The height is the speed
A speed-time graph looks like a distance-time graph, with time along the horizontal axis. The difference is the vertical axis: it shows the speed. Each point on the graph says how fast the traveler was going at that moment, not how far they had gone.
That changes what a flat line means. A horizontal line at 60 says that the speed is 60 km/h the whole time: the car is moving at a steady speed, not standing still. A car that has stopped has a speed of 0, so its graph runs along the time axis.
A steady speed of 60 km/h: the height of the line stays at 60 while the time goes on. Each column of the grid is 1 hour, and each row is 5 km/h.
The gradient is the acceleration
A line that climbs means the speed itself is increasing. Start from rest, a speed of 0, and let the speed rise steadily to 80 km/h over 4 hours. The line climbs from 0 to 80, and its gradient is rise ÷ run = 80 ÷ 4 = 20: the speed goes up by 20 km/h in every hour.
The rate at which the speed changes is called acceleration, and on a speed-time graph it is the gradient. A steeper line means a greater acceleration. A line sloping down means the speed is falling, which is deceleration, and its gradient is negative.
The unit of acceleration is a speed per unit of time. Here it is km/h per hour. With speeds in meters per second and time in seconds, an acceleration is in meters per second, per second, written . A speed that rises by in 4 seconds has an acceleration of .
The speed rises by 20 km/h in each hour, from 0 to 80 km/h at the marked point, 4 hours in. The gradient, 80 ÷ 4 = 20, is the acceleration.
The area is the distance
Distance = speed × time. At a steady 60 km/h for 2 hours, a car covers 60 × 2 = 120 km.
On the graph, that product is an area. Under the line at 60, from 0 to 2 hours, is a rectangle 2 hours wide and 60 km/h high, and its area is 60 × 2 = 120. The area under a speed-time graph is the distance traveled.
The shaded rectangle is 2 hours wide and 60 km/h high. Its area, 60 × 2 = 120, is the 120 km covered.
Under a sloping line
The area is still the distance when the speed changes. Under the line that climbs from 0 to 80 km/h in 4 hours, the shape is a triangle with a base of 4 and a height of 80. Its area is , so 160 km is covered in those 4 hours.
The half has a reason. The speed rose steadily from 0 to 80 km/h, so the average speed was halfway between them, 40 km/h, and 40 × 4 = 160. A steady 80 km/h for the whole 4 hours would have covered 80 × 4 = 320 km, the area of the whole rectangle, and the triangle is half of that rectangle.
A journey that speeds up, keeps a steady speed and then slows down has a graph with a triangle, a rectangle and another triangle underneath it. Find the area of each piece and add them to get the whole distance.
Under the climbing line, from 0 to 4 hours, the shaded triangle has an area of , so 160 km is covered.
The usual mistakes
Reading a flat line as a stop. That is true on a distance-time graph. On a speed-time graph, flat at 60 means a steady 60 km/h.
Taking the whole rise as the rise each hour. The speed climbs from 0 to 80 in 4 hours, so it rises 80 ÷ 4 = 20 km/h in each hour, not 80.
Reading the distance off the height. The height is the speed. The distance is the area under the graph: at a steady 60 km/h for 2 hours it is 60 × 2 = 120 km, not 60, and not 60 + 2 = 62.
Forgetting the half under a sloping line. 4 × 80 = 320 km is the distance at a steady 80 km/h. Speeding up steadily from 0 to 80 km/h covers half of that, 160 km.
Worked example: A Train Between Two Stations on a Speed-Time Graph
Question A train leaves a station and its speed rises steadily from 0 m/s to 20 m/s in the first 10 seconds. It keeps this speed for the next 30 seconds, and then slows steadily to rest at the next station in a further 20 seconds. The speed-time graph joins (0, 0), (10, 20), (40, 20) and (60, 0). (a) Find the acceleration of the train in the first 10 seconds. (b) Find the distance between the two stations.
1.The acceleration is the gradient of the first part of the graph. The speed rises by 20 m/s in 10 seconds, so the acceleration is 2010 = 2 m/s2. (a) The acceleration is 2 m/s2.
(a) On a speed-time graph the gradient is the acceleration. In the first 10 seconds the speed rises by 20 m/s, so the acceleration is 2010 = 2 m/s2. 2.The distance traveled is the area under the graph. Vertical lines at 10 seconds and at 40 seconds split this area into a triangle, a rectangle and another triangle.
The distance traveled is the area under the graph. The two dashed lines split it into a triangle, a rectangle and a triangle. 3.The first triangle has area 12 × 10 × 20 = 100. The rectangle is 40 − 10 = 30 seconds wide, so its area is 30 × 20 = 600. The second triangle is 20 seconds wide, so its area is 12 × 20 × 20 = 200.
The areas are 12 × 10 × 20 = 100, 30 × 20 = 600 and 12 × 20 × 20 = 200. 4.(b) The distance between the stations is 100 + 600 + 200 = 900 m. Check: the whole shape is a trapezium with parallel sides of 60 and 30 and a height of 20, and 12 × (60 + 30) × 20 = 900.
(b) The train travels 100 + 600 + 200 = 900 m. The whole shape is a trapezium, and 12 × (60 + 30) × 20 = 900 as well.
Answer: (a) 2 m/s2; (b) 900 m
Common mistakes
- Multiplying the greatest speed by the whole time, 20 × 60 = 1200 m. The train travels at 20 m/s for only 30 of the 60 seconds. While it speeds up and slows down it covers less ground, which is why those parts are triangles.
- Reading the height of the graph at the end, 0, as the distance. The height of a speed-time graph is a speed. The distance is the area under the graph, not a height on it.