Graphs

The straight line —

y = mx + c

where m is the gradient and c the y-intercept. Most graph questions in physics reduce to identifying which physical quantity plays the part of m, and which of c.

Finding a gradient — choose two points on the line, not two data points, as far apart as possible, and use

m = (y₂ − y₁) ÷ (x₂ − x₁)

Using a large triangle matters: reading errors are a smaller fraction of a large Δy and Δx, so the gradient is more reliable.

Units of a gradient — divide the unit on the y-axis by the unit on the x-axis. A graph of potential difference in volts against current in amperes has a gradient in V A⁻¹, which is the ohm.

Intercepts — the y-intercept is where the line crosses at x = 0. It has physical meaning only if x = 0 lies within the measured range; otherwise you are extrapolating and should say so.

Linearising — rearrange the physics until a straight line is expected, then plot whatever combination makes the equation match y = mx + c.

Example: for a gas at constant temperature, pV = constant. Plotting p against V gives a curve, which is hard to judge. Rearranged as p = constant × (1/V), plotting p against 1/V gives a straight line through the origin of gradient equal to that constant.

A checklist for linearising — write the equation, decide what you can plot on each axis so that the equation matches y = mx + c, then say what the gradient and intercept will represent before drawing anything. Doing this first stops you plotting the wrong pair of variables.

Area under a graph — often a physical quantity, and the units tell you which. On a velocity–time graph, m s⁻¹ × s = m, so the area is displacement. On a force–extension graph, N × m = J, so it is work done.

Tangents — for a curved graph, the gradient at a point is the gradient of the tangent there. The tangent must touch the curve at that point and match its direction, and the triangle used should span most of the tangent's length.