Maths bridge › Gradients and areas under graphs
Gradients and areas under graphs
What a gradient and an area mean physically, and how to extract both with their units.
The two questions every graph answers
The gradient is the rate of change of the y quantity with the x quantity, and its unit is the y unit divided by the x unit. The area under the line is the product of the two quantities accumulated along the x axis, with the product unit. Before calculating either, say what it means for those axes; that sentence is often a mark by itself.
On a curve, the gradient at a point is the gradient of the tangent drawn there, not of a chord between two convenient points. Draw the tangent long, and read its rise and run from widely separated points to keep the percentage error down.
Worked and handed over
WORKED EXAMPLE
A velocity-time graph, both ways
A velocity-time graph rises in a straight line from 2.0 m s−1 to 8.0 m s−1 over 4.0 s. Find the acceleration and the distance covered.
Gradient first: rise over run is (8.0 − 2.0)/4.0 = 1.5 m s−2, and the unit came from the axes: velocity over time.
The area under the line is a trapezium: ½ × (2.0 + 8.0) × 4.0 = 20 m. Area under velocity-time is displacement, and the unit m s−1 × s = m confirms it.
YOUR TURN
Say it, then find it
A graph plots charge (in coulombs) on the y axis against time (in seconds) on the x axis for a discharging capacitor. State what the gradient of the tangent at any point represents, with its unit, before opening the working.
Show the working
The gradient is the rate of change of charge with time, which is the current, in coulombs per second: amperes.
Its sign is negative during discharge, because the charge is falling; the magnitude of the tangent's gradient is the size of the current at that instant.
Where physics leans on this: Motion graphs and SUVAT · Charging and discharging · Density and Hooke's law. All eight skills: the maths bridge.