Analysing Results

Straight-line graphs — a straight line means the two plotted quantities are related by

y = mx + c

where m is the gradient and c the y-intercept. Almost all A-level analysis reduces to identifying which physical quantity plays the part of m and which of c.

Finding a gradient — pick two points on the line, not two data points, as far apart as the line allows. Use the largest triangle that fits, because the uncertainty in the gradient depends on the size of the triangle you read it from.

gradient = Δy ÷ Δx

The gradient has units — divide the units of the y-axis by the units of the x-axis. A graph of extension in metres against force in newtons has a gradient in m N⁻¹, which is the reciprocal of the spring constant.

Example: a force–extension graph for a spring has force on the y-axis and extension on the x-axis, so the gradient is the spring constant k in N m⁻¹. Swap the axes and the gradient becomes 1/k. Always work out what the gradient means from the axes rather than from memory.

Proportionality — two quantities are directly proportional only if the graph is a straight line through the origin. A straight line that misses the origin shows a linear relationship, which is not the same claim.

Linearising a relationship — a curve is hard to interpret, so rearrange the physics until a straight line is expected. Compare your equation with y = mx + c and plot whatever combination makes it match.

Example: for a gas at constant temperature, pV = constant. A graph of p against V is a curve. Rearranged as p = constant × (1/V), plotting p against 1/V should give a straight line through the origin whose gradient is that constant — and a straight line is far easier to judge than a curve.

Intercepts — the y-intercept is the value of y when x is zero, read where the line crosses the axis. It only has physical meaning if x = 0 is actually on your graph; otherwise you are extrapolating, and you should say so.

Extrapolation — extending the line beyond the measured range. It can be genuinely powerful: extrapolating a pressure–temperature graph back to zero pressure gives an estimate of absolute zero, at about −273 °C, without ever cooling anything near it. But it assumes the relationship continues to hold outside the range you tested, which is an assumption, not a measurement.

Area under a graph — often a physical quantity. The area under a velocity–time graph is displacement; under a force–extension graph it is the work done. Check by multiplying the units of the two axes: m s⁻¹ × s gives m, which confirms displacement.