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Mathematical Skills - A-Levels Physics

Mathematical Skills C.0 – Arithmetic & Computation

(i) C.0 – arithmetic and numerical computation

RefMathematical SkillsExemplification of mathematical skill in the context of A Level Physics
C.0.1Recognise and make use of appropriate units in calculationsStudents may be tested on their ability to:

• identify the correct units for physical properties such as m s−1\boldsymbol{\text{m}\ \text{s}^{-1}}, the unit for velocity
• convert between units with different prefixes, e.g. cm3\boldsymbol{\text{cm}^3} to m3\boldsymbol{\text{m}^3}
C.0.2Recognise and use expressions in decimal and standard formStudents may be tested on their ability to:

• use physical constants expressed in standard form such as c=3.00×108 m s−1\boldsymbol{c = 3.00 \times 10^8\ \text{m}\ \text{s}^{-1}}
C.0.3Use ratios, fractions and percentagesStudents may be tested on their ability to:

• calculate efficiency of devices
• calculate percentage uncertainties in measurements
C.0.4Estimate resultsStudents may be tested on their ability to:

• estimate the effect of changing experimental parameters on measurable values
C.0.5Use calculators to find and use power, exponential and logarithmic functionsStudents may be tested on their ability to:

• solve for unknowns in decay problems such as N=N0e−λt\boldsymbol{N = N_0 e^{-\lambda t}}
C.0.6Use calculators to handle sin⁡x\boldsymbol{\sin x}, cos⁡x\boldsymbol{\cos x}, tan⁡x\boldsymbol{\tan x} when x\boldsymbol{x} is expressed in degrees or radiansStudents may be tested on their ability to:

• calculate the direction of resultant vectors

Mathematical Skills C.1 – Handling Data

(ii) C.1 – handling data

RefMathematical SkillsExemplification of mathematical skill in the context of A Level Physics
C.1.1Use an appropriate number of significant figuresStudents 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
C.1.2Find arithmetic meansStudents may be tested on their ability to:

• calculate a mean value for repeated experimental readings
C.1.3Understand simple probabilityStudents may be tested on their ability to:

• understand probability in the context of radioactive decay
C.1.4Make order of magnitude calculationsStudents may be tested on their ability to:

• evaluate equations with variables expressed in different orders of magnitude
C.1.5Identify uncertainties in measurements and use simple techniques to determine uncertainty when data are combined by addition, subtraction, multiplication, division and raising to powersStudents may be tested on their ability to:

• determine the uncertainty where two readings for length need to be added together

Mathematical Skills C.2 – Algebra

(iii) C.2 – algebra

RefMathematical SkillsExemplification of mathematical skill in the context of A Level Physics
C.2.1Understand and use the symbols: =\boldsymbol{=}, <\boldsymbol{<}, ≪\boldsymbol{\ll}, ≫\boldsymbol{\gg}, >\boldsymbol{>}, ∝\boldsymbol{\propto}, ≈\boldsymbol{\approx}, Δ\boldsymbol{\Delta}Students may be tested on their ability to:

• recognise the significance of the symbols in the expression F∝Δp/Δt\boldsymbol{F \propto \Delta p / \Delta t}
C.2.2Change the subject of an equation, including non-linear equationsStudents may be tested on their ability to:

• rearrange E=mc2\boldsymbol{E = mc^2} to make m\boldsymbol{m} the subject
C.2.3Substitute numerical values into algebraic equations using appropriate units for physical quantitiesStudents may be tested on their ability to:

• calculate the momentum p\boldsymbol{p} of an object by substituting the values for mass m\boldsymbol{m} and velocity v\boldsymbol{v} into the equation p=mv\boldsymbol{p = mv}
C.2.4Solve algebraic equations, including quadratic equationsStudents may be tested on their ability to:

• solve kinematic equations for constant acceleration such as v=u+at\boldsymbol{v = u + at} and s=ut+12at2\boldsymbol{s = ut + \frac{1}{2}at^2}
C.2.5Use logarithms in relation to quantities that range over several orders of magnitudeStudents may be tested on their ability to:

• recognise and interpret real-world examples of logarithmic scales

Mathematical Skills C.3 &ndash; Graphs

(iv) C.3 &ndash; graphs (C.3.1 to C.3.6)

