Salam Academy
Sign inCreate account
Contents · 0% complete
  1. Learning objectives
  2. Core idea
  3. Key formulae
  4. Worked examples
  5. Explore the graph
  6. Common mistakes
  7. Exam tip
  8. Guided practice
  9. Practice questions
  10. Challenge
  11. Summary
  12. Quick quiz
  13. Same topic, different board
  14. Past paper questions
Learn7367Paper 3 · Mechanics optionDimensional analysisSI units, prefixes and estimates
Dimensional analysis · Subtopic 1 of 1

SI Units, Prefixes and Estimates

Every quantity has a magnitude and a unit. All SI units are built from six base units (kg, m, s, A, K, mol) used in this course. Writing a derived unit in base units lets you check whether an equation is homogeneous. Prefixes and powers of ten keep numbers manageable, and good physicists can estimate any everyday quantity to the right order of magnitude.

AQA 7367 Paper 3Cambridge 9702 Paper 1Cambridge 9702 Paper 2Edexcel YPH11 WPH11AQA 7408 Paper 1≈ 30 minRevision 5 · updated 2026-10-27
Practise this topicTimed testAsk Salam Tutor
Section 1 of 14

Learning objectives

  • Recall the SI base quantities and their units.
  • Express derived units in base units.
  • Use base units to check that an equation is homogeneous.
  • Use prefixes from pico to tera, and make sensible estimates.
Pause and think

A student writes E=mvE=mv for kinetic energy. Without remembering the correct formula, how can you tell it must be wrong?

Section 2 of 14

Core idea

Start hereCatching a wrong formula without solving anything

Suppose you half-remember a formula as s=ut+ats=ut+at. Before plugging in numbers, check the units: utut is metres, but atat is metres per second. You can't add metres to metres per second, so the formula is wrong. Units are a free error-detector.

The big idea

Quantities = number × unit. Six SI base units: kg, m, s, A, K, mol. A valid equation is homogeneous: every term has the same base units.

1. Base and derived units

  • Base: kg, m, s, A, K, mol.
  • Derived: N == kg m s−2^{-2}; J == kg m2^2 s−2^{-2}; W == J s−1^{-1}; C == A s; V == J C−1^{-1}.
Building derived units from defining equations
  1. F=ma⇒N=kg m s−2F=ma\Rightarrow\text{N}=\text{kg m s}^{-2}
    Mass × acceleration.
  2. W=Fs⇒J=kg m2s−2W=Fs\Rightarrow\text{J}=\text{kg m}^2\text{s}^{-2}
    Force × distance.
  3. p=FA⇒Pa=kg m−1s−2p=\frac FA\Rightarrow\text{Pa}=\text{kg m}^{-1}\text{s}^{-2}
    Divide by m².
Worked exampleUnits of a drag constant

In F=kv2F=kv^2, find the base units of kk.

Goal
Units of kk.
Strategy
Make kk the subject; substitute base units.
  1. k=Fv2k=\frac F{v^2}
    Rearrange.
  2. kg m s−2m2s−2=kg m−1\frac{\text{kg m s}^{-2}}{\text{m}^2\text{s}^{-2}}=\text{kg m}^{-1}
    Cancel s−2^{-2} and one m.
Answerkg m−1^{-1}.
✓ Checkk×v2k\times v^2: kg m−1^{-1} × m² s−2^{-2} = kg m s−2^{-2} = N ✓.

2. Homogeneity

Worked exampleCheck s=ut+12at2s=ut+\frac12at^2

Is it homogeneous?

Goal
Yes/no.
Strategy
Units of each term.
  1. s: ms:\ \text{m}
    Left side.
  2. ut: m s−1⋅s=mut:\ \text{m s}^{-1}\cdot\text{s}=\text{m}
    First term.
  3. at2: m s−2⋅s2=mat^2:\ \text{m s}^{-2}\cdot\text{s}^2=\text{m}
    Second term.
AnswerHomogeneous.
✓ CheckThe 12\frac12 is unitless: homogeneity can't confirm it ✓.
Edge caseHomogeneous but wrong s=ut+at2s=ut+at^2 is homogeneous yet wrong: units can't check pure numbers.

3. Prefixes and estimates

Prefixes: each step on this scale is ×1000
−10−55100.511.522.530power of tenpnμm(unit)kMGTnano: light wavelengthsgiga: radio frequencies
1 nm=10−91\text{ nm}=10^{-9} m; 1 GHz=1091\text{ GHz}=10^{9} Hz. Also centi (10−210^{-2}) and deci (10−110^{-1}).
  • 450450 nm =4.5×10−7=4.5\times10^{-7} m; 2.42.4 GHz =2.4×109=2.4\times10^9 Hz.
  • Estimates: adult ~70 kg; room height ~3 m; walking ~1.5 m s−1^{-1}; lamp ~10–100 W.
Common misconceptionPrefixes on squared units 11 mm² =(10−3)2=(10^{-3})^2 m² =10−6=10^{-6} m², not 10−310^{-3}.
Pause and think

Convert 5 mm² to m².

