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

This guide teaches 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 lists Section P under Paper 2, with vectors, kinematics, forces and moments, so this content is examined there. A calculator is required in every 7357 paper.

Section P is short, but every mechanics answer depends on it. A correct method with the wrong units, or with grams left as grams, gives a wrong number.

Use this guide with the Quantities and units revision notes and the Quantities and units practice questions. The course hub is at /boards/aqa/a-level/mathematics/, the printable checklist at /checklists/aqa/a-level/mathematics/, and you can find your weak spots first with the free 10-minute diagnostics.

What Section P covers

Ref	What you must be able to do
P1 (fundamental)	Understand and use the fundamental quantities and units in the SI system: length, time, mass
P1 (derived)	Understand and use the derived quantities and units: velocity, acceleration, force, weight, moment

Appendix B of the specification lists formulae you must use without them being provided. For mechanics it includes weight = mass × g and Newton’s second law in the form F = ma. Both appear below, because they are what give force and weight their units.

Notation the specification uses

Appendix A of the specification sets out the mechanics notation. Use it in your answers.

Symbol	Meaning
kg	kilogram
m	metre
km	kilometre
m/s or m s⁻¹	metres per second (velocity)
m/s² or m s⁻²	metres per second per second (acceleration)
N	newton
N m	newton metre (moment of a force)
F	force or resultant force
t, s, u, v, a	time, displacement, initial velocity, (final) velocity, acceleration
g	acceleration due to gravity

Watch the two uses of “s” and “m”. The unit s means seconds, while the variable s means displacement. The unit m means metres, while the variable m usually means mass. Write units after numbers and variables in your working so the two never clash.

P1: fundamental quantities and units

The SI system builds every unit in this course from three fundamental (base) quantities:

Quantity	SI unit	Symbol
length	metre	m
time	second	s
mass	kilogram	kg

The kilogram, not the gram, is the SI unit of mass. A question may give you grams, centimetres, minutes, hours or kilometres. Convert to kg, m and s before you substitute into any formula.

Useful conversions:

• 1 km = 1000 m; 1 m = 100 cm
• 1 kg = 1000 g
• 1 minute = 60 s; 1 hour = 3600 s
• 1 km h⁻¹ = 1000 m ÷ 3600 s = 1/3.6 m s⁻¹, so divide by 3.6 to go from km h⁻¹ to m s⁻¹, and multiply by 3.6 to go back

Worked example 1. Convert each quantity to SI units: (a) 3.2 km, (b) 450 g, (c) 1 hour 20 minutes, (d) 90 km h⁻¹, (e) 360 cm s⁻².

(a) 3.2 km = 3.2 × 1000 = 3200 m
(b) 450 g = 450 ÷ 1000 = 0.45 kg
(c) 1 h 20 min = 80 min = 80 × 60 = 4800 s
(d) 90 km/h = 90 × 1000 m ÷ 3600 s = 25 m s⁻¹
(e) 360 cm s⁻² = 360 ÷ 100 = 3.6 m s⁻²

In (e) only the length unit changes. The “per second per second” part is already SI.

To check a speed conversion, compare sizes. A metre is much shorter than a kilometre, but a second is much shorter than an hour, and the second effect wins: the same speed is always a smaller number in m s⁻¹ than in km h⁻¹. If you had multiplied by 3.6 you would get 324 m s⁻¹, bigger than 90, so you would know to stop and check.

P1: derived quantities and units

A derived quantity is built from the fundamental ones. Its unit follows from its definition.

Quantity	Definition	SI unit	In base units
velocity	rate of change of displacement	m s⁻¹	m s⁻¹
acceleration	rate of change of velocity	m s⁻²	m s⁻²
force	mass × acceleration (F = ma)	newton, N	kg m s⁻²
weight	the force of gravity on a mass (W = mg)	newton, N	kg m s⁻²
moment	force × perpendicular distance from the point	newton metre, N m	kg m² s⁻²

Velocity and speed

Velocity is displacement per unit time, so its unit is metres per second. Speed has the same unit. Velocity has a direction and speed does not, but that difference does not change the unit.

Acceleration

Acceleration is the change in velocity per unit time: (m s⁻¹) ÷ s = m s⁻². Read it as “metres per second, every second”. An acceleration of 2 m s⁻² means the velocity goes up by 2 m s⁻¹ each second.

Force and the newton

Newton’s second law, F = ma, defines the unit of force. One newton is the force that gives a mass of 1 kg an acceleration of 1 m s⁻². So

1 N = 1 kg × 1 m s⁻² = 1 kg m s⁻².

Large forces are often given in kilonewtons: 1 kN = 1000 N.

