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SPM Mathematics: How to Master Graphs of Motion
The trick is knowing what gradient and area mean on each graph: on a distance-time graph gradient is speed, while on a speed-time graph gradient is acceleration and area is distance.
Two graphs, two meanings
A distance-time graph and a speed-time graph look similar but say different things, and confusing them is the number-one error here. On a distance-time graph the gradient is speed and a flat line means stationary; on a speed-time graph the gradient is acceleration and the area under the line is distance travelled.
Label which graph you are reading before you calculate anything.
Area means breaking it into shapes
To find distance from a speed-time graph, split the area into triangles and trapeziums and add them up; this is where careful arithmetic earns full marks. Watch the units: if speed is in metres per second and time in seconds, distance comes out in metres.
A quick sketch of the shapes you are adding prevents missing a section.
Checking the Axes Before You Trust the Numbers
Before calculating anything, check what each axis is actually measuring. Distance might be given in metres or in kilometres, and time might be in seconds, minutes, or hours, mixing these up is one of the most common ways to lose marks even when every formula and method is correct.
If the question gives speed in km/h but asks for an answer in m/s, convert first, before the gradient or area calculation, not after. Also check where the graph actually starts: many questions have an object already moving at the start of the graph rather than starting from rest, or the horizontal axis beginning at a value other than zero, and missing this changes every reading taken from the graph.
For the curved sections that occasionally appear in SPM questions, remember you're almost always only asked to describe the motion in words, speeding up at a changing rate, or slowing down at a changing rate, rather than calculate an exact value, since finding a precise gradient or area from a curve is beyond what's expected at this level.
Common Question Patterns in Paper 2
Most Paper 2 motion-graph questions describe a multi-stage journey: a vehicle speeds up for a few seconds, travels at a constant speed for a while, then slows to a stop, and you're given some information but not all of it. The usual tasks are to complete a missing value on the graph, calculate the speed or acceleration for a specific stage, find the total distance travelled, or find the average speed for the whole journey.
A useful habit is to label each stage on the graph itself, Stage 1, Stage 2, Stage 3, as soon as you read the question, and write down what you know and what you need to find for each one. This turns one intimidating multi-part question into three or four small, familiar calculations.
Diagrams are almost always drawn to scale on the grid lines even when a value is missing, so cross-checking a calculated answer against how it looks on the graph is a quick way to catch an error before you move on.
Frequently asked questions
What's the difference between a distance-time graph and a speed-time graph?
A distance-time graph shows how far an object has travelled at each moment, and its gradient gives speed. A speed-time graph shows how fast an object is moving at each moment, and its gradient gives acceleration, while the area under the graph gives distance travelled.
How do you find the distance travelled from a speed-time graph?
Find the area of the region under the graph between the two times you're interested in. Break the region into simple shapes, rectangles and triangles are the most common, calculate each area separately, then add them together.
Why is average speed not the same as the average of the speeds shown on the graph?
Average speed is defined as total distance divided by total time for the whole journey, not the arithmetic mean of the speeds at different stages. If a journey spends unequal amounts of time at each speed, simply averaging the speed values gives the wrong answer, you must calculate total distance and total time separately first.