Speed at an instant from a tangent (Higher)

A speed camera beside a road does not care about your average speed. It reads what you were doing in one instant. Now put a car that is speeding up onto a distance-time graph. The line is a curve, with no straight bit anywhere to take a gradient from. Every straight line you could lay across that curve gives a different speed. Only one of them would agree with the camera.

This is video two of five in Motion graphs and acceleration. It builds on Distance-time graphs, so if a straight-line gradient feels shaky, start with that one.

This one is Higher tier, and only one of the three exam boards sets it, so each board's row is shown here. If it is not on your paper, skip to Acceleration and velocity-time graphs, because nothing after this depends on it.

On a distance-time graph, the gradient is the speed. On a curve, add the tangent: the gradient of the tangent is the speed at that moment. A tangent is a straight line that touches the curve at one point, and only there. Before drawing one, it helps to see why a curve needs it. Here is a train pulling out of a station. After ten seconds it has gone twenty-five metres, after twenty seconds a hundred, after thirty seconds two hundred and twenty-five, and after forty seconds four hundred. The curve gets steeper as it goes, so its gradient is different at every point. The train is speeding up, and its speed is not one number any more. Join the two ends of the curve with a straight line. Its gradient is four hundred metres divided by forty seconds, which is ten metres per second, the average speed over the whole trip. At thirty seconds, is the train going faster or slower than ten metres per second? Faster. The slope at thirty seconds is steeper than that joining one, so the train is beating ten metres per second there. One examiner's report on a Higher paper records this: fewer than half the 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 carries straight over to curves. Name the quantity before you describe the shape: the gradient of a tangent is the speed at that instant.

Mark a point P on the curve, at thirty seconds and two hundred and twenty-five metres. Three straight lines pass through P. Line one runs from P back to the corner. Line two crosses the curve at P and meets it again further up. Line three grazes P, with the curve falling away on both sides. Which line is the tangent to the curve at P: one, two or three? Line three. It touches the curve at P without crossing it, so it has the curve's own steepness at that one point. Now take the gradient of the two others. Line one gives two hundred and twenty-five metres over thirty seconds, which is seven and a half metres per second. Line two gives a hundred and seventy-five metres over ten seconds, which is seventeen and a half metres per second. Line one's seven and a half is a real speed. Which speed is it? The average from the start up to P. It is a true number, but it is not the one the camera reads. Line one undershoots, because it borrows the slower stretch before P. Line two overshoots, because it borrows the steeper stretch after P. They are wrong in opposite directions, and only the tangent uses the steepness at P itself.

Drawing a good tangent is a ruler skill. Lay the ruler on P, then rock it gently until the curve falls away by the same amount on both sides. Then draw the line long, right across the grid, because a long tangent leaves room for a big triangle. Here the tangent crosses the time axis at fifteen seconds, and reaches three hundred and seventy-five metres at forty seconds. Both corners sit on gridlines. Pause and find that tangent's gradient, in metres per second. Fifteen metres per second. That is the train's speed at P. The change in distance is three hundred and seventy-five metres take away zero, so three hundred and seventy-five metres. The change in time is forty seconds take away fifteen, so twenty-five seconds. Three hundred and seventy-five metres divided by twenty-five seconds gives that fifteen metres per second, with metres over seconds giving the unit. Fifteen sits between the two wrong answers, seven and a half and seventeen and a half, which is where the true speed has to be. Here is why the tangent is drawn long. Take a tiny triangle, two seconds wide, and misread its height by six metres. You get eighteen metres per second instead of fifteen. Make the same six-metre slip on the big triangle, and you get fifteen point two. The big triangle shares the slip over twenty-five seconds, so the answer barely moves. Two careful people will draw slightly different tangents, so this method gives an estimate. A careful answer close to fifteen is right, even if it is not exactly fifteen. A last check, on the unit. Metres divided by seconds is metres per second, so this is a speed, even though the train is speeding up. An acceleration would come out in metres per second squared, and nothing on these axes can produce that. That is why it needs a different graph, in Acceleration and velocity-time graphs.

Three questions to finish, each one answered once you have had a go. On a curved distance-time graph, the gradient of what gives the speed at one moment? The tangent. The gradient of the tangent is the speed at that instant. A new moment on the same curve. The same train's tangent at ten seconds crosses the time axis at five seconds, and reaches a hundred and seventy-five metres at forty seconds. What is the speed at ten seconds, in metres per second? Five metres per second. The run is thirty-five seconds, and the rest is the usual division. Now the unit. A tangent answer in metres per second: speed or acceleration? A speed. An acceleration would need metres per second squared. So for a curve, rock the ruler till the gaps match, draw it long, then put the triangle on it. Back at the speed camera on the thirty-second mark, it reads fifteen metres per second. That is the tangent's number, not the average of ten.

If this one is in the bag, a thumbs up marks it done, so you never have to sit through it twice. If not, save it and come back. Draw one tangent with a real ruler on real paper and the method sticks.

Next in the chapter: Acceleration and velocity-time graphs.

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