Mathematical Requirements - A-Levels Physics

5e. Mathematical requirements

In order to be able to develop their skills, knowledge and understanding in A Level Physics, learners need to have been taught, and to have acquired competence in, the appropriate areas of mathematics relevant to the subject as indicated in the table of coverage below.

The assessment of quantitative skills will include at least 40% Level 2 (or above) mathematical skills for physics. These skills will be applied in the context of the relevant physics.

All mathematical content will be assessed within the lifetime of the specification. Skills shown in bold type will only be tested in the full A level course, not the standalone AS level course. This list of examples is not exhaustive and is not limited to Level 2 examples.

Mathematical requirements

Ref.	Mathematical skill to be assessed	Exemplification in the context of A Level Physics	Areas of specification
M0 - Arithmetic and numerical computation
M0.1	Recognise and make use of appropriate units in calculations.	Learners may be tested on their ability to: • identify the correct units for physical properties such as $\mathrm{m\,s^{-1}}$ , the unit for velocity; • convert between units with different prefixes e.g. $\mathrm{cm^3}$ to $\mathrm{m^3}$ .	1.1.2(b), 2.1.1(a), 3.1.1(a), 3.2.4(a)
M0.2	Recognise and use expressions in decimal and standard form.	Learners may be tested on their ability to use physical constants expressed in standard form such as $c = 3.00 \times 10^8\,\mathrm{m\,s^{-1}}$ .	1.1.3(c), 4.1.2(b)
M0.3	Use ratios, fractions and percentages.	Learners may be tested on their ability to: • calculate efficiency of devices; • calculate percentage uncertainties in measurements.	3.3.3(c), 6.3.3(f), 6.5.3(e)
M0.4	Estimate results.	Learners may be tested on their ability to estimate the effect of changing experimental parameters on measurable values.	2.1.1(b), 6.4.1(c)
M0.5	Use calculators to find and use power, exponential and logarithmic functions.	Learners may be tested on their ability to solve for unknowns in decay problems such as $N = N_0e^{-\lambda t}$ .	3.3.2(a), 3.4.2(b), 6.1.3(c), 6.4.3(g), 6.5.1(d)
M0.6	Use calculators to handle $\sin x$ , $\cos x$ and $\tan x$ when $x$ is expressed in degrees or radians.	Learners may be tested on their ability to calculate the direction of resultant vectors.	2.3.1(c)(d), 3.1.3(b)
M1 - Handling data
M1.1	Use an appropriate number of significant figures.	Learners may be tested on their ability to: • report calculations to an appropriate number of significant figures given raw data quoted to varying numbers of significant figures; • understand that calculated results can only be reported to the limits of the least accurate measurement.	1.1.3(c), 3.2.1(a)
M1.2	Find arithmetic means.	Learners may be tested on their ability to calculate a mean value for repeated experimental readings.	1.1.3(a)
M1.3	Understand simple probability.	Learners may be tested on their ability to understand probability in the context of radioactive decay.	1.1.4(d), 6.4.3(a)
M1.4	Make order of magnitude calculations.	Learners may be tested on their ability to evaluate equations with variables expressed in different orders of magnitude.	3.1.1(a), 5.5.3(m), 6.4.1(c)
M1.5	Identify uncertainties in measurements and use simple techniques to determine uncertainty when data are combined by addition, subtraction, multiplication, division and raising to powers.	Learners may be tested on their ability to determine the uncertainty where two readings for length need to be added together.	1.1.4(d), 2.2.1(c)(d)
M2 - Algebra
M2.1	Understand and use the symbols: $=$ , $<$ , $\ll$ , $\gg$ , $>$ , $\propto$ , $\sim$ , $\Delta$ .	Learners may be tested on their ability to recognise the significance of the symbols in the expression $F \propto \Delta p/\Delta t$ .	3.2.4(c), 3.5.1(c)
M2.2	Change the subject of an equation, including non-linear equations.	Learners may be tested on their ability to rearrange $E = mc^2$ to make $m$ the subject.	3.1.2(a), 4.2.5(a), 5.3.1(e)
M2.3	Substitute numerical values into algebraic equations using appropriate units for physical quantities.	Learners may be tested on their ability to calculate the momentum $p$ of an object by substituting the values for mass $m$ and velocity $v$ into the equation $p = mv$ .	4.3.3(c), 4.5.2(c), 5.4.2(a)
M2.4	Solve algebraic equations, including quadratic equations.	Learners may be tested on their ability to solve kinematic equations for constant acceleration such as $v = u + at$ and $s = ut + \frac{1}{2}at^2$ .	3.1.2(a), 5.2.2(c)
M2.5	Use logarithms in relation to quantities that range over several orders of magnitude.	Learners may be tested on their ability to recognise and interpret real world examples of logarithmic scales.	6.1.3(c), 6.4.3(g)
