Mathematical and Data Handling Skills
What this domain tests
Unit consistency, vector resolution, graph linearisation, logarithmic and exponential manipulation, and the interpretation of gradients and intercepts in physical terms. These skills are marked across every other domain, so a correct physics argument with sloppy units still drops marks.
Units, vectors and scalars
- Check homogeneity: every term in a valid equation has the same base units. Use this to test a remembered formula.
- SI base units: kg, m, s, A, K, mol, cd. Derived: N = kg m s⁻², J = N m, W = J s⁻¹, Pa = N m⁻², C = A s, V = W A⁻¹, F = C V⁻¹, T = kg s⁻² A⁻¹.
- Resolve a vector: Fx = F cosθ, Fy = F sinθ. Add vectors by components, then magnitude = √(Fx² + Fy²) and direction from tanθ = Fy/Fx with the quadrant determined from both component signs.
Linearising relationships
- y = axⁿ → log y = n log x + log a: plot log y against log x; gradient n, intercept log a.
- y = ae^(−kx) → ln y = −kx + ln a: plot ln y against x; gradient −k.
- T = 2π√(l/g) → plot T² against l; gradient = 4π²/g, so g = 4π²/gradient.
- 1/f = 1/u + 1/v → plot 1/v against 1/u; gradient −1, intercepts 1/f.
- Area under a graph has physical meaning (v–t gives displacement, p–V gives work).
Worked example
A student plots T² against l for a simple pendulum and obtains a straight line of gradient 4.05 s² m⁻¹. Find g.
From T = 2π√(l/g), squaring gives T² = (4π²/g)l, so the gradient is 4π²/g. g = 4π²/gradient = 39.48/4.05 = 9.75 m s⁻². The numerical difference from 9.81 m s⁻² is about 0.6%. Agreement must be assessed against an actual uncertainty budget; the numerical difference alone does not establish consistency.
Traps that cost marks
- Forgetting to convert cm to m before computing a gradient - the gradient's units then betray you.
- Taking the logarithm of a quantity with units; take the log of the numerical value only, or of a ratio.
- Gradients read from the steepest and shallowest lines through the error bars when a best-fit gradient is requested.
- Forcing a line through the origin when the intercept is physically meaningful (for example, a systematic offset).
- Rounding intermediate answers and carrying the error forward.
- Treating a curved graph as linear over the whole range.