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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

Learn9702Paper 2 · AS Level Structured QuestionsPhysical quantities and unitsSI units, prefixes and estimates

Physical quantities and units · Subtopic 1 of 3

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.

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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 $E=mv$ 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 $s=ut+at$ . Before plugging in numbers, check the units: $ut$ is metres, but $at$ 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 $^{-2}$ ; J $=$ kg m $^2$ s $^{-2}$ ; W $=$ J s $^{-1}$ ; C $=$ A s; V $=$ J C $^{-1}$ .

Building derived units from defining equations

$F=ma\Rightarrow\text{N}=\text{kg m s}^{-2}$
Mass × acceleration.
$W=Fs\Rightarrow\text{J}=\text{kg m}^2\text{s}^{-2}$
Force × distance.
$p=\frac FA\Rightarrow\text{Pa}=\text{kg m}^{-1}\text{s}^{-2}$
Divide by m².

Worked exampleUnits of a drag constant

In $F=kv^2$ , find the base units of $k$ .

Goal: Units of $k$ .
Strategy: Make $k$ the subject; substitute base units.

$k=\frac F{v^2}$
Rearrange.
$\frac{\text{kg m s}^{-2}}{\text{m}^2\text{s}^{-2}}=\text{kg m}^{-1}$
Cancel s $^{-2}$ and one m.

Answerkg m

^{-1}

✓ Check

k\times v^2

: kg m

^{-1}

× m² s

^{-2}

= kg m s

^{-2}

= N ✓.

2. Homogeneity

Worked exampleCheck $s=ut+\frac12at^2$

Is it homogeneous?

Goal: Yes/no.
Strategy: Units of each term.

$s:\ \text{m}$
Left side.
$ut:\ \text{m s}^{-1}\cdot\text{s}=\text{m}$
First term.
$at^2:\ \text{m s}^{-2}\cdot\text{s}^2=\text{m}$
Second term.

AnswerHomogeneous.

✓ CheckThe

\frac12

is unitless: homogeneity can't confirm it ✓.

Edge caseHomogeneous but wrong

s=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

1\text{ nm}=10^{-9}

1\text{ GHz}=10^{9}

Hz. Also centi (

10^{-2}

) and deci (

10^{-1}

$450$ nm $=4.5\times10^{-7}$ m; $2.4$ GHz $=2.4\times10^9$ Hz.
Estimates: adult ~70 kg; room height ~3 m; walking ~1.5 m s $^{-1}$ ; lamp ~10–100 W.

Common misconceptionPrefixes on squared units

1

mm²

=(10^{-3})^2

m²

=10^{-6}

m², not

10^{-3}

Pause and think

Convert 5 mm² to m².

Summary

Six base units.
Homogeneity (can't check numbers).
Square/cube prefixes with the unit.

Homogeneous: Same base units on both sides.
Base unit: One of the six SI fundamental units.

HistoryThe 2019 redefinition. Since May

2019

, 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

Newton

1\ \text{N}=1\ \text{kg m s}^{-2}

Joule and watt

1\ \text{J}=1\ \text{kg m}^2\,\text{s}^{-2},\quad1\ \text{W}=1\ \text{kg m}^2\,\text{s}^{-3}

Pascal

1\ \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

=\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=kv^2

. Find the SI base units of

k

Example 2 · Finding powers0 / 2 steps

The speed of a wave on a string is

v=T^x\mu^y

, where

T

is tension and

\mu

is mass per unit length. Find

x

and

y

Section 5 of 14

Explore the graph

Powers of ten

Loading the interactive graph…

Try this

The axis is the power of ten of a length in metres. The fixed points mark a nucleus ( $10^{-15}$ ), an atom ( $10^{-10}$ ), a person ( $10^0$ ), the Earth ( $10^7$ ), the Sun ( $10^9$ ) and our galaxy ( $10^{21}$ ).
Move $k$ : the row shows the length $10^k$ m. How many powers of ten separate an atom from its nucleus?
Which prefix would you use for $10^{-9}$ m? For $10^9$ W?

Physics covers more than $36$ 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.

Incorrect

\mu

=10^{-3}

Correct

10^{-6}

m is milli, μ is micro.

IncorrectWriting "kg/m/s²"

Correctkg m⁻¹ s⁻²

Use negative indices.

what's the SI base unit of force?

Newton

I'm derived

\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

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?

Salam Academy Original Practice · Proposed mark scheme

SA-9702-PQ1-0031 markL1 FoundationCalculatorPrefixes & estimates

Convert a current of

250

μA into amperes.

Salam Academy Original Practice · Proposed mark scheme

SA-9702-PQ1-0041 markL1 FoundationCalculatorPrefixes & estimates

Convert

72

km h⁻¹ into m s⁻¹.

Salam Academy Original Practice · Proposed mark scheme

SA-9702-PQ1-0051 markL2 StandardCalculatorSI units

The pascal expressed in SI base units is

Salam Academy Original Practice · Proposed mark scheme

SA-9702-PQ1-0061 markL2 StandardCalculatorSI units

Which equation is not homogeneous? (

s

displacement,

u

v

velocities,

a

acceleration,

t

time)

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?

Salam Academy Original Practice · Proposed mark scheme

SA-9702-PQ1-0081 markL2 StandardCalculatorSI units

The drag force on a car is

F=kv^2

. The SI base units of

k

are

Salam Academy Original Practice · Proposed mark scheme

SA-9702-PQ1-0092 marksL2 StandardCalculatorPrefixes & estimates

A Wi-Fi signal has a frequency of

5.0

GHz. Calculate its wavelength. (

c=3.00\times10^8

m s⁻¹)

Salam Academy Original Practice · Proposed mark scheme

SA-9702-PQ1-0101 markL1 FoundationCalculatorPrefixes & estimates

Calculate the number of seconds in one year (

365

days), in units of

10^7

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=T^x\mu^y

, where

T

is the tension and

\mu

is the mass per unit length. Use base units to determine

x

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

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: AS & A Level Physics 9702
Where: AS Level (Papers 1 and 2) · 1.1 Physical quantities; 1.2 SI units
Expected: Understand that a physical quantity is a numerical magnitude and a unit; make reasonable estimates of physical quantities; recall the SI base quantities and units (kg, m, s, A, K, mol); express derived units as products or quotients of base units; check the homogeneity of equations; use the prefixes p, n, μ, m, c, d, k, M, G, T.
Calculator: A calculator is allowed; a data and formulae list is printed in every paper.
Typical style: "Express the unit of the Young modulus in SI base units" and "estimate the kinetic energy of a sprinting athlete".
Open in: 9702 · Paper 1 9702 · Paper 2

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.

9702/22 · Oct/Nov 2025Question 111 marks
Units, scalars and vectors, upthrust on a balloon, and combining percentage uncertainties.
Question paper, page 4 Mark scheme
9702/22 · Oct/Nov 2024Question 17 marks
Vectors, SI base units from a drag formula, and forces at terminal velocity.
Question paper, page 3 Mark scheme
9702/22 · May/June 2024Question 110 marks
SI base quantities, intensity, percentage uncertainty and efficiency.
Question paper, page 4 Mark scheme
9702/22 · Oct/Nov 2023Question 19 marks
SI base quantities, uniform acceleration and a velocity–time graph.
Question paper, page 5 Mark scheme
9702/22 · May/June 2023Question 16 marks
Pressure, SI base units of a constant, and percentage uncertainty.
Question paper, page 3 Mark scheme

All 9702 past paper questions by topic

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Cambridge International 9702, 2025–2027 (official specification).