cannot be added to a speed.
Homogeneity is a check, not a proof
For v = u + at, both velocities have unit m s⁻¹ and at has unit (m s⁻²)s = m s⁻¹. For s = ut + 1/2 at², each term has unit m. Both equations pass this unit check.
A dimensionally consistent equation can still have a wrong numerical factor, sign, or physical assumption. Homogeneity is a useful necessary check; it does not establish that an equation describes the situation correctly.
Pause and explain
What does dimensional homogeneity establish?
Worked example
Test whether s = ut + 1/2 at is dimensionally homogeneous, given that s is displacement, u is velocity, a is acceleration and t is time.
Show the worked solution
- The left side s has unit m.
- ut has unit (m s⁻¹)s = m.
- at has unit (m s⁻²)s = m s⁻¹, so it cannot be added to a length.
Answer No. The final term has unit m s⁻¹, not m, so the equation is not homogeneous.
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Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Check the units in v = u + at.
Give me a hint
Replace each symbol with its SI unit.
Compare my reasoning
- v and u are velocities with unit m s⁻¹.
- a t has unit (m s⁻²)(s).
- Simplify the product.
The equation is homogeneous: every term has unit m s⁻¹.
Look for these in your work
- Both velocity terms have matching units
- Acceleration multiplied by time reduces to m s⁻¹
Check the units in E = 1/2 mv², where E is energy.
Give me a hint
Use kg for mass and m s⁻¹ for velocity.
Compare my reasoning
- m v² has unit kg × (m s⁻¹)².
- Square the velocity unit.
- Compare with J = kg m² s⁻².
kg m² s⁻² = J, so the equation is dimensionally homogeneous.
Look for these in your work
- Velocity unit squared correctly
- Energy base unit identified
A learner proposes x = vt + 1/2 at², where x is displacement and v is velocity. Does this equation pass a unit check? Explain one limit of your conclusion.
Give me a hint
Check that each term has the dimensions of length.
Compare my reasoning
- x has unit m.
- vt has unit (m s⁻¹)s = m.
- at² has unit (m s⁻²)s² = m.
Yes: every term has unit m. This only establishes dimensional consistency; it does not prove the equation's assumptions or numerical coefficients are correct.
Look for these in your work
- Each term compared with m
- Limit of dimensional analysis stated
Common mistakes
- Checking only the symbols and not their units
- Adding terms with different dimensions
- Treating a successful unit check as proof that an equation is physically correct
What can you explain now?
What is the SI base-unit form of acceleration?
Compare with the explanation
m s⁻²
Acceleration is change in velocity per time: (m s⁻¹)/s.
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Make it stick
What do you compare in a homogeneity check, and what can a successful check not tell you?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.