The gradient of a straight-line graph is (change in y) ÷ (change in x), and its unit is the y-unit divided by the x-unit. The number alone is not a full answer: you also state the unit and what the gradient tells you about the physics. It is used across physics investigations and explanations, after you have chosen the axes.
How do you find and interpret a gradient?
- Draw the line of best fit: a ruler line with about equal scatter of points above and below it.
- Choose two points on the line (not necessarily data points) that are far apart, and mark a large right-angled triangle.
- Read the change in y and the change in x from the axes, with their units.
- Divide, then write the unit as y-unit per x-unit.
- Connect to the model: what physical quantity does m equal?
Worked example
Invented data. A student hangs loads from a spring and records the extension. The graph has load F (N) on the y-axis and extension x (cm) on the x-axis.
| F / N | 0 | 1.0 | 2.0 | 3.0 | 4.0 | 5.0 |
|---|---|---|---|---|---|---|
| x / cm | 0 | 2.1 | 3.9 | 6.1 | 8.0 | 9.9 |
The line of best fit passes through the origin and through (8.0 cm, 4.0 N).
Step 1: Change in F = 4.0 − 0 = 4.0 N. Change in x = 8.0 − 0 = 8.0 cm.
Step 2: Gradient = 4.0 N ÷ 8.0 cm = 0.50 N/cm.
Step 3: Convert to N/m: 1 cm = 0.01 m, so 0.50 N/cm = 0.50 N ÷ 0.01 m = 50 N/m.
Step 4: Meaning: the gradient is the force needed per unit extension, the spring constant k = 50 N/m. Check against the table: 2.0 N gives about 3.9 cm, and 50 N/m × 0.039 m = 1.95 N, which is close to 2.0 N.
The mistake to watch for
A student writes the working correctly but then states: “k = 0.50 N/m.”
Mistaken answer: 4.0 N ÷ 8.0 cm = 0.50 N/m
The number was worked out with centimetres, so the unit must be N/cm. Writing N/m without converting makes the spring 100 times too soft.
The correction is to carry the unit through the division, then convert deliberately. A second common slip is to divide x by F, which gives 2.0 cm/N. That is the inverse of the spring constant, so it answers a different question.
Check yourself
1. A speed-time graph is a straight line from 2 m/s at 0 s to 14 m/s at 6 s. Find the gradient with its unit and say what it means.
Show answer
Gradient = (14 − 2) m/s ÷ 6 s = 12 ÷ 6 = 2 m/s². It is the acceleration of the object.
2. A distance-time line passes through (10 s, 20 m) and (30 s, 80 m). Find the gradient and what it means.
Show answer
Gradient = (80 − 20) m ÷ (30 − 10) s = 60 ÷ 20 = 3.0 m/s. It is the constant speed.
3. A current-voltage graph for a resistor has I on the y-axis. The line passes through the origin and (6.0 V, 0.30 A). Find the gradient, then the resistance.
Show answer
Gradient = 0.30 A ÷ 6.0 V = 0.050 A/V. This is 1/R, so R = 1 ÷ 0.050 = 20 Ω. Check: V = IR gives 0.30 × 20 = 6.0 V.
Where this leads next
The gradient is only as reliable as the data behind it, so the next lesson looks at systematic and random limitations. The bounds and rounding explainer shows how rounded readings create an interval, and the reasoning board practises choosing a relationship before calculating.
Students who divide correctly but lose the unit or the meaning can get quick feedback in online one-to-one Physics tuition.