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Speed-time graphs

Change and motion · level 4, lesson 3 of 6

A question to start

Here is a graph with a flat line on it, at the height 20.

Sam says: "Flat means stopped. The car is parked." Lee says: "No, flat means the car is going steadily at 20 metres per second." They are looking at the same line. Who is right?

Both of them could be, depending on one thing they forgot to look at. And once you know which kind of graph it is, you can find out something surprising from it: how far the car went, by measuring an area.

The idea

A new kind of graph

In level 3 you met distance-time graphs: time across, distance up. Now meet the speed-time graph: time across, speed up the side.

Each point now says "at this time, I was going at this speed". It does not tell you where you are, only how fast you are going.

So who was right in the puzzle? If the graph is distance-time, a flat line means the distance is not changing: Sam is right, the car is stopped. If it is speed-time, a flat line at 20 means the speed is not changing and is 20 m/s: Lee is right. Always read the labels on the axes before anything else.

Reading the shapes

On a speed-time graph:

The line is... It means...
flat, above the time axis moving at a steady speed
flat, on the time axis (speed 0) stopped
sloping up speeding up
sloping down slowing down (not going backwards!)
steeper the speed is changing faster
steady speedstoppedspeeding upslowing down

The steepness of a speed-time graph tells you how quickly the speed is changing. A sports car that reaches motorway speed in a few seconds has a steep line; a heavy lorry has a gentle one. The amount of speed gained each second has a name, acceleration, and it gets its own lesson in level 5.

Distance is the area under the graph

Now the surprise. A car goes at a steady 20 metres per second (m/s) for 10 seconds. How far does it go?

From the rate lessons: 20 metres in each second, for 10 seconds, is \(20 \times 10 = 200\) metres.

Now look at its speed-time graph. It is a flat line at height 20, from time 0 to time 10. The region under the line (down to the time axis) is a rectangle, 10 wide and 20 tall. Its area is width × height \(= 10 \times 20 = 200\).

024681012 time (seconds) area = 10 × 20 = 200so 200 metres 05101520 m/s

The same number, 200, twice. That is no accident:

  • distance = speed × time;
  • on the graph, speed is the height of the rectangle and time is its width;
  • so distance = height × width = area of the rectangle.

The units agree as well: the area is (metres per second) × (seconds), and the "per second" and the "seconds" cancel, leaving metres.

The distance travelled is the area under the speed-time graph. This is one of the most important ideas in the whole topic: it comes back in level 5 for changing speeds and in level 8 as the heart of calculus.

A trip in parts: add the areas

If a trip has several steady parts, the region under the graph is several rectangles. Add their areas.

A bus drives at 10 m/s for 60 seconds, stops at a bus stop for 30 seconds, then drives at 15 m/s for 60 seconds. (We pretend it changes speed instantly; really it takes a few seconds, but the idea is the same.)

0306090120150 time (seconds) 600 m900 mstop 05101520 m/s
  • First part: \(10 \times 60 = 600\) metres.
  • The stop: speed 0, so the "rectangle" has no height and no area: 0 metres. Of course: a stopped bus goes nowhere.
  • Last part: \(15 \times 60 = 900\) metres.
  • Total: \(600 + 0 + 900 = 1500\) metres.

The same trip, two graphs

Here is the bus trip as a distance-time graph, for comparison. Each steady part of the speed-time graph (a flat line) is a sloping straight line on the distance-time graph, and the stop (on the axis) is a flat line.

0306090120150 time (seconds) 040080012001600 m

The two graphs tell the same story in two languages:

  • steepness of the distance-time graph = height of the speed-time graph (both are the speed);
  • height of the distance-time graph = area under the speed-time graph (both are the distance so far).

These two facts are the two halves of calculus, seen here with nothing but rectangles.

Worked examples

1. Reading the bus graph. How far did the bus go, and what was its average speed?

Distance: total area \(= 1500\) m. Time: 150 s. Average speed: \(1500 \div 150 = 10\) m/s. The stop pulled the average down: while moving, the bus did 10 m/s and 15 m/s.

2. A train. A train runs at a steady 25 m/s for 2 minutes. How far does it go?

Careful with units: the speed is per second, so change 2 minutes into \(2 \times 60 = 120\) seconds. Area \(= 25 \times 120 = 3000\) metres, which is 3 km.

