Contents · 0% complete

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

Learn4PH1Paper 2P · Paper 2 (all content)Forces and motionPhysical quantities and measurement

Forces and motion · Subtopic 1 of 6

Physical Quantities and Measurement

Every physical quantity is a number with a unit. Measure carefully, and measure many small things together to reduce errors. Scalars have size only; vectors have size and direction, so two vectors at right angles combine by Pythagoras.

Cambridge 5054 Paper 1Cambridge 5054 Paper 2Cambridge 0625 Papers 2, 4Cambridge 0625 Papers 1, 3Edexcel 4PH1 Paper 1PEdexcel 4PH1 Paper 2P≈ 25 minRevision 5 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Measure length, volume and time, choosing suitable instruments.
Measure small quantities accurately by measuring many and dividing.
Use SI units and prefixes such as kilo, milli and micro.
Distinguish scalars and vectors, and find the resultant of two perpendicular vectors.

Pause and think

How could you find the thickness of one sheet of paper using only a $30$ cm ruler marked in millimetres?

Section 2 of 14

Core idea

Start hereTiming a swing with a stopwatch

Your reaction time is about 0.2 s, but one pendulum swing lasts only about 1.6 s. Time a single swing and your error is over 10%. Time 20 swings and the same 0.2 s error is spread over all of them. Good measurement is about outsmarting your own limits.

The big idea

Choose an instrument with the right precision, read at eye level, repeat and average, and measure multiples to shrink errors. Scalars have size; vectors have size and direction.

1. Instruments and precision

Metre rule: $1$ mm. Vernier calipers: $0.1$ mm. Micrometer: $0.01$ mm.
Stopwatch: reaction time (~0.2 s) dominates.
Measuring cylinder: read the bottom of the meniscus at eye level.

Reading a measurement at eye level avoids parallax error

Look straight across at the mark. Looking from above or below makes the reading seem higher or lower than it really is.

Worked exampleTiming a pendulum

20 oscillations take $32.0$ s.

Goal: Period and its uncertainty.
Strategy: Divide the time and the error by 20.

$T=\frac{32.0}{20}=1.60\text{ s}$
One oscillation.
$\frac{0.2}{20}=0.01\text{ s}$
Reaction error shared over 20 swings.

Answer

T=1.60\pm0.01

✓ CheckOne swing timed alone would be

\pm0.2

s: 20 times worse ✓.

Worked exampleThickness of one sheet

200 sheets measure $2.0$ cm.

Goal: One sheet.
Strategy: Measure a multiple.

$2.0\div200=0.010\text{ cm}$
Divide.
$=0.10\text{ mm}$
Convert.

Answer

0.10

mm.

✓ CheckFar below a ruler's 1 mm precision: only possible via a stack ✓.

Common misconceptionZero error If a micrometer reads

0.02

mm when closed, subtract

0.02

mm from every reading.

2. Scalars and vectors

Scalars: distance, speed, mass, energy, time, temperature.
Vectors: displacement, velocity, acceleration, force, momentum.

Adding two forces at right angles

Draw the vectors head to tail. The resultant goes from the start to the finish:

\sqrt{4^2+3^2}=5

Worked example3 N east and 4 N north

Resultant force.

Goal: Size and direction.
Strategy: Right-angled vectors: Pythagoras and tan.

$\sqrt{3^2+4^2}=5\text{ N}$
Pythagoras.
$\tan^{-1}\tfrac43=53.1^\circ$
North of east.

Answer

5

N at

53.1^\circ

north of east.

✓ CheckBetween the extremes

4-3=1

and

4+3=7

✓.

Pause and think

Why is a zero error important on a micrometer?

Summary

Match instrument to precision.
Multiples, repeats, averages.
Vectors add by drawing or Pythagoras.

Parallax error: Misreading from viewing at an angle.
Vector: Quantity with size and direction.

HistoryThe metre and the Earth. In

1791

France defined the metre as one ten-millionth of the distance from the North Pole to the equator through Paris. Two astronomers spent seven years surveying from Dunkirk to Barcelona to measure it. Today the metre is defined by the speed of light.

Fun factA $125$ million dollar unit mistake. In

1999

NASA lost the Mars Climate Orbiter because one team used pound-force seconds and another used newton seconds. The spacecraft flew too low and burned up in the Martian atmosphere. Units matter.

Section 3 of 14

Key formulae

Period from many swings

T=\frac{\text{time for }n\text{ oscillations}}{n}

Resultant of perpendicular vectors

R=\sqrt{a^2+b^2},\quad\tan\theta=\frac ba

Prefixes

\text{k}=10^3,\ \text{M}=10^6,\ \text{m}=10^{-3},\ \mu=10^{-6}

TrickDirection test. To decide if something is a vector, ask: "does it make sense to say which way?" "

20

N to the left" makes sense, so force is a vector. "

20

kg to the left" doesn't, so mass is a scalar.

Section 4 of 14

Worked examples

Example 1 · Period of a pendulum0 / 2 steps

A pendulum makes

20

complete oscillations in

36.0

s. Find its period.

