IGCSE Physics / Practical and maths skills
Gradients, tangents and areas under graphs
Edexcel International GCSE Physics (4PH1), chapter 29: Practical and maths skills, specification point M4B, M4D, M4E, M4F. Free lesson.
Read the revision notes · IGCSE Physics past papers
A graph’s gradient and area can represent useful physics quantities. You need to calculate them with the right method and units, then use the graph type to decide what they mean.
How to find a gradient and intercept
A gradient is the change up divided by the change across. Use a large triangle on a straight line so the changes are easy to read. For a straight-line graph, y = mx + c: m is the gradient and c is the intercept, where the line crosses the vertical axis. Include the gradient’s unit, found by dividing the vertical-axis unit by the horizontal-axis unit. On a velocity-time graph, velocity is in m/s and time is in seconds, so the gradient is in m/s².
What a velocity-time graph tells you
For a cyclist whose velocity rises steadily from 4 m/s to 16 m/s in 6 s, the graph’s gradient is the acceleration. The intercept is the starting velocity. The area under the graph gives the distance travelled, because velocity multiplied by time gives distance. Split the area into shapes you can calculate, such as a rectangle and a triangle. Check the units: m/s × s = m.
How to find the gradient of a curve
A curve’s gradient changes from point to point. To find it at one particular point, draw a tangent there: a straight line that just touches the curve and has the same slope at that point. Use a large triangle on the tangent, not on the curve. On a distance-time graph, the gradient gives the speed at that instant. A line joining two points on the curve gives an average instead. For the trolley at 4 s, the tangent rises 16 m in 4 s, so its gradient is 4 m/s. This is the trolley’s speed at exactly 4 s.
What gradients and areas mean on different graphs
The meaning of a gradient or area depends on the graph’s axes. On a distance-time graph, the gradient is speed and the area is not used. On a velocity-time graph, the gradient is acceleration and the area is distance travelled. On a current-voltage graph with current on the vertical axis, gradient = I ÷ V = 1 ÷ resistance. So resistance = 1 ÷ gradient. You can also find resistance from a point on the graph using R = V ÷ I.
How to estimate area by counting squares
You can estimate the area under a curve by counting the squares it covers. First work out the value of one square by multiplying its height by its width. Then multiply by the number of squares covered. If each square is 1 m/s high and 2 s wide, one square represents 2 m of distance. About 34 squares therefore represent about 68 m.