On a distance-time graph, the gradient of each straight section is the speed, and a flat section means the object is stationary. On a speed-time graph, the gradient of each straight section is the acceleration, a flat section means constant speed, and the area under the graph gives the distance travelled. Mixing these two up, especially reading a flat speed-time section as "stopped", is the single most common exam error on this topic.
Both graphs plot time on the horizontal axis, both are usually drawn with straight-line sections, and both can look almost identical on paper. The difference is entirely in what the vertical axis measures: distance on one, speed on the other. Because the shapes look so similar, students often carry the rule from one graph straight over to the other, and that is where the marks are lost.
The mistake that costs the most marks is this: on a speed-time graph, a flat horizontal section does not mean the object has stopped. It means the object is travelling at a constant, unchanging speed. On a distance-time graph, a flat section genuinely does mean the object is stationary, because distance is not increasing. Learn which axis you are reading before you touch the gradient or the area.
Distance-time graph
gradient = speed
flat section = stationary
steeper line = faster
Speed-time graph
gradient = acceleration
flat section = constant speed
area under graph = distance
For a distance-time graph, gradient is calculated the usual way: gradient = change in distance ÷ change in time. The units work out to distance per time, which is exactly what speed is. For a speed-time graph, gradient is change in speed ÷ change in time, which gives units of speed per time, exactly what acceleration is. A negative gradient on a speed-time graph means deceleration (slowing down), not negative distance. This is the same gradient idea used in conversion graphs and linear models, just applied to time instead of a general x-axis.
Area under a speed-time graph works because area = (a value on the vertical axis) × (a value on the horizontal axis) = speed × time, and speed × time is distance. Split the region under the graph into triangles, rectangles and trapeziums, find each area separately, then add them together.
A cyclist's journey is recorded on a distance-time graph. In the first 30 minutes she travels from 0 km to 20 km. In the next 30 minutes she travels from 20 km to 50 km. Find her speed in each stage, and her average speed for the whole journey.
Solution
speed = 20 ÷ 0.5 = 40 km/hspeed = 30 ÷ 0.5 = 60 km/haverage speed = total distance ÷ total time = 50 ÷ 1 = 50 km/hQuick check: notice 50 km/h sits between 40 km/h and 60 km/h here only because both stages took the same amount of time. If the stages had different durations, the simple average of the two speeds would give the wrong answer. Always go back to total distance over total time.
A car's speed-time graph has three straight sections. From 0 to 8 seconds the car accelerates uniformly from 0 m/s to 16 m/s. From 8 to 20 seconds the car travels at a constant 16 m/s. From 20 to 28 seconds the car decelerates uniformly from 16 m/s to 0 m/s. Find the acceleration in the first stage and the total distance travelled.
Solution
acceleration = 16 ÷ 8 = 2 m/s²deceleration = 16 ÷ 8 = 2 m/s² (speed decreasing)½ × base × height = ½ × 8 × 16 = 64 mbase × height = 12 × 16 = 192 m½ × base × height = ½ × 8 × 16 = 64 m64 + 192 + 64 = 320 mExam tip: the whole shape under this graph is a trapezium, so you could also use the trapezium area formula directly: ½ (a + b) h, where the two parallel sides are the 12 s and 28 s widths and the height is 16 m/s. That gives ½ (12 + 28) × 16 = 320 m, matching the split-into-shapes method. Use whichever method you find faster; both are equally acceptable in the exam.
The step students get wrong
Reading a flat section on a speed-time graph as "the object has stopped". A flat section on a speed-time graph means the speed is not changing, so the object is moving at a steady, constant speed. Only a flat section at zero on the vertical axis of a speed-time graph means the object has actually stopped. Before answering any question on this topic, check which axis the vertical axis is measuring: distance or speed. That single check prevents most of the errors on this topic.
Before you interpret any gradient or shaded area, write "d" or "s" next to the vertical axis on your working, standing for distance or speed. This costs a few seconds and removes the single biggest source of error: applying the distance-time rules to a speed-time graph, or vice versa.
Does a curved line on a distance-time graph mean anything different?
A curved line means the speed is changing (the object is accelerating or decelerating), because the gradient is not constant along a curve. O-Level E-Maths questions on this topic almost always use straight-line sections, so you are expected to read constant speeds and constant accelerations, not changing ones. Finding the speed at one exact instant on a curved section instead needs the method for the gradient of a curve, rather than a single straight-line gradient.
What does a negative gradient mean on each graph?
On a distance-time graph, a negative gradient means the object is travelling back towards the starting point, so distance from the start is decreasing. On a speed-time graph, a negative gradient means the object is decelerating, that is, slowing down. Neither graph shows negative distance or negative speed in a typical O-Level context.
Can I use area under a distance-time graph for anything?
No. Area under a distance-time graph has no standard physical meaning in the E-Maths syllabus (it would multiply distance by time), so do not calculate it. Area only carries meaning, distance travelled, on a speed-time graph, because speed × time = distance.
How is this different from the average speed mistake?
They are closely related. The average speed guide covers why you cannot simply average two speeds together; this guide covers reading the graphs themselves. Worked example 1 above uses exactly that average speed rule, applied to a distance-time graph.
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Mr. Gan helps students build the habit of checking the axis first, so gradient and area questions stop costing easy marks.
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