S1.3 Mathematics

Choose the mathematics from the relationship

First identify what changes, what is held constant and whether the model is additive, proportional, exponential or geometric. Rearrange symbols before substituting numbers; this exposes the dependence and reduces calculator-entry errors.

Worked check — rearrangement and proportion

From $E=\tfrac12mv^2$, $v=\sqrt{2E/m}$. If $m$ is unchanged and $E$ becomes four times larger, $v$ becomes $\sqrt4=2$ times larger. This scale check should agree with the calculated value.

Estimate before accepting a result

Keep guard digits until the end, compare the answer with the nearest order of magnitude and neglect an effect only when you can explain why it is small. A proportionality is not an equation until its constant is included.

A vector needs magnitude and direction

Draw its arrow to scale when a scale diagram is required, label its magnitude, direction and point of application, and choose axes before resolving it. Scalars such as mass and energy have magnitude only; force, velocity and momentum are vectors.

Worked resolution

For a $10.0\,\text{N}$ force at $30^\circ$ above the positive horizontal, $F_x=10.0\cos30^\circ=8.66\,\text{N}$ and $F_y=10.0\sin30^\circ=5.00\,\text{N}$. The signs change if the chosen directions change; the physical vector does not.

Free-body diagram method

Choose one object, draw only forces acting on it at the required point of application or centre of mass, then add up to three coplanar vectors head-to-tail or by components. Subtraction means adding the reversed vector; multiplying by a negative scalar reverses direction.

Do not mix an interaction pair

The force exerted by the object on its surroundings belongs on the surroundings' diagram, not on the object's own free-body diagram.

Make every number carry a compatible unit

Convert prefixes before substitution and use the symbols defined in the guide/data booklet. The SI base units most often used in physics are metre (m), kilogram (kg), second (s), ampere (A) and kelvin (K); mole (mol) and candela (cd) complete the SI base set.

Use units as an equation check

For $E=\tfrac12mv^2$, the right side has units $\text{kg}\,\text{m}^2\,\text{s}^{-2}=\text{J}$, so it can represent energy. This unit check can reject an expression, but matching units alone do not prove that its numerical factor or physics is correct.

Round once, at the end

Keep guard digits during working. Report the final value and its uncertainty to compatible decimal places, with the uncertainty usually at one significant figure (or two when needed to avoid misleading rounding). Units such as eV, ly, pc, hour, day and year may be used where the syllabus context makes them appropriate.

Uncertainty states a range, not a mistake

Write a measured result as $x\pm\Delta x$ in matching units. The absolute uncertainty is $\Delta x$; fractional uncertainty is $\Delta x/x$; percentage uncertainty is $(\Delta x/x)\times100\%$.

Worked propagation

For $L=(2.00\pm0.01)\,\text{m}$ and $W=(1.00\pm0.01)\,\text{m}$, $A=LW=2.00\,\text{m}^2$. The fractional uncertainty is $0.01/2.00+0.01/1.00=0.015$, or $1.5\%$. Therefore $\Delta A=0.015\times2.00=0.03\,\text{m}^2$, so $A=(2.00\pm0.03)\,\text{m}^2$.

Powers multiply fractional uncertainty

If $V\propto r^3$ and $r$ has $2\%$ uncertainty, $V$ has $3\times2\%=6\%$ uncertainty under the syllabus propagation rule. For addition or subtraction, add absolute uncertainties instead—never percentage uncertainties.

Report sensible precision

Round the uncertainty first, then round the measured value to the same decimal place. A small difference between two measured values can have a large percentage uncertainty even when both original measurements look precise.

Build a graph that exposes the relationship

Put the independent variable on the horizontal axis, label both axes with quantity and unit, choose scales that use the plotting area, and show uncertainty bars where available. Use a line or curve of best fit for the trend; do not join each point.

Extract meaning from the fit

Calculate gradient from two well-separated points on the best-fit line and include its units. Use maximum- and minimum-gradient acceptable lines through the uncertainty bars to estimate gradient uncertainty; apply the same idea to intercepts. Interpolate within the measured range cautiously; extrapolation relies on the model continuing beyond the evidence.

Example — test an exponential claim

For exponential decay, equal time intervals should give approximately the same multiplicative factor (or a constant half-life). Alternatively, a suitable logarithmic transformation should be linear. Agreement must be judged with the uncertainty bars, not from visual closeness alone.

Interpret, do not merely describe

A gradient is a rate of change; a changing gradient shows a changing rate; an intercept is the predicted value at zero input; maxima/minima are turning points; and an area under a graph is an accumulated quantity only when the product of the axis units represents that quantity.