0580 · 2.9Graphs in practical situations

Reading axes and graph features

0:00A horizontal line can represent someone standing still, or someone travelling steadily along a road. The difference comes from what the axes measure. Practical graphs turn measurements into a picture of a situation. By the end of this video, you will understand how to draw that picture and recover information from its height, gradient and area. Begin every graph by reading both axis labels and their units. The horizontal axis usually shows the quantity we choose or record first, such as time.

0:27The vertical axis shows the corresponding measurement, such as distance or speed. Each plotted point connects two values from the same observation, so neither coordinate should be interpreted alone. You need three familiar mathematical ideas here. Coordinates locate points, gradient measures steepness, and simple shape areas measure regions. Gradient equals the change in the vertical quantity divided by the corresponding change in the horizontal quantity. We will use rectangles, triangles and trapeziums, but each calculation will have a practical meaning rather than being geometry for its own sake.

Drawing and reading conversion graphs

1:00A conversion graph expresses the same measurement in two different units. For this example, use the given conversion of five miles to eight kilometres. Ten miles therefore corresponds to sixteen kilometres, and fifteen miles corresponds to twenty-four kilometres. These points lie on a straight line through the origin because multiplying the miles by a fixed conversion factor gives kilometres. To draw this graph from the data, label the axes with miles and kilometres.

1:24Choose scales with equal numerical increases represented by equal spacing on each axis. The horizontal and vertical scales do not have to match each other. Plot every corresponding pair carefully, then use a ruler to draw the straight conversion line through them. An exam-style question asks you to use the graph to convert twelve point five miles into kilometres. Locate twelve point five on the horizontal axis, halfway between ten and fifteen.

1:48Move vertically to the conversion line, then horizontally to the kilometres axis. The reading is twenty kilometres, using the conversion relationship given in this question. For the reverse conversion, start with twenty-eight kilometres on the vertical axis. Move horizontally to the line, then vertically down to the miles axis. This gives seventeen point five miles. Reversing the direction of reading does not require a different graph because the same line connects both units.

2:13Reading between known data values is called interpolation. Extending a graph beyond the given data is called extrapolation. Extending a fixed unit conversion can be reasonable because its relationship stays unchanged. For other practical situations, such as temperature changing during a day, assuming the same pattern continues beyond the measurements may be unreliable.

Distance-time graphs and speed

2:32Now consider a walker whose total distance travelled is recorded over nine minutes. At the start, the distance is zero. After two minutes, it is one hundred and eighty metres, and it remains unchanged at five minutes. By nine minutes, the walker has travelled four hundred and twenty metres altogether. Suppose the question states that the walker travels at constant speed between readings and stops when the distance stays unchanged.

2:54Plot each time with its corresponding distance, using the scales shown. Join consecutive points with straight sections because those sections match the stated motion. Without that information, the measurements alone would not prove that the speed stayed constant between observations. The first rising section represents movement at constant speed. The horizontal section represents a stop because time passes without any increase in distance travelled. The final rising section represents movement again.

3:21A distance time graph is not a drawing of the route, so a rising line does not mean the walker is climbing a hill. The gradient of a distance time graph gives speed. Its vertical change measures distance travelled during an interval, while its horizontal change measures the time taken. Dividing these quantities gives distance travelled per unit time. A steeper section therefore represents a greater speed, provided you are comparing sections using the same axis scales.

3:46For the first section, the distance increases by one hundred and eighty metres during two minutes. Dividing one hundred and eighty by two gives ninety metres per minute. The stop lasts 5 − 2, which is three minutes. These answers describe different features of the journey, so include the appropriate units with each one. For the final section, subtract one hundred and eighty from four hundred and twenty to obtain two hundred and forty metres.

4:08The elapsed time is 9 − 5, giving four minutes. Dividing two hundred and forty by four gives sixty metres per minute. To express this in metres per second, divide by sixty, giving one metre per second. Notice that we subtracted the coordinates before dividing. 420 ÷ 9 would describe the average speed over the whole journey, not the final section. The final section begins after earlier movement and a stop.

4:33A large gradient triangle makes its changes in distance and time clear, reducing errors when reading the graph. Always check whether the vertical axis shows total distance travelled or distance from a particular place. Total distance travelled cannot decrease, even during a return journey. Distance from home can decrease when someone moves towards home. On that graph, a falling section indicates a return, and the magnitude of its gradient gives the speed.

4:58A curved distance time graph shows that speed is changing. If the curve becomes steeper as time passes, the object is accelerating because its speed is increasing. A curve that flattens out shows deceleration instead, with the speed falling. The height tells us the distance travelled so far, not how fast the object is moving at that moment. To estimate the speed at one instant, estimate the gradient of a tangent at that point.

5:21A tangent is a straight line with the same local direction as the curve at the chosen point. Imagine placing a ruler so that it follows the curve's direction there. Its gradient represents the rate of change at that instant, rather than an average over a longer interval. Suppose the question asks you to estimate the speed at four seconds. Draw the tangent at the marked point, where the distance is eight metres.

5:43Choose two well-separated points on your tangent, not two arbitrary points on the curve. Here, the tangent passes through two seconds at zero metres and six seconds at sixteen metres