Distance-time graphs

Picture a distance-time graph of a walk to school. The line climbs like a hill, levels off for a while, then climbs again, more steeply. But the road is flat the whole way, and you never once turned back. If the line is not the shape of the road, then what is it the shape of?

A distance-time graph has time along the bottom and distance up the side. Time is t, measured in seconds. Distance is s, measured in metres. Up the side is not height above the ground. It is the distance travelled so far, counted from the start, so the line climbs on a flat road as every step adds a little more. Everything in this video is on Foundation and Higher papers alike, so it is for everyone. Here is a delivery robot. Every ten seconds, someone wrote down how far it had gone: zero, eight, sixteen, twenty-four metres, then twenty-four, and twenty-four again. Pick a scale that uses most of the grid, then plot each reading as a point. The first four points sit on a straight line rising from the corner, and the last three sit level. From thirty to fifty seconds the line is flat. Is the robot cruising, stopped, or reversing? Stopped. The clock keeps running, but the distance never changes, so the robot is standing still. It cannot be reversing, either. Going backwards still adds to the distance travelled, and distance already travelled cannot be un-travelled. So a distance-time line only ever goes up or stays flat. That is the walk to school too. The level stretch is not flat road. It is you, waiting at a crossing, while the seconds tick by. Read the other shapes the same way. A straight sloping line means a steady speed. A steeper line means a faster speed, because more metres pass each second. And a curved line means the speed is changing. Finding the speed at one moment on a curve has its own Higher video, Speed at an instant from a tangent. The line is not a map of the road. It is a record of distance piling up against the clock.

How steep a line is has a name: its gradient. The gradient is how far the line rises, divided by how far it runs along. To measure it, draw a triangle on the line. For the robot's first thirty seconds, run the triangle from the corner at zero up to the point at thirty seconds. The rise is the change in distance. Twenty-four metres take away zero metres is twenty-four metres. The run is the change in time. Thirty seconds take away zero seconds is thirty seconds. Now divide the rise by the run. Twenty-four metres divided by thirty seconds is nought point eight metres per second, and metres on top of seconds makes metres per second. Why is a distance-time graph's gradient a speed, and not something else? Because speed is distance divided by time, and the gradient is a change in distance over a change in time. It is the same sum, so it gives the same thing. Two habits keep the triangle accurate. Draw it big, across most of the line, because a small slip in reading matters far less when it is shared over a big change. And put its corners where the line crosses gridlines, so every value you read is exact, rather than a guess between two lines. So: change over change, with a big triangle on the line and its corners on gridlines. One examiner's report on a Foundation paper records that most candidates were able to interpret the distance-time graph to calculate the speed of the object. Many successful candidates wrote down their workings to calculate the gradient or wrote down the equation speed equals distance divided by time. The working is what carried them, so the habit is to write the two changes down every time, with their units, before you divide. The other half is saying what the gradient means. When a question asks, the sentence to write is this: on a distance-time graph, the gradient is the speed. Here is an answer a student might write instead: the distance is going up as time goes on. It is marked wrong. Every word of that answer is true. So what is wrong with it? It describes the picture. The question asked what the gradient is, and describing the shape of a line never names a quantity. A different report, on a Foundation paper, records this: only about ten per cent of students were able to recall that the gradient of a distance-time graph represents speed. Many responses gave descriptions of the distance increasing with time, or the steepness of the line. The fix is to name the quantity first. Write the word speed, then its value and its unit, before you describe how the line looks.

Now two cyclists set off together, and both of their lines go on the same distance-time axes, labelled A and B. Now, from the picture alone, which cyclist is faster, and how can you tell? Cyclist A. A's line is steeper, so A covers more metres in every second. Check it with two triangles, both from zero up to twenty seconds. By twenty seconds, A has covered a hundred and twenty metres, and B has covered eighty. For A, a hundred and twenty metres divided by twenty seconds is six metres per second. For B, eighty metres divided by twenty seconds is four metres per second. Six is more than four, so the sums agree with the picture. The steeper line is the faster speed.

A last graph, and this line does not start at the corner. A train's line begins at twenty seconds, when it has already covered three hundred metres. It ends at sixty seconds, at eleven hundred metres. The triangle goes on the line as before, and the one new step is that neither corner sits at zero. Which is its speed: eighteen point three, twenty, or twenty-seven point five metres per second? The right one is twenty metres per second, the middle answer. The change in distance is eleven hundred take away three hundred, which is eight hundred metres. The change in time is sixty take away twenty, which is forty seconds. Eight hundred metres divided by forty seconds is twenty metres per second. Eighteen point three is eleven hundred divided by sixty. It treats the end point as if the line had come all the way from the corner. So what is that eighteen point three, if it is not the speed on this line? It is an average over the whole first minute, including twenty seconds before this line begins, when the train was going slower. And twenty-seven point five is eleven hundred divided by forty. That is the end distance over the change in time, two things that do not belong together. Through the corner, the shortcut happens to agree. Off the corner, it quietly gives the wrong number, and only change over change works every time.

Time to see what has stuck. Each question comes first, then the answer, starting with the sentence itself. On a distance-time graph, what does the gradient give you? The speed. On a distance-time graph, the gradient is the speed. Next one. A bus has gone two hundred metres at ten seconds, and five hundred metres at thirty seconds. What is the bus's speed, in metres per second, over that stretch? Fifteen metres per second. The change is three hundred over twenty, and neither corner is at zero. A different picture now. On a distance-time graph, a line runs perfectly flat for a whole minute. What was the object doing for that minute? Standing still. The clock ran on and the distance did not change. Any straight stretch works the same way: its speed is change over change. Back on the walk to school, the hill was never the road. The steep climb is where you walked fastest, and the flat stretch is the crossing where you stood still.

If you could teach this one to a friend now, give it a thumbs up, so you know you can skip it next time. Still wobbly? Save it for later instead. Three graphs in, reading them starts to feel normal.

Next in the chapter: Speed at an instant from a tangent.

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