Summary

  • Six base units.
  • Homogeneity (can't check numbers).
  • Square/cube prefixes with the unit.
Key words
Homogeneous
Same base units on both sides.
Base unit
One of the six SI fundamental units.
HistoryThe 2019 redefinition. Since May 20192019, all SI base units are defined by fixing the values of fundamental constants: the speed of light, Planck's constant, the elementary charge, the Boltzmann constant and the Avogadro constant. The kilogram was the last unit to be defined by a physical object.
Fun factFermi's piano tuners. Enrico Fermi loved asking students: "How many piano tuners are there in Chicago?" Breaking it into estimates (population, pianos per household, tunings per year, tunings per tuner per day) gets surprisingly close. These are now called Fermi problems.
Section 3 of 14

Key formulae

Newton1 N=1 kg m s−21\ \text{N}=1\ \text{kg m s}^{-2}
Joule and watt1 J=1 kg m2 s−2,1 W=1 kg m2 s−31\ \text{J}=1\ \text{kg m}^2\,\text{s}^{-2},\quad1\ \text{W}=1\ \text{kg m}^2\,\text{s}^{-3}
Pascal1 Pa=1 N m−2=1 kg m−1 s−21\ \text{Pa}=1\ \text{N m}^{-2}=1\ \text{kg m}^{-1}\,\text{s}^{-2}
TrickBuild units from formulae you know. You don't need to memorise base units for every quantity. Pick a simple defining equation and substitute: pressure =FA→kg m s−2m2=kg m−1s−2=\dfrac FA\to\dfrac{\text{kg m s}^{-2}}{\text{m}^2}=\text{kg m}^{-1}\text{s}^{-2}.
Section 4 of 14

Worked examples

Example 1 · Units of a constant0 / 2 steps
Drag force F=kv2F=kv^2. Find the SI base units of kk.
    Example 2 · Finding powers0 / 2 steps
    The speed of a wave on a string is v=Txμyv=T^x\mu^y, where TT is tension and μ\mu is mass per unit length. Find xx and yy.
      Section 5 of 14

      Explore the graph

      Powers of ten
      Loading the interactive graph…
      Try this
      1. The axis is the power of ten of a length in metres. The fixed points mark a nucleus (10−1510^{-15}), an atom (10−1010^{-10}), a person (10010^0), the Earth (10710^7), the Sun (10910^9) and our galaxy (102110^{21}).
      2. Move kk: the row shows the length 10k10^k m. How many powers of ten separate an atom from its nucleus?
      3. Which prefix would you use for 10−910^{-9} m? For 10910^9 W?

      Physics covers more than 3636 orders of magnitude in length alone.

      In the real worldEngineers check units first. Spreadsheets and simulation software used in engineering often track units automatically. A homogeneity check is the quickest way to catch a formula typed in wrongly before it costs time or money.
      Section 6 of 14

      Common mistakes

      IncorrectNewton is a base unit
      CorrectIt's derived: kg m s⁻²
      Only six base units are used here.
      IncorrectHomogeneous means correct
      CorrectIt's necessary, not sufficient
      Numerical factors can still be wrong.
      Incorrectμ\muA =10−3=10^{-3} A
      Correct10−610^{-6} A
      m is milli, μ is micro.
      IncorrectWriting "kg/m/s²"
      Correctkg m⁻¹ s⁻²
      Use negative indices.
      Me

      what's the SI base unit of force?

      Newton

      I'm derived

      kg m s−2\text{kg m s}^{-2}

      we were here first

      Section 7 of 14

      Exam tip

      Exam tipShow the substitution. For "express in base units", write the defining equation, substitute units in brackets and simplify step by step. A bare final answer may lose method marks if it's wrong.
      Section 8 of 14

      Guided practice

      SA-9702-PQ1-0011 markL1 FoundationCalculatorSI units
      The newton expressed in SI base units is
      Choose one option

      Sign in or create a free account to check answers, get hints and track your progress.

      Salam Academy Original Practice · Proposed mark scheme
      Section 9 of 14

      Practice questions

      SA-9702-PQ1-0021 markL1 FoundationCalculatorSI units
      Which is not an SI base unit?
      Choose one option

      Sign in or create a free account to check answers, get hints and track your progress.

      Salam Academy Original Practice · Proposed mark scheme
      SA-9702-PQ1-0031 markL1 FoundationCalculatorPrefixes & estimates
      Convert a current of 250250 μA into amperes.

      Sign in or create a free account to check answers, get hints and track your progress.

