What you'll learn
Travel graphs — distance–time graphs and speed–time graphs — turn motion into a picture you can read and measure, and they form part of the CSEC Mathematics syllabus under Relations, Functions and Graphs. They let you describe a journey, calculate speed and acceleration, and work out distance travelled, all from the shape of a line. In this guide you will learn how to interpret each type of graph, what the gradient and the area underneath represent, and how to handle journeys with several stages including stops. These graphs appear in Paper 1 and Paper 2, often in a real-world context such as a bus route or a cyclist's trip, and reward careful reading of axes and units.
Key terms and definitions
Distance–time graph — a graph with time on the horizontal axis and distance on the vertical axis.
Speed–time graph — a graph with time on the horizontal axis and speed on the vertical axis.
Gradient (slope) — the steepness of a line, representing speed (on a distance–time graph) or acceleration (on a speed–time graph).
Speed — distance travelled per unit time, e.g. metres per second or km/h.
Acceleration — the rate of change of speed per unit time.
Deceleration — negative acceleration; slowing down.
Area under the graph — on a speed–time graph, the area represents the distance travelled.
Core concepts
Reading a distance–time graph
On a distance–time graph, the vertical axis shows how far an object is from a starting point. A straight, sloping line means constant speed; the steeper the line, the faster the motion. A horizontal line means the object is stationary (distance not changing) — a stop. A line sloping back down to zero means returning towards the start. The gradient of the line gives the speed: speed = distance ÷ time = change in distance ÷ change in time.
Calculating speed from a distance–time graph
To find the speed during any stage, pick two points on that stage and compute the gradient: speed = (change in distance) ÷ (change in time). For a whole journey with stops, average speed = total distance ÷ total time, including the time spent stationary.
Reading a speed–time graph
On a speed–time graph, the vertical axis shows speed. A horizontal line now means constant speed (not a stop). A line sloping upward means acceleration (speeding up); sloping downward means deceleration (slowing down). A line returning to the time-axis means the object has stopped (speed zero).
Gradient and area on a speed–time graph
Two measurements matter. The gradient gives acceleration: acceleration = change in speed ÷ change in time. The area under the line gives the distance travelled. For sections that are rectangles or triangles, use ½ × base × height for triangles and base × height for rectangles; for a trapezium, use ½(a + b)h. Adding the areas of each section gives the total distance.
Multi-stage journeys
Real problems combine several stages: accelerate, travel at constant speed, decelerate, stop. Read each section separately, calculate what is asked (speed, acceleration or distance) for that section, then combine. Always check the axis units (seconds vs hours, metres vs kilometres) before calculating.
Worked examples
Example 1: Speed from a distance–time graph (Paper 2 style)
A cyclist travels 30 km in the first 2 hours, rests for 1 hour, then returns the 30 km in 1.5 hours. Find the speed on the outward leg and the average speed for the whole trip.
Outward speed = 30 ÷ 2 = 15 km/h. Total distance = 30 + 30 = 60 km; total time = 2 + 1 + 1.5 = 4.5 hours. Average speed = 60 ÷ 4.5 ≈ 13.3 km/h (including the rest).
Example 2: Acceleration from a speed–time graph (Paper 2 style)
A car speeds up from 0 to 20 m/s in 8 seconds. Find its acceleration.
Acceleration = change in speed ÷ time = (20 − 0) ÷ 8 = 2.5 m/s².
Example 3: Distance as area under a speed–time graph (Paper 2 style)
A train accelerates from rest to 30 m/s in 10 s, travels at 30 m/s for 20 s, then decelerates to rest in 6 s. Find the total distance.
The graph is a trapezium. Acceleration phase (triangle): ½ × 10 × 30 = 150 m. Constant phase (rectangle): 20 × 30 = 600 m. Deceleration phase (triangle): ½ × 6 × 30 = 90 m. Total distance = 150 + 600 + 90 = 840 m.
Common mistakes and how to avoid them
Confusing the two graph types. A horizontal line means a stop on a distance–time graph but constant speed on a speed–time graph. Always read the vertical-axis label first.
Forgetting stops in average speed. Average speed uses total time, including time spent stationary, not just moving time.
Mixing up gradient and area. On a speed–time graph the gradient is acceleration and the area is distance — do not swap them.
Ignoring units. Check whether time is in seconds or hours and distance in metres or kilometres before dividing; convert if necessary.
Misusing area formulas. Use ½ × base × height for triangular sections and base × height for rectangles; split awkward shapes into these pieces.
Exam technique for Travel Graphs
Identify the graph type immediately. Read the vertical axis: distance or speed. This decides what gradient and horizontal lines mean.
Work stage by stage. Treat each straight segment separately, then combine results.
Use gradient for rates, area for distance on speed–time graphs; use gradient for speed on distance–time graphs.
Watch units and convert. Keep time and distance consistent before calculating, and state units in the answer.
Sketch or annotate. Mark each section's values on the graph; it makes multi-stage calculations clearer and earns method marks.
Quick revision summary
Travel graphs picture motion. On a distance–time graph, the vertical axis is distance: a sloping line means constant speed (steeper = faster), a horizontal line means a stop, and the gradient gives speed. Average speed for a whole journey is total distance ÷ total time, including stops. On a speed–time graph, the vertical axis is speed: a horizontal line now means constant speed, an upward slope means acceleration and a downward slope deceleration. Here the gradient gives acceleration and the area under the line gives distance travelled — use triangle (½ × base × height), rectangle (base × height) and trapezium areas, adding the sections for a multi-stage journey. The commonest error is confusing the two graph types, so always read the vertical-axis label first. Work stage by stage, keep units consistent, and annotate the graph to secure method marks.