AQA A-Level Mathematics: P: Quantities and units in mechanics (7357) – Revision Notes

These notes cover Section P: Quantities and units in mechanics (P1) of the AQA A-level Mathematics (7357) specification, version 1.3 (31 January 2018), for A-level exams from June 2018 onwards. The specification places Section P in Paper 2, and a calculator is required in every 7357 paper. For full explanations and worked examples, start with the Quantities and units study guide; these notes are for quick recall in the final weeks.

Practise afterwards with the Quantities and units practice questions. Course hub: /boards/aqa/a-level/mathematics/. Printable checklist: /checklists/aqa/a-level/mathematics/. Find your weak spots with the free 10-minute diagnostics.

P1 at a glance

Ref	You must be able to
P1	Understand and use the fundamental SI quantities and units: length, time, mass
P1	Understand and use the derived quantities and units: velocity, acceleration, force, weight, moment

Fundamental quantities

Quantity	SI unit	Symbol	Common non-SI forms you must convert
length	metre	m	km, cm, mm
time	second	s	minutes, hours
mass	kilogram	kg	g, tonnes

The SI unit of mass is the kilogram, not the gram.

Derived quantities

Quantity	Defined by	Unit	Base units
velocity (and speed)	displacement ÷ time	m s⁻¹	m s⁻¹
acceleration	change in velocity ÷ time	m s⁻²	m s⁻²
force	F = ma	N	kg m s⁻²
weight	W = mg	N	kg m s⁻²
moment	force × perpendicular distance	N m	kg m² s⁻²

Appendix B of the specification lists weight = mass × g and F = ma among the formulae you must use without being given them.

1 N is the force that gives 1 kg an acceleration of 1 m s⁻². 1 kN = 1000 N.

Notation (Appendix A)

• m/s and m s⁻¹ both mean metres per second; m/s² and m s⁻² both mean metres per second per second.
• N m is the newton metre, the unit of the moment of a force.
• g is the acceleration due to gravity; F is a force or resultant force.
• The letters s and m are both units (second, metre) and variables (displacement, mass). Keep units next to numbers so they never get confused.

Method: convert before you substitute

1. List every given quantity with its unit.
2. Convert each one to kg, m, s (or m s⁻¹, m s⁻², N).
3. Substitute into the formula.
4. Give the answer with its SI unit, rounded sensibly.

Key conversion factors:

From	To	Do
km	m	× 1000
cm	m	÷ 100
g	kg	÷ 1000
tonnes	kg	× 1000
minutes	s	× 60
hours	s	× 3600
km h⁻¹	m s⁻¹	÷ 3.6
m s⁻¹	km h⁻¹	× 3.6
cm s⁻²	m s⁻²	÷ 100

Sense check: the same speed is always a smaller number in m s⁻¹ than in km h⁻¹.

Reminder: 126 km h⁻¹ ÷ 3.6 = 35 m s⁻¹. 0.8 tonnes = 800 kg.

Mass versus weight

Mass	Weight
amount of matter in a body	force of gravity on the body
kilograms, kg	newtons, N
the same everywhere	W = mg, so it changes if g changes
scalar	a force, acting vertically downwards

The specification says g may be taken as constant and is used to varying degrees of accuracy, but g is not a universal constant: it depends on location. Use the value of g given in the question, exactly.

Reminder: a 5 kg box has weight 5 × 9.8 = 49 N with g = 9.8 m s⁻². At a place where g = 9.78 m s⁻² its weight is 48.9 N. Its mass is 5 kg in both places.

Moment in one line

moment = F × d, where d is the perpendicular distance (in metres) from the point to the line of action of the force. Unit: N m. Calculations with moments in equilibrium belong to Section S.

Reminder: 25 N at right angles, 12 cm from a hinge: d = 0.12 m, moment = 3 N m.

Three short worked reminders

A. Force given in kN. A resultant force of 2.4 kN acts on a car of mass 800 kg.

F = 2.4 kN = 2400 N
a = F ÷ m = 2400 ÷ 800 = 3 m s⁻²

Leaving F as 2.4 gives 0.003 m s⁻², which is 1000 times too small.

