The big idea: A straight line is easy to read — its gradient is a single number. So when a law is curved (like P getting bigger as V gets smaller), we re-plot it as a straight line.
Pick what to put on each axis so the graph comes out straight. This is linearizing.
Once it is straight you can do two things
- Read a physics quantity off the gradient.
- Test whether the data really obey the law.
What 'directly proportional' looks like: Two quantities are directly proportional when their graph is a straight line that passes through the origin (0, 0).
So to test a 'proportional' claim, you check both: is the line straight, and does it go through the origin?
Here is a curved law made straight. The gas law PV = constant is a curve if you plot P against V — but if you instead plot P against 1/V (one over the volume), the points fall on a straight line through the origin:
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
Read it like y = mx + c: Compare your plotted variables to the line equation Y = mX + c:
- the thing on the up axis is Y - the thing on the across axis is X - the gradient m and intercept c are then physics quantities.
Free preview
This is the free notes preview
You're reading the free notes. Aimnova Pro unlocks the full study experience — and you can try it with your first topic free to keep:
- FlashcardsLock in vocabulary and key terms with spaced repetition.
- Practice questionsAnswer exam-style questions and get instant AI marking.
- Mock exams & past-paper vaultSit full mocks and see exactly how examiners award marks.
- Personalised study planA daily plan built around your exam date and weak areas.
Method: rearrange the law into the form Y = mX + c. Whatever multiplies the variable becomes the gradient.
Example: the gas law: The gas law PV = k rearranges to P = k × (1/V) — so plotting P (up) against 1/V (across) gives a straight line whose gradient is k.
- the quantity plotted on the vertical axis (chosen so the graph is straight)
- the quantity plotted on the horizontal axis
- the gradient — equals a physics constant you are trying to find
- the vertical intercept — usually 0 for a 'directly proportional' law
The gradient is a physics quantity: After linearizing, the gradient is never 'just a number' — it equals a constant in the law (a spring constant, a refractive index, a gas constant…). Always say what the gradient represents and give its units.
A student measures the time period T of a pendulum for several lengths L. Theory says T = k√L (T is proportional to the square root of L). The student wants a straight-line graph through the origin so they can find k from the gradient. (a) State what to plot on each axis. (b) State what the gradient represents.
Model answer plan
See the answer broken into clear points, with guidance on how each point earns marks.
Get feedback like a real examiner
Submit your answers and get instant feedback — what you did well, what's missing, and exactly what to write to score full marks.
Powers and reciprocals straighten many laws, but not an exponential or a law with an unknown power. Taking logarithms does.
Which log graph?
| Law | Take logs | Plot | Gradient | Intercept |
|---|---|---|---|---|
| y = A e−kx | ln y = ln A − kx | ln y against x | −k | ln A |
| y = k xⁿ | log y = n log x + log k | log y against log x | n (the power) | log k |
Testing for exponential change without logs: An exponential loses the same fraction in equal times. Read the time to halve from several starting values on the graph: if it is constant, the change is exponential.
A ball is dropped and the distance d it falls in time t is measured. A graph of log(d / m) against log(t / s) is a straight line of gradient 2.0 and intercept 0.69. Determine the power of t in the relationship and the constant k in d = k tⁿ.
Model answer plan
See the answer broken into clear points, with guidance on how each point earns marks.
How this is tested: Paper 1B almost always builds a whole question around one experiment, and linearizing is the part that ties it together.
What they ask
- State what to plot to get a straight line.
- Find a missing point's coordinates so you can plot it.
- Explain how the graph decides whether a proposed law holds.
- Show (with two data rows) that two quantities are not directly proportional.
The classic trap: Assuming any straight-ish line means 'proportional'. It only counts as proportional if the line also passes through the origin.
An experiment measures how the depth d that a marker sinks depends on the water pressure P pushing on it. Theory predicts d = k√P, so the student plans to plot depth against root-pressure (√P). The graph below is straight and passes through the origin, supporting the law:
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
One table row reads pressure P = 9.0 kPa and depth d = 4.6 cm. For the depth-versus-root-pressure graph, determine the coordinates of the point this row should be plotted at.
Model answer plan
See the answer broken into clear points, with guidance on how each point earns marks.
Suppose a classmate instead claims depth is directly proportional to pressure itself (d ∝ P). Using these two rows — P = 4.0 kPa with d = 3.1 cm, and P = 9.0 kPa with d = 4.6 cm — show that depth and pressure are not directly proportional.
Model answer plan
See the answer broken into clear points, with guidance on how each point earns marks.