Motion graphs translate movement into geometry. On a distance-time graph, gradient represents speed. On a speed-time graph, the vertical value is speed itself, while area under the graph represents distance travelled.
This thirty-ninth Secondary 4 Mathematics Learning Guide develops motion graphs as one interpretation system. It belongs to the Secondary Mathematics Hub and S1–S4 Capability Map.
The aim is to distinguish position, distance, speed, gradient and area so that the graph is read according to its axes rather than by visual intuition alone.
Distance-time graphs: gradient is speed
If distance is plotted vertically and time horizontally, gradient is:
speed = change in distance / change in time.
A steeper upward line means a greater speed. A horizontal line means distance is unchanged, so the object is at rest relative to the reference point.
Worked Example 1 | Constant speed from a straight segment
An object travels from 0 km at 0 h to 150 km at 2.5 h.
Speed=150/2.5=60 km/h.
Because the segment is straight, the speed is constant over that interval.
Horizontal distance-time segment means rest
If distance stays at 80 km from t=2 h to t=2.5 h, the object covers no additional distance during that half-hour.
Speed=0 km/h during the horizontal segment.
Curved distance-time graph means changing speed
If the curve becomes steeper, speed is increasing. If it becomes less steep, speed is decreasing. A tangent can estimate the speed at one particular instant.
See Gradient of Curves: Tangents, Local Rate of Change and Graphical Estimation.
Worked Example 2 | Average speed over a curved interval
A distance-time curve shows 20 km at t=0.5 h and 95 km at t=2 h. Find average speed over that interval.
Average speed=(95−20)/(2−0.5)=75/1.5=50 km/h.
This is the secant gradient over the interval, not the local speed at either endpoint.
Speed-time graphs: height is speed
On a speed-time graph, the vertical coordinate gives speed directly. A horizontal segment therefore means constant speed, not rest unless the line lies on speed=0.
Worked Example 3 | Constant speed on a speed-time graph
A speed-time graph is horizontal at 18 m/s from t=5 s to t=15 s.
The object travels at 18 m/s for 10 seconds.
Distance travelled during this interval is the rectangular area:
18×10=180 m.
Area under a speed-time graph gives distance
Because speed has units distance/time, multiplying speed by time gives distance. Geometrically, that multiplication is represented by area under the graph.
Worked Example 4 | Triangular area
Speed increases uniformly from 0 to 20 m/s over 8 s.
The area under the graph is a triangle:
Distance=1/2×8×20=80 m.
Worked Example 5 | Trapezium area
Speed rises linearly from 10 m/s to 22 m/s over 6 s. Find distance travelled.
Distance=1/2(10+22)×6=96 m.
This is the area of a trapezium under the speed-time segment.
Gradient of a speed-time graph describes change of speed
The gradient of a speed-time graph is change in speed per unit time. A positive gradient means speed is increasing; zero means constant speed; negative means speed is decreasing.
This guide treats that idea graphically and numerically, without requiring A-Math calculus.
Worked Example 6 | Rate of change of speed
Speed increases from 8 m/s to 20 m/s in 4 s.
Gradient=(20−8)/4=3 m/s².
The unit m/s² means speed changes by 3 m/s each second over the straight-line segment.
Returning toward the start on a distance-from-start graph
If the vertical axis is distance from a fixed starting point, a downward segment means the object is getting closer to the starting point.
The gradient is negative because the plotted distance-from-start is decreasing, even though physical speed itself is not negative.
Worked Example 7 | Return journey
Distance from start falls from 120 km at 3 h to 60 km at 4 h.
Graph gradient=(60−120)/(4−3)=−60 km/h.
The negative sign describes decreasing distance from the start. The physical speed magnitude is 60 km/h.
Average speed for a whole journey
Average speed is total distance travelled divided by total elapsed time. Rest time remains part of elapsed time unless the problem explicitly asks for average speed while moving.
Worked Example 8 | Include a rest period
A cyclist travels 30 km in 1 h, rests for 0.5 h, then travels another 20 km in 0.5 h.
Total distance=50 km.
Total elapsed time=2 h.
Average speed=25 km/h.
Piecewise graphs require interval-by-interval reading
Many examination graphs contain several stages: acceleration-like increase, constant speed, slowing, rest or return. Treat each segment according to its axes and then combine distances or times as required.
Worked Example 9 | Three-stage speed-time journey
A vehicle increases speed from 0 to 12 m/s over 4 s, travels at 12 m/s for 6 s, then decreases to 0 over 3 s.
Stage 1 distance=1/2×4×12=24 m.
Stage 2 distance=6×12=72 m.
Stage 3 distance=1/2×3×12=18 m.
Total distance=114 m.
Graph scale and units can change everything
Always inspect whether time is measured in seconds, minutes or hours and whether distance is metres or kilometres. A gradient of 2 km/min is not 2 km/h.
Convert units before comparison when necessary.
Worked Example 10 | Convert graph-derived speed
A distance-time graph gives gradient 0.8 km/min. Express this in km/h.
0.8×60=48 km/h.
Common failure modes
| Error | Cause | Repair |
|---|---|---|
| Horizontal line on speed-time graph called rest | Graph types confused | Read the vertical axis first |
| Uses area under distance-time graph as distance | Area rule transferred to wrong graph | Area under speed-time gives distance |
| Uses gradient of speed-time graph as speed | Height and slope confused | Vertical coordinate gives speed |
| Rest time omitted from whole-journey average speed | Elapsed time misread | Use total journey time unless told otherwise |
| Negative distance-time gradient treated as negative physical speed | Direction and speed magnitude mixed | Interpret what the vertical quantity represents |
| Units left inconsistent | Axes read visually only | Write units beside every derived rate |
Independent practice
- A straight distance-time segment goes from (1 h,40 km) to (4 h,220 km). Find speed.
- A speed-time graph is horizontal at 15 m/s for 12 s. Find distance travelled.
- Speed rises uniformly from 0 to 24 m/s in 6 s. Find distance travelled.
- Speed falls from 18 m/s to 6 m/s over 4 s. Find the gradient.
- A journey covers 90 km in 1.5 h, rests 0.5 h, then covers 60 km in 1 h. Find average speed over the whole elapsed time.
Explained answers
1. (220−40)/(4−1)=180/3=60 km/h.
2. 15×12=180 m.
3. 1/2×6×24=72 m.
4. (6−18)/4=−3 m/s².
5. Total distance=150 km; elapsed time=3 h; average speed=50 km/h.
Final thought
Motion graphs are reliable when the axes are read before the shape. On distance-time graphs, slope carries speed. On speed-time graphs, height carries speed and area carries distance.
Read the axes first, then decide whether the question needs height, gradient, area or total elapsed time.
Return to the Secondary Mathematics Hub.
