Interpreting distance-time and velocity-time graphs, finding gradients as rates and areas as totals, and estimating gradients of curves at Higher tier.

What this dot point is asking

AQA wants you to read and interpret distance-time and velocity-time graphs, use the gradient as a rate of change and the area under a graph as an accumulated total, and at Higher tier estimate the gradient of a curve with a tangent. These graphs turn real-world motion into geometry, and the two key ideas (gradient is a rate, area is a total) recur in many applied questions.

Distance-time graphs

A distance-time graph plots distance from a start point against time. The gradient of each segment is the speed, because speed is distance divided by time. A straight sloping line means constant speed; a horizontal line means the object is stationary (distance not changing); a steeper line means a higher speed. A line returning to the axis shows the object travelling back toward the start.

To find the speed on a segment, compute the change in distance divided by the change in time over just that segment. Be careful to exclude any rest periods, which are separate flat sections.

Velocity-time graphs

A velocity-time graph plots speed against time, and it carries two pieces of information at once.

Estimating gradients and areas under curves at Higher tier

When a graph is curved, the rate of change is not constant. To estimate the rate at a particular point, draw a tangent (a straight line just touching the curve there) and find its gradient using two points on the tangent. The gradient of a tangent on a distance-time curve gives the instantaneous speed, and on a velocity-time curve gives the instantaneous acceleration.

To estimate the area under a curve (and so the distance from a velocity-time curve), divide the region into thin strips and treat each as a trapezium, then add the strip areas. More, narrower strips give a closer estimate. Counting grid squares is an alternative for a quick approximation.

Interpreting graphs in context

Exam questions increasingly ask you to describe what a graph means in words, not just calculate. A distance-time graph that gets steeper shows acceleration; a flat section shows a stop. On a velocity-time graph, a downward-sloping line means deceleration (negative acceleration), and the object is still moving forward until the velocity reaches zero. Other real-world graphs follow the same logic: on a graph of water depth against time as a container fills, a steep section means the depth rises fast (a narrow part of the container) and a shallow section means it rises slowly (a wide part). Reading the gradient as a rate is the unifying idea across all of these.

Average versus instantaneous rate

A common distinction is between an average rate over an interval and an instantaneous rate at a moment. The average speed over a journey is the total distance divided by the total time, which is the gradient of the straight chord joining the start and end points on a distance-time graph. The instantaneous speed at a single instant is the gradient of the tangent at that point. For a straight-line graph the two are equal, but for a curve they differ, and recognising which one a question wants (a chord for average, a tangent for instantaneous) is essential at Higher tier.