M3 - Graphs
M3.1	Translate information between graphical, numerical and algebraic forms.	Learners may be tested on their ability to calculate Young modulus for materials using stress-strain graphs.	1.1.3(d), 1.2.1(g), 3.4.2(a)(d)
M3.2	Plot two variables from experimental or other data.	Learners may be tested on their ability to plot graphs of extension of a wire against force applied.	1.1.3(d), 3.4.1(d)(i), 3.4.2(d)
M3.3	Understand that $y = mx + c$ represents a linear relationship.	Learners may be tested on their ability to rearrange and compare $v = u + at$ with $y = mx + c$ for velocity-time graphs in constant acceleration problems.	1.1.3(d), 3.1.2(a)
M3.4	Determine the slope and intercept of a linear graph.	Learners may be tested on their ability to read off and interpret intercept point from a graph e.g. the initial velocity in a velocity-time graph.	1.1.3(d), 3.1.1(c)
M3.5	Calculate rate of change from a graph showing a linear relationship.	Learners may be tested on their ability to calculate acceleration from a linear velocity-time graph.	3.1.1(d)
M3.6	Draw and use the slope of a tangent to a curve as a measure of rate of change.	Learners may be tested on their ability to draw a tangent to the curve of a displacement-time graph and use the gradient to approximate the velocity at a specific time.	3.1.1(b)
M3.7	Distinguish between instantaneous rate of change and average rate of change.	Learners may be tested on their ability to understand that the gradient of the tangent of a displacement-time graph gives the velocity at a point in time which is a different measure to the average velocity.	3.1.1(a)(c)
M3.8	Understand the possible physical significance of the area between a curve and the x axis and be able to calculate it or estimate it by graphical methods as appropriate.	Learners may be tested on their ability to recognise that for a capacitor the area under a voltage-charge graph is equivalent to the energy stored.	3.5.1(e), 6.1.2(a)
M3.9	Apply the concepts underlying calculus (but without requiring the explicit use of derivatives or integrals) by solving equations involving rates of change, e.g. $\frac{\Delta x}{\Delta t} = -\lambda x$ using a graphical method or spreadsheet modelling.	Learners may be tested on their ability to determine $g$ from distance-time plot, projectile motion.	3.1.1(a), 3.5.1(c), 5.3.1(d), 6.1.3(d), 6.3.3(d), 6.4.3(g)
M3.10	Interpret logarithmic plots.	Learners may be tested on their ability to obtain time constant for capacitor discharge by interpreting plot of log $V$ against time.	6.1.3(c)
M3.11	Use logarithmic plots to test exponential and power law variations.	Learners may be tested on their ability to use logarithmic plots with decay law of radioactivity / charging and discharging of a capacitor.	6.1.3(e), 6.5.1(d)
M3.12	Sketch relationships which are modelled by $y = k/x$ , $y = kx^2$ , $y = k/x^2$ , $y = kx$ , $y = \sin x$ , $y = \cos x$ , $y = e^{\pm x}$ , and $y = \sin^2x$ , $y = \cos^2x$ as applied to physical relationships.	Learners may be tested on their ability to sketch relationships between pressure and volume for an ideal gas.	3.4.2(b), 4.2.3(c), 5.3.1(d), 6.1.3(c), 6.4.3(f)
M4 - Geometry and trigonometry
M4.1	Use angles in regular 2D and 3D structures.	Learners may be tested on their ability to interpret force diagrams to solve problems.	3.2.3(f)
M4.2	Visualise and represent 2D and 3D forms including two-dimensional representations of 3D objects.	Learners may be tested on their ability to draw force diagrams to solve mechanics problems.	2.3.1(c), 3.2.3(f)
M4.3	Calculate areas of triangles, circumferences and areas of circles, surface areas and volumes of rectangular blocks, cylinders and spheres.	Learners may be tested on their ability to calculate the area of the cross section to work out the resistance of a conductor given its length and resistivity.	3.1.1(d), 3.2.4(a), 3.5.1(e)
M4.4	Use Pythagoras' theorem, and the angle sum of a triangle.	Learners may be tested on their ability to calculate the magnitude of a resultant vector, resolving forces into components to solve problems.	2.3.1(c), 3.2.3(f)
M4.5	Use $\sin$ , $\cos$ and $\tan$ in physical problems.	Learners may be tested on their ability to resolve forces into components.	2.3.1(d), 3.1.3(b)
M4.6	Use of small angle approximations including $\sin\theta \approx \theta$ , $\tan\theta \approx \theta$ , $\cos\theta \approx 1$ for small $\theta$ where appropriate.	Learners may be tested on their ability to calculate fringe separations in interference patterns.	4.4.3(g), 5.5.3(a)
M4.7	Understand the relationship between degrees and radians and translate from one to the other.	Learners may be tested on their ability to convert angle in degrees to angle in radians.	5.2.1(a), 5.3.1(a)