Example 2 · Resultant force0 / 2 steps

Forces of

5

N and

12

N act at right angles. Find the resultant.

Section 5 of 14

Explore the graph

Adding two perpendicular vectors

Loading the interactive graph…

Try this

The horizontal arrow is a vector of size $a$ ; the vertical arrow has size $b$ . The diagonal is the resultant.
With $a=3$ , $b=4$ , the resultant is $5$ (the last row shows its size). Try $a=6$ , $b=8$ .
Can the resultant ever be smaller than the larger vector? Why not, when they're at right angles?

Perpendicular vectors add like the sides of a right-angled triangle.

In the real worldAircraft and crosswinds. A plane flying north at

200

m/s in a

50

m/s easterly wind actually moves at

\sqrt{200^2+50^2}\approx206

m/s, slightly east of north. Pilots point the nose into the wind so the resultant takes them where they want to go.

Section 6 of 14

Common mistakes

IncorrectTiming one swing of a pendulum

CorrectTime

20

and divide

Reaction time is then a small fraction of the reading.

IncorrectReading a measuring cylinder from above

CorrectEye level with the meniscus

Avoids parallax error.

Incorrect

3

N and

4

N at right angles give

7

Correct

5

Add perpendicular vectors by Pythagoras.

IncorrectSpeed is a vector

CorrectSpeed is a scalar; velocity is a vector

Speed has no direction.

I'll measure the thickness of one sheet of paper with a ruler

Ruler

my smallest division is 1 mm

measure 100 sheets and divide by 100. Science.

Section 7 of 14

Exam tip

Exam tipUnits on every answer. A correct number with no unit, or the wrong unit, usually loses the final mark. Write the unit every time, and convert to SI (m, kg, s) before calculating.

Section 8 of 14

Guided practice

SA-5054-MF1-0011 markL1 FoundationCalculatorMeasurement

Convert

2.5

km into metres.

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-5054-MF1-0021 markL1 FoundationCalculatorScalars & vectors

Which quantity is a vector?

Salam Academy Original Practice · Proposed mark scheme

SA-5054-MF1-0032 marksL1 FoundationCalculatorMeasurement

A pendulum makes

20

complete oscillations in

36.0

s. Calculate its period.

Salam Academy Original Practice · Proposed mark scheme

SA-5054-MF1-0042 marksL1 FoundationCalculatorMeasurement

A stack of

50

identical sheets of paper is

6.0

mm thick. Calculate the thickness of one sheet, in mm.

Salam Academy Original Practice · Proposed mark scheme

SA-5054-MF1-0051 markL1 FoundationCalculatorMeasurement

The water level in a measuring cylinder rises from

45

cm³ to

62

cm³ when a stone is lowered in. Find the volume of the stone, in cm³.

Salam Academy Original Practice · Proposed mark scheme

SA-5054-MF1-0062 marksL1 FoundationCalculatorScalars & vectors

A force of

3.0

N acts east and a force of

4.0

N acts north on the same object. Calculate the size of the resultant force.

Salam Academy Original Practice · Proposed mark scheme

SA-5054-MF1-0071 markL1 FoundationCalculatorScalars & vectors

Forces of

6

N to the right and

2

N to the left act on a box. State the size of the resultant force.

Salam Academy Original Practice · Proposed mark scheme

SA-5054-MF1-0082 marksL2 StandardCalculatorScalars & vectors

For forces of

3.0

N east and

4.0

N north, calculate the angle between the resultant and the east direction, in degrees.

Salam Academy Original Practice · Proposed mark scheme

SA-5054-MF1-0091 markL1 FoundationCalculatorMeasurement

Convert a mass of

450

g into kilograms.

Salam Academy Original Practice · Proposed mark scheme

SA-5054-MF1-0101 markL1 FoundationCalculatorScalars & vectors

Which list contains only scalar quantities?

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-5054-MF1-0113 marksL3 ExamCalculatorScalars & vectors

A boat points due north and moves through the water at

4.0

m/s. A current flows due east at

3.0

m/s. Calculate (a) the boat's resultant speed (b) the angle its path makes with north, in degrees.

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Every quantity needs a number and a unit.
Measure many small things together, then divide.
Scalars: size only. Vectors: size and direction.
Perpendicular vectors: $R=\sqrt{a^2+b^2}$ .

Section 12 of 14

Quick quiz

Q1 · SA-5054-MF1-0121 markL1 FoundationCalculatorMeasurement

A student times

20

oscillations of a pendulum rather than one. The main reason is to

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: International GCSE Physics 4PH1
Where: Section 1 Forces and motion: units
Expected: SI units and prefixes, scalars and vectors.
Calculator: Calculator allowed in both papers.
Typical style: Structured questions with a list of equations to recall; Paper 2 adds extra content.
Open in: 4PH1 · Paper 1P 4PH1 · Paper 2P

Section 14 of 14

Past paper questions

1 question (8 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

0625/42 · Oct/Nov 2025Question 18 marks
Scalars and vectors, and the velocity and momentum of a thrown ball.
Question paper, page 3 Mark scheme

All 0625 past paper questions by topic

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Pearson Edexcel 4PH1, 2017 specification (official specification).