      Salam Academy Original Practice · Proposed mark scheme
      SA-9702-PQ1-0041 markL1 FoundationCalculatorPrefixes & estimates
      Convert 7272 km h⁻¹ into m s⁻¹.

      Sign in or create a free account to check answers, get hints and track your progress.

      Salam Academy Original Practice · Proposed mark scheme
      SA-9702-PQ1-0051 markL2 StandardCalculatorSI units
      The pascal expressed in SI base units is
      Choose one option

      Sign in or create a free account to check answers, get hints and track your progress.

      Salam Academy Original Practice · Proposed mark scheme
      SA-9702-PQ1-0061 markL2 StandardCalculatorSI units
      Which equation is not homogeneous? (ss displacement, uu, vv velocities, aa acceleration, tt time)
      Choose one option

      Sign in or create a free account to check answers, get hints and track your progress.

      Salam Academy Original Practice · Proposed mark scheme
      SA-9702-PQ1-0071 markL1 FoundationCalculatorPrefixes & estimates
      Which is the best estimate of the power of a domestic electric kettle?
      Choose one option

      Sign in or create a free account to check answers, get hints and track your progress.

      Salam Academy Original Practice · Proposed mark scheme
      SA-9702-PQ1-0081 markL2 StandardCalculatorSI units
      The drag force on a car is F=kv2F=kv^2. The SI base units of kk are
      Choose one option

      Sign in or create a free account to check answers, get hints and track your progress.

      Salam Academy Original Practice · Proposed mark scheme
      SA-9702-PQ1-0092 marksL2 StandardCalculatorPrefixes & estimates
      A Wi-Fi signal has a frequency of 5.05.0 GHz. Calculate its wavelength. (c=3.00×108c=3.00\times10^8 m s⁻¹)

      Sign in or create a free account to check answers, get hints and track your progress.

      Salam Academy Original Practice · Proposed mark scheme
      SA-9702-PQ1-0101 markL1 FoundationCalculatorPrefixes & estimates
      Calculate the number of seconds in one year (365365 days), in units of 10710^7.

      Sign in or create a free account to check answers, get hints and track your progress.

      Salam Academy Original Practice · Proposed mark scheme

      More questions on every skill in this subtopic are in the question bank under Practise.

      Section 10 of 14

      Challenge

      SA-9702-PQ1-0113 marksL3 ExamCalculatorSI units
      The speed of a transverse wave on a stretched string is v=Txμyv=T^x\mu^y, where TT is the tension and μ\mu is the mass per unit length. Use base units to determine xx.

      Sign in or create a free account to check answers, get hints and track your progress.

      Salam Academy Original Practice · Proposed mark scheme
      Section 11 of 14

      Summary

      • Base units: kg, m, s, A, K, mol.
      • Derived units from defining equations: N == kg m s⁻².
      • Correct equations are homogeneous (but homogeneous isn't always correct).
      • Prefixes p to T; estimate using known anchors.
      Section 12 of 14

      Quick quiz

      Q1 · SA-9702-PQ1-0121 markL1 FoundationCalculatorPrefixes & estimates
      The prefix "n" (nano) represents
      Choose one option

      Sign in or create a free account to check answers, get hints and track your progress.

      Salam Academy Original Practice · Proposed mark scheme
      Section 13 of 14

      Same topic, different board

      Specification
      A-level Physics 7408
      Where
      3.1.1 Use of SI units and their prefixes; 3.1.3 Estimation of physical quantities
      Expected
      Fundamental (base) units, prefixes, and orders-of-magnitude estimates.
      Calculator
      Calculator allowed; a data and formulae booklet is provided.
      Typical style
      Papers mix multiple choice, short answer and extended response.
      Open in
      7367 · Paper 37408 · Paper 1
      Section 14 of 14

      Past paper questions

      5 questions (43 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

      1. 9702/22 · Oct/Nov 2025Question 111 marks

        Units, scalars and vectors, upthrust on a balloon, and combining percentage uncertainties.

        Question paper, page 4Mark scheme
      2. 9702/22 · Oct/Nov 2024Question 17 marks

        Vectors, SI base units from a drag formula, and forces at terminal velocity.

        Question paper, page 3Mark scheme
      3. 9702/22 · May/June 2024Question 110 marks

        SI base quantities, intensity, percentage uncertainty and efficiency.

        Question paper, page 4Mark scheme
      4. 9702/22 · Oct/Nov 2023Question 19 marks

        SI base quantities, uniform acceleration and a velocity–time graph.

        Question paper, page 5Mark scheme
      5. 9702/22 · May/June 2023Question 16 marks

        Pressure, SI base units of a constant, and percentage uncertainty.

        Question paper, page 3Mark scheme

      All 9702 past paper questions by topic

      Salam Academy original content · Content status: reviewed and published. Syllabus mapping: AQA 7367, First teaching 2017 (official specification).