Worked example 2. A car of mass 1.2 tonnes (1 tonne = 1000 kg) speeds up from rest to 54 km h⁻¹ in 10 s with constant acceleration. Find the resultant force on the car.

mass:      1.2 tonnes = 1200 kg
velocity:  54 km/h = 54 ÷ 3.6 = 15 m s⁻¹
acceleration = change in velocity ÷ time
             = (15 − 0) ÷ 10 = 1.5 m s⁻²
F = ma = 1200 × 1.5 = 1800 N

Every quantity was converted before substitution. Using 1.2 for the mass, or 54 for the velocity, gives a force that is wrong by a factor of 1000 or 3.6.

Weight

Weight is a force, so it is measured in newtons. Mass is the amount of matter in a body and is measured in kilograms. They are linked by

W = mg,

where g is the acceleration due to gravity. The specification says g may be taken as constant and its value is used to varying degrees of accuracy. It also says g is not a universal constant: it depends on location. A body’s mass stays the same wherever it is, while its weight changes if g changes.

Use the value of g the question tells you. If it says “take g = 9.8 m s⁻²”, use exactly 9.8.

Worked example 3. A parcel has mass 2.5 kg. Find its weight using (a) g = 9.8 m s⁻², (b) g = 9.78 m s⁻². Comment on the mass in each case.

(a) W = mg = 2.5 × 9.8  = 24.5 N
(b) W = mg = 2.5 × 9.78 = 24.45 N, or 24.5 N to 3 s.f.

The mass is 2.5 kg in both cases. Only the weight depends on g.

Never write a mass in newtons or a weight in kilograms. “The box weighs 3 kg” is everyday language, but in mechanics the box has mass 3 kg and weight 3g N.

Moment

The moment of a force about a point measures its turning effect. Its size is

moment = force × perpendicular distance from the point to the line of action of the force.

Force is in newtons and distance in metres, so the unit is the newton metre, N m. The calculations with moments (balancing, equilibrium of rigid bodies) belong to Section S; here you need the quantity and its unit.

Worked example 4. A force of 30 N acts at right angles to a spanner, 40 cm from the centre of a nut. (a) Find the moment of the force about the centre of the nut. (b) Find the force needed at 60 cm from the centre to give the same moment.

(a) distance = 40 cm = 0.4 m
    moment = 30 × 0.4 = 12 N m
(b) F × 0.6 = 12
    F = 12 ÷ 0.6 = 20 N

Had you used 40 instead of 0.4, the answer would be 1200, which is the moment in N cm, not in N m.

Using units to check your work

Every term in a correct equation has the same units. This gives you a fast check on a formula you have written down, and a way to find the units of a constant. Treat it as a checking habit; Section P does not list it as a separate content statement.

Worked example 5. A model says the air resistance on a cyclist is R = kv², where R is in newtons and v is the speed in m s⁻¹.

(a) Find the SI units of k. (b) R = 72 N when v = 24 m s⁻¹. Find k. (c) Use the model to find R when the speed is 90 km h⁻¹.

(a) k = R ÷ v²
    units: N ÷ (m s⁻¹)² = kg m s⁻² ÷ (m² s⁻²) = kg m⁻¹
(b) k = 72 ÷ 24² = 72 ÷ 576 = 0.125 kg m⁻¹
(c) 90 km/h = 25 m s⁻¹
    R = 0.125 × 25² = 78.125 = 78.1 N (3 s.f.)

In (c), substituting v = 90 gives R = 1012.5 N, more than ten times too large. The model only works with v in the units k was found in.

The same check catches a misremembered formula. In v² = u² + 2as, each term has units (m s⁻¹)² = m² s⁻², and 2as has units m s⁻² × m = m² s⁻². The units match. If you wrote v = u + as, the left side is in m s⁻¹ but as is in m² s⁻², so the formula cannot be right.

Accuracy

Your answers can only be as precise as your data. If g is given to two significant figures (9.8), an answer such as 24.45 N says more than the data justify. Give answers to a sensible degree of accuracy, follow any instruction in the question, and keep full calculator values in your working until the last line so rounding errors do not build up.

Common errors

• Substituting grams, centimetres, minutes or km h⁻¹ straight into F = ma, W = mg or a moment calculation.
• Multiplying by 3.6 when converting km h⁻¹ to m s⁻¹. You divide, because m s⁻¹ values are smaller.
• Giving a weight in kg, or a mass in N.
• Writing the unit of acceleration as m s⁻¹, or the unit of moment as N or N m⁻¹.
• Treating g as a fixed universal constant, or using 9.81 when the question says 9.8.
• Rounding an intermediate value (such as a converted speed) and carrying the rounded value forward.
• Missing units on a final answer, or giving N cm when the question works in metres.
• Mixing up the variable m (mass) with the unit m (metre), or the variable s (displacement) with the unit s (second).

Next steps

• Condensed recall: Quantities and units revision notes.
• Test yourself: Quantities and units practice questions.
• Forces as vectors and F = ma in component form: Vectors study guide.
• Modelling assumptions and checking answers: Overarching themes.
• All free 10-minute diagnostics.
• Course hub: /boards/aqa/a-level/mathematics/. Checklist: /checklists/aqa/a-level/mathematics/.

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.