3. A cyclist (a peek ahead). A cyclist speeds up steadily from 0 to 8 m/s in 4 seconds, rides at 8 m/s for 6 seconds, then brakes steadily to a stop in 2 seconds.

024681012 time (seconds) 16488 0246810 m/s
  • 0 to 4 s: sloping up, speeding up. 4 to 10 s: flat, steady at 8 m/s. 10 to 12 s: sloping down, braking. The braking line is steeper than the starting line: the speed changes faster when braking (8 m/s lost in 2 s, compared with 8 m/s gained in 4 s).
  • Distance in the middle: rectangle, \(8 \times 6 = 48\) m.
  • Distance while speeding up: the speed rose steadily from 0 to 8, so on average it was 4 m/s (halfway). In 4 seconds: \(4 \times 4 = 16\) m. And the triangle's area is \(\frac{1}{2} \times 4 \times 8 = 16\). The area rule still works!
  • Distance while braking: average speed 4 m/s for 2 s: 8 m, which is the small triangle, \(\frac{1}{2} \times 2 \times 8\).
  • Total: \(16 + 48 + 8 = 72\) metres.

You will see why the area rule works for every shape in level 5.

4. A lift. A lift goes from the ground floor to the top floor. Its speed-time graph starts at 0, slopes up, is flat for most of the trip, slopes down to 0. Story: the lift starts from rest, speeds up, travels steadily, slows down and stops at the top. Nothing in the graph says "up" or "down": a speed-time graph shows only how fast.

Try it

  1. A speed-time graph is a flat line at 12 m/s from 0 to 30 seconds. What is happening? How far does it go?
  2. Part of a speed-time graph lies right on the time axis. What does that mean?
  3. A car drives at 20 m/s for 15 seconds, then at 30 m/s for 10 seconds. Sketch the speed-time graph and find the total distance.
  4. A person walks at 1.5 m/s for 4 minutes. How far do they walk?
  5. A speed-time graph rises steadily from 0 to 10 m/s in 5 seconds, is flat for 10 seconds, then falls steadily to 0 in 5 seconds. Describe the motion, and find the distance using areas.
  6. Ana says: "The speed-time graph goes down, so the car is reversing." Is she right?
  7. Car A goes at 20 m/s for 30 seconds. Car B goes at 15 m/s. How long must car B drive to go the same distance?
  8. Challenge: a ball rolls from rest, speeding up steadily to 6 m/s in 3 seconds. How far has it rolled? Explain in two ways.
Answers
  1. Moving at a steady 12 m/s. Distance = area \(= 12 \times 30 = 360\) metres.
  2. The speed is 0 there: the object is stopped.
  3. Two flat pieces: at 20 from 0 to 15 s, at 30 from 15 to 25 s. Distance: \(20 \times 15 + 30 \times 10\) \(= 300 + 300 = 600\) metres.
  4. 4 minutes is 240 seconds. \(1.5 \times 240 = 360\) metres.
  5. It speeds up from rest, goes steadily at 10 m/s, then slows to a stop. Areas: triangle \(\frac{1}{2} \times 5 \times 10 = 25\), rectangle \(10 \times 10 = 100\), triangle 25. Total 150 metres.
  6. No. Going down on a speed-time graph means the speed is getting smaller: the car is slowing down, still moving forwards. (A speed-time graph never shows direction.)
  7. Car A goes \(20 \times 30 = 600\) m. Car B needs \(600 \div 15 = 40\) seconds.
  8. (1) The speed rises steadily from 0 to 6, so the average speed is 3 m/s; \(3 \times 3 = 9\) metres. (2) The area under the graph is a triangle with base 3 and height 6: \(\frac{1}{2} \times 3 \times 6 = 9\). Both give 9 metres.

Common mistake

Reading a speed-time graph as a distance-time graph. On a distance-time graph, flat means stopped. On a speed-time graph, flat means steady speed, and only a line on the time axis means stopped. Read the axis labels first, every time.

Mixing minutes and seconds in the area. If the speed is in metres per second, the time across must be in seconds before you multiply. 25 m/s for 2 minutes is \(25 \times 120\), not \(25 \times 2\).

Where it leads