Form 4 · Relationship and Algebra
Graphs of Motion
Graphs of motion read distance and speed from a picture, gradients become speeds and areas become distances.
What is Graphs of Motion?
This chapter connects graphs to movement. On a distance–time graph the gradient is speed; on a speed–time graph the gradient is acceleration and the area underneath is distance travelled.
You learn to draw these graphs, read values off them, and interpret what a shape says about a journey.
Content standards (DSKP)
The DSKP KSSM sets these content standards for this chapter:
- 7.1 Distance-Time Graphs
- 7.2 Speed-Time Graphs
The key ideas
Gradient means rate
A steeper line means a faster rate, speed on a distance–time graph, acceleration on a speed–time graph.
Area under a speed–time graph
The area beneath the line is the distance travelled, usually found by splitting it into triangles and rectangles.
Reading a journey from the shape
A flat line means stationary; a downward slope on a distance graph means returning. The shape tells the story.
How this chapter is examined
A Paper 2 favourite: a speed–time graph where you find acceleration from a gradient and total distance from the area, often across several stages of a journey. Real-life contexts run throughout, so read the units carefully.
Formulae given in the exam for this chapter
Common mistakes to avoid
- Confusing a distance–time graph with a speed–time graph
- Forgetting units when reading a gradient
- Missing a part of the area when the graph has several stages
A flat line means different things on the two graphs
Because a distance–time graph and a speed–time graph look similar, the same shape can mislead you if you do not check the vertical axis first. A horizontal (flat) line on a distance–time graph means the distance is not changing, so the object is stationary, parked, resting, waiting.
But a horizontal line on a speed–time graph means the speed is not changing, so the object is moving at a constant speed, which is the opposite of standing still. In the same way, a straight sloping line on a distance–time graph is constant speed, while a straight sloping line on a speed–time graph is constant acceleration.
Always read the axis label before you interpret a shape; the picture alone does not tell you what is happening.
Splitting the area under a speed–time graph
On a speed–time graph the distance travelled is the area between the line and the time axis, and the reliable way to find it is to slice that area into simple shapes you already know how to measure. A stage where speed is constant gives a rectangle, area = base × height.
A stage of uniform acceleration from or to rest gives a triangle, area = ½ × base × height. A stage that speeds up or slows down between two non-zero speeds gives a trapezium, area = ½ × (sum of the two parallel sides) × width.
Work out each piece separately and add them for the total distance, and take care never to skip a stage, because a missed slice is a missed chunk of distance. You are expected to know these area formulas; they are not printed on the exam formula sheet.
Average speed is total distance over total time
A trap that costs many students marks is averaging the speeds instead of using distance and time. Average speed is defined as the total distance travelled divided by the total time taken, never the mean of the individual stage speeds.
If a car covers 80 m in one stage and 240 m in another, and the whole journey takes 25 seconds, the average speed is (80 + 240 + …) ÷ 25, not the average of the stage speeds. The two only agree by accident.
So whenever a question asks for average speed across a multi-stage journey, first add up every part of the distance (usually the whole area under a speed–time graph), then add up every part of the time, and divide once at the end. Keep the units consistent, and remember the average speed will normally sit somewhere between the slowest and fastest stages, which is a quick sanity check.
A worked exam-style example
This example is a classic speed–time journey where you find acceleration from a gradient, distance from areas, and then the average speed.
- (a) Acceleration is the gradient of the speed–time line: a = (change in speed) ÷ (time taken) = (20 − 0) ÷ 8.
- So a = 20 ÷ 8 = 2.5 m/s².
- (b) Total distance is the whole area under the speed–time graph, so split it into the three stages.
- Stage 1 (accelerating, a triangle): area = ½ × 8 × 20 = 80 m.
- Stage 2 (constant speed, a rectangle): area = 12 × 20 = 240 m.
- Stage 3 (decelerating, a triangle): area = ½ × 5 × 20 = 50 m.
- Total distance = 80 + 240 + 50 = 370 m.
- (c) Total time = 8 + 12 + 5 = 25 s.
- Average speed = total distance ÷ total time = 370 ÷ 25 = 14.8 m/s.
How to study this chapter
Study Graphs of Motion
Frequently asked questions
How this chapter is examined
SPM Mathematics assesses this chapter across Mathematics Paper 1 (Objective) and Mathematics Paper 2 (Subjective), drawing on the DSKP content standards above. Paper 2 gives marks for working, so showing every step matters.
Common mistakes to avoid
Confusing a distance–time graph with a speed–time graph; Forgetting units when reading a gradient; Missing a part of the area when the graph has several stages.
Formulae given in the exam for this chapter
Yes, Distance between two points, Average speed, Gradient (two points), Gradient (intercepts) appear on the formula sheet the exam provides. Anything else in this chapter you are expected to know.
How do I tell a distance–time graph from a speed–time graph?
Read the vertical axis label first. If it says distance or displacement, the gradient gives speed and the height shows how far from the start.
If it says speed or velocity, the gradient gives acceleration and the area beneath gives distance. The two graphs look alike, so always check the axis before you calculate anything.
Do I use the gradient or the area?
The gradient gives a rate: speed from a distance–time graph, acceleration from a speed–time graph. The area under the line gives an accumulated total: on a speed–time graph the area is the distance travelled.
Ask what the question wants, a rate points to the gradient, a total distance points to the area.
A section slopes down to zero, is that negative distance?
No. On a speed–time graph a downward slope means the object is decelerating, but it is still moving forward, so the area beneath it is still positive distance you add on.
Only the gradient turns negative during deceleration; the area stays a real distance travelled, never a subtraction.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)· SPM: Format Pentaksiran mulai 2021, Matematik (1449)
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