RefMathematical SkillsExemplification of mathematical skill in the context of A Level Physics
C.3.1Translate information between graphical, numerical and algebraic formsStudents may be tested on their ability to:

• calculate Young modulus for materials using stress-strain graphs
C.3.2Plot two variables from experimental or other dataStudents may be tested on their ability to:

• plot graphs of extension of a wire against force applied
C.3.3Understand that y=mx+c\boldsymbol{y = mx + c} represents a linear relationshipStudents may be tested on their ability to:

• rearrange and compare v=u+at\boldsymbol{v = u + at} with y=mx+c\boldsymbol{y = mx + c} for velocity-time graph in constant acceleration problems
C.3.4Determine the slope and intercept of a linear graphStudents 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
C.3.5Calculate rate of change from a graph showing a linear relationshipStudents may be tested on their ability to:

• calculate acceleration from a linear velocity-time graph
C.3.6Draw and use the slope of a tangent to a curve as a measure of rate of changeStudents 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

(iv) C.3 &ndash; graphs (C.3.7 to C.3.12)

RefMathematical SkillsExemplification of mathematical skill in the context of A Level Physics
C.3.7Distinguish between instantaneous rate of change and average rate of changeStudents 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
C.3.8Understand the possible physical significance of the area between a curve and the x\boldsymbol{x} axis and be able to calculate it or estimate it by graphical methods as appropriateStudents 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
C.3.9Apply the concepts underlying calculus (but without requiring the explicit use of derivatives or integrals) by solving equations involving rates of change, e.g. Δx/Δt=−λx\boldsymbol{\Delta x / \Delta t = -\lambda x} using a graphical method or spreadsheet modellingStudents may be tested on their ability to:

• determine g\boldsymbol{g} from distance-time plot, projectile motion
C.3.10Interpret logarithmic plotsStudents may be tested on their ability to:

• obtain time constant for capacitor discharge by interpreting plot of log V\boldsymbol{V} against time
C.3.11Use logarithmic plots to test exponential and power law variationsStudents may be tested on their ability to:

• use logarithmic plots with decay law of radioactivity/charging and discharging of a capacitor
C.3.12Sketch relationships which are modelled by y=k/x\boldsymbol{y = k/x}, y=k/x2\boldsymbol{y = k/x^2}, y=kx\boldsymbol{y = kx}, y=sin⁡x\boldsymbol{y = \sin x}, y=cos⁡x\boldsymbol{y = \cos x}, y=e±x\boldsymbol{y = e^{\pm x}}, and y=sin⁡2x\boldsymbol{y = \sin^2 x}, y=cos⁡2x\boldsymbol{y = \cos^2 x} as applied to physical relationshipsStudents may be tested on their ability to:

• sketch relationships between pressure and volume for an ideal gas

Mathematical Skills C.4 &ndash; Geometry & Trigonometry

(v) C.4 &ndash; geometry and trigonometry

RefMathematical SkillsExemplification of mathematical skill in the context of A Level Physics
C.4.1Use angles in regular 2D and 3D structuresStudents may be tested on their ability to:

• interpret force diagrams to solve problems
C.4.2Visualise and represent 2D and 3D forms including two-dimensional representations of 3D objectsStudents may be tested on their ability to:

• draw force diagrams to solve mechanics problems
C.4.3Calculate areas of triangles, circumferences and areas of circles, surface areas and volumes of rectangular blocks, cylinders and spheresStudents 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
C.4.4Use Pythagoras' theorem, and the angle sum of a triangleStudents may be tested on their ability to:

• calculate the magnitude of a resultant vector, resolving forces into components to solve problems
C.4.5Use sin, cos and tan in physical problemsStudents may be tested on their ability to:

• resolve forces into components
C.4.6Use of small angle approximations including sin⁡θ≈θ\boldsymbol{\sin \theta \approx \theta}, tan⁡θ≈θ\boldsymbol{\tan \theta \approx \theta}, cos⁡θ≈1\boldsymbol{\cos \theta \approx 1} for small θ\boldsymbol{\theta} where appropriateStudents may be tested on their ability to:

• calculate fringe separations in interference patterns
C.4.7Understand the relationship between degrees and radians and translate from one to the otherStudents may be tested on their ability to:

• convert angle in degrees to angle in radians