B. Mixed time units. A runner covers 400 m in 1 minute 20 seconds.

time = 60 + 20 = 80 s
average speed = 400 ÷ 80 = 5 m s⁻¹
in km/h: 5 × 3.6 = 18 km h⁻¹

C. Mass in grams. A trolley of mass 1500 g speeds up steadily from 0.5 m s⁻¹ to 1.7 m s⁻¹ in 3 s.

m = 1500 ÷ 1000 = 1.5 kg
a = (1.7 − 0.5) ÷ 3 = 0.4 m s⁻²
F = ma = 1.5 × 0.4 = 0.6 N

Fast sense checks

• A speed in m s⁻¹ is always smaller than the same speed in km h⁻¹.
• A weight in newtons is roughly ten times the mass in kilograms when g is about 9.8.
• A moment worked out with centimetres is 100 times the moment in N m.
• If an answer is out by a factor of 1000, 100, 60 or 3.6, look for an unconverted unit.

Method: check units in a formula

1. Write each term in base units (kg, m, s).
2. Every term added or subtracted, and both sides of =, must match.
3. For an unknown constant, rearrange and divide the units.

Reminder: in s = ut + ½at², the units are m, (m s⁻¹)(s) = m and (m s⁻²)(s²) = m. All three match.

Reminder: if a resistance is R = cv, then c = R/v has units N ÷ m s⁻¹ = kg m s⁻² × s m⁻¹ = kg s⁻¹.

Must-know distinctions

• Fundamental vs derived: length, time and mass are fundamental; velocity, acceleration, force, weight and moment are derived from them.
• Mass vs weight: kg vs N; W = mg.
• Velocity vs acceleration units: m s⁻¹ vs m s⁻².
• Moment vs force: N m vs N. A moment needs a distance.
• kN vs N: multiply kN by 1000 before using F = ma.
• Unit s vs variable s: seconds vs displacement.

Quick self-test

1. Name the SI units of length, mass and time.
2. Convert 5.4 km to metres.
3. Convert 2 hours 15 minutes to seconds.
4. Convert 45 km h⁻¹ to m s⁻¹.
5. Convert 3 m s⁻¹ to km h⁻¹.
6. Write the newton in base units.
7. Find the weight of a 60 kg person, taking g = 9.81 m s⁻².
8. Find the resultant force needed to give a 4 kg mass an acceleration of 2.5 m s⁻².
9. A 16 N force acts at right angles to a lever, 35 cm from the pivot. Find its moment.
10. A label says “weight 9.8 kg”. Correct the physics.
11. Write the newton metre in base units.
12. Which of these is not a correct unit of acceleration: m s⁻², m/s², m s⁻¹ per s, N kg⁻¹, m s⁻¹?

Answers

1. metre (m), kilogram (kg), second (s)
2. 5.4 × 1000 = 5400 m
3. 2 × 3600 + 15 × 60 = 8100 s
4. 45 ÷ 3.6 = 12.5 m s⁻¹
5. 3 × 3.6 = 10.8 km h⁻¹
6. 1 N = 1 kg m s⁻²
7. 60 × 9.81 = 588.6 = 589 N (3 s.f.)
8. F = ma = 4 × 2.5 = 10 N
9. 16 × 0.35 = 5.6 N m
10. 9.8 kg is a mass. The body’s weight is 9.8g N, which is 96.04 N or 96.0 N (3 s.f.) with g = 9.8 m s⁻².
11. N m = kg m s⁻² × m = kg m² s⁻²
12. m s⁻¹ is a velocity unit. (N kg⁻¹ = kg m s⁻² ÷ kg = m s⁻², so it is a correct acceleration unit.)

Where marks are usually lost

• Using a mass in grams, or a distance in centimetres, in F = ma, W = mg or a moment.
• Multiplying by 3.6 instead of dividing when converting km h⁻¹ to m s⁻¹.
• Leaving a force in kN inside F = ma, which makes the acceleration 1000 times too small.
• Stating a weight in kg or a mass in N.
• Giving acceleration in m s⁻¹, or a moment in N.
• Using 9.81 when the question says g = 9.8, or the other way round.
• Rounding a converted speed early (for example 50 km h⁻¹ to 13.9 m s⁻¹) and carrying the rounded value on.
• Using the slant distance instead of the perpendicular distance in a moment.
• Missing units on the final answer.

Next steps

• Full teaching and worked examples: Quantities and units study guide.
• Exam-style practice: Quantities and units practice questions.
• Forces in component form: Vectors revision notes.
• All free 10-minute diagnostics.

Official syllabus

AQA A-level Mathematics (7357) specification, version 1.3, 31 January 2018, for A-level exams June 2018 onwards: Section 3.17 P: Quantities and units in mechanics (P1), with Appendix A: mathematical notation (mechanics) and Appendix B: mathematical formulae and identities.