Time along, distance up
A distance-time graph shows a journey. Time goes along the horizontal axis, and the distance from the starting point goes up the vertical axis. Each point on the graph says how far from the start the traveler was at that moment.
A cyclist sets off and rides 20 km in every hour. After 1 hour she is 20 km from the start, after 2 hours 40 km, and after 3 hours 60 km. Plotted, those points lie on a straight line that climbs 20 km for every hour across.
Time in hours along the bottom, distance in kilometers up the side. Each hour, the line climbs 20 km: the cyclist is 20 km out after 1 hour, 40 km after 2 hours and 60 km after 3 hours.
The gradient is the speed
Speed is the distance traveled divided by the time taken. On the graph, the distance traveled is the rise and the time taken is the run, so the speed is rise ÷ run, which is the gradient of the line. From 0 to 3 hours the line rises 60 km, and 60 ÷ 3 = 20, so the speed is 20 km/h.
A straight line has the same gradient all along it, so a straight line means a steady speed: the same distance is covered in every hour. A steeper line covers more distance in each hour, so it is a faster speed. The graph of a car at 40 km/h climbs 40 km in each hour, twice as steeply as the cyclist's.
Two steady speeds from the same start, on a grid whose columns are 1 hour and whose rows are 5 km. One line climbs 40 km in each hour and the other 20 km, so the steeper line is the faster one.
A flat stretch is a stop
After 3 hours the cyclist rests for an hour. The time keeps going, but her distance from the start stays at 60 km, so the graph runs level: a horizontal line at 60 from 3 hours to 4 hours.
A flat stretch means the distance is not changing, so she has stopped. Its gradient is 0, and her speed is 0. A flat line on a distance-time graph never means a steady speed: a steady speed is a straight line that climbs.
Coming back
Then she rides home. Now her distance from the start gets smaller, so the graph slopes down. From 4 hours to 6 hours it falls from 60 km to 0: she covers 60 km in 2 hours, so her speed is 60 ÷ 2 = 30 km/h.
The gradient of this piece is −30. The minus sign says that the distance from the start is decreasing, that is, she is heading back. Her speed is 30 km/h, the size of the gradient. The ride home is the steepest piece of the graph, so it is the fastest part of the journey.
The whole ride: out at 20 km/h for 3 hours, stopped from 3 hours to 4 hours, then home at 30 km/h, reaching the start at 6 hours.
The usual mistakes
Reading a flat stretch as a steady speed. On a distance-time graph, flat means the distance is not changing, so the traveler has stopped.
Reading the speed off the height. The height is the distance from the start, not the speed. At 3 hours the cyclist is 60 km from the start, but her speed is 20 km/h.
Reading the climb over the wrong time. The speed is the climb in one hour. Over 2 hours the line climbs 40 km, and that is still 20 km/h, because 40 ÷ 2 = 20.
Reading a downward slope as slowing down. A line that slopes down means the traveler is moving back toward the start. Slowing down on the way out would make the line climb less and less steeply.
Worked example: A Bicycle Ride Out, a Rest and the Ride Home on a Distance-Time Graph
Question Mei cycles from her home to a lake, rests there, and cycles home along the same road. The graph of her distance from home, in km, against the time since she left, in hours, joins the points (0, 0), (2, 36), (3, 36) and (4.5, 0) with straight lines. (a) Find her speed on the way out and her speed on the way back. (b) Find her average speed for the whole outing, including the rest.
1.The graph has three straight parts. It rises from (0, 0) to (2, 36), stays level until (3, 36), and falls to (4.5, 0). The gradient of each part is the speed on that part.
On a distance-time graph the gradient of a part is the speed on that part. The graph rises, stays level, then falls to zero. 2.On the way out Mei covers 36 km in 2 hours, so the gradient is 362 = 18. Her speed is 18 km/h.
On the way out Mei rides 36 km in 2 hours, so her speed is 362 = 18 km/h. 3.From 2 hours to 3 hours the graph is level, so she rests for 1 hour. On the way back she covers 36 km in 4.5 − 3 = 1.5 hours, so her speed is 361.5 = 24 km/h. The gradient of this part is −24, which is negative because her distance from home is decreasing. (a) She rides at 18 km/h on the way out and at 24 km/h on the way back.
(a) The level part is a rest of 1 hour. On the way back she rides 36 km in 1.5 hours, which is 361.5 = 24 km/h. The gradient is −24 because she is coming closer to home. 4.For the whole outing, the total distance is 36 + 36 = 72 km, and the total time, including the rest, is 4.5 hours.
For the whole outing the distance is 36 + 36 = 72 km and the time, with the rest, is 4.5 hours. 5.(b) The average speed is 724.5 = 16 km/h. Check: 16 × 4.5 = 72.
(b) The average speed is 724.5 = 16 km/h.
Answer: (a) 18 km/h on the way out and 24 km/h on the way back; (b) 16 km/h
Common mistakes
- Finding the mean of the two speeds, 18 + 242 = 21 km/h. Mei spends different lengths of time at each speed, and an hour at rest, so the average speed must be the total distance divided by the total time.
- Using 4.5 hours as the time for the ride back. The point (4.5, 0) gives the time since she left home. The ride back starts at 3 hours, so it takes 4.5 − 3 = 1.5 hours.