Gradients in Kinematic Graphs

Reading a gradient in context

A gradient compares a change up or down the graph with the corresponding change across it. For a straight section, choose two clear points on that section and calculate

$\text{gradient}=\frac{\text{change in vertical value}}{\text{change in horizontal value}}=\frac{y_2-y_1}{x_2-x_1}.$

The axis labels tell you what the answer means. If the vertical axis is measured in metres and the horizontal axis in seconds, the gradient has units of metres per second. A horizontal section has gradient zero; a downward section has a negative gradient. On a graph with several straight sections, calculate each gradient separately because the rate may change.

Distance–time graphs

On a graph of distance from a starting point against time, the magnitude of a straight section's gradient gives speed. The steeper the section, the faster the distance changes. A horizontal section means the object stays the same distance from the start, so it is stationary. A downward section means it is travelling back towards the start: its gradient is negative, although its speed is positive.

Data for Gradients on a distance-from-start graph

In this journey, the distance from the start rises from 0 km to 6 km during the first 2 hours. The gradient is $(6-0)/(2-0)=3$ km/h, so the speed is 3 km/h. From 2 to 3 hours the distance stays at 6 km: the gradient and speed are both zero. From 3 to 5 hours it falls from 6 km to 2 km. The gradient is $(2-6)/(5-3)=-2$ km/h; the traveller's speed is 2 km/h towards the start. If an axis instead shows , its values cannot fall as the journey continues, so check its label before describing a downward line.

Velocity–time graphs

Velocity includes direction. On a velocity–time graph, a straight section's gradient is the acceleration: change in velocity divided by change in time. When velocity is in m/s and time is in s, the gradient is in m/s². A rising section shows positive acceleration. A horizontal section means constant velocity and zero acceleration. If velocity remains positive while the line falls, the object is slowing down: its acceleration is negative, and the size of that negative acceleration is its deceleration.

Data for Gradients on a velocity–time graph

In this graph, velocity rises from 0 to 12 m/s over the first 4 seconds. Its gradient is $(12-0)/(4-0)=3$ m/s²: velocity increases by 3 m/s each second. Between 4 and 8 seconds it stays at 12 m/s, so acceleration is zero. Between 8 and 10 seconds the gradient is $(4-12)/(10-8)=-4$ m/s², describing a deceleration of 4 m/s² while the object is still moving in the positive direction.

The same method beyond motion

The calculation does not change when the axes describe money rather than movement. For example, if a straight-line graph of total hire cost rises from £40 at 0 miles to £140 at 200 miles, its gradient is $(140-40)/(200-0)=£0.50$ per mile. This means each additional mile adds 50p to the cost. Always name the quantity changing, not just the numerical gradient.