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Gradients and areas under graphs

A gradient is the change in y over the change in x, so its unit is the y-axis unit over the x-axis unit. The area under a graph has the two units multiplied. What each means physically depends on the axes: on a velocity-time graph the gradient is acceleration and the area is displacement.

Gradient and area

The gradient is dy/dx and has units equal to the y-axis unit divided by the x-axis unit. The area between a curve and the x-axis is the integral of y with respect to x and has units equal to the product of the axis units. The physical meaning of each depends on the plotted quantities, so state what the gradient or area means for those axes before calculating it.

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.

A displacement-time curve rising ever more steeply. A tangent touches it at point P and gives the gradient at P. A dashed chord joins P to a later point R; it is steeper than the tangent and gives only the average gradient between P and R. — FIG. 1On a displacement-time graph the tangent at P gives the velocity at that instant. The chord from P to R gives only the average velocity between the two, and it is steeper here because the curve bends upwards.

Both readings in practice

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.

Velocity-time graph: a straight line from 2.0 m/s at time zero to 8.0 m/s at 4.0 s. A triangle under the line has a run of 4.0 s and a rise of 6.0 m/s, so the gradient is 1.5 m/s squared, the acceleration. The trapezium under the line is shaded, with area 20 m, the displacement. — FIG. 2The example drawn to scale. Rise over run on the amber triangle is the acceleration; the shaded trapezium is the displacement.

Gradient first. Rise over run is (8.0 − 2.0)/4.0 = 1.5 m s⁻², and the unit came straight off 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.

GUIDED PRACTICE

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, measured in coulombs per second, otherwise known as 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.

Gradient and area on common graphs

Graph (y against x)	Gradient	Gradient unit	Area under the graph	Area unit
displacement-time	velocity	m s⁻¹	not used
velocity-time	acceleration	m s⁻²	displacement	m
acceleration-time	not used		change in velocity	m s⁻¹
momentum-time	resultant force	N	not used
force-time	not used		impulse (change in momentum)	N s
force-extension	stiffness k, on the straight section	N m⁻¹	work done in stretching	J
stress-strain	Young modulus, on the straight section	Pa	work done per unit volume	J m⁻³
charge-time	current	A	not used
current-time	not used		charge	C
charge-p.d. for a capacitor	capacitance	F	energy stored	J
power-time	not used		energy transferred	J

Where physics leans on this: Motion graphs and SUVAT · Charging and discharging · Density and Hooke's law. All eight skills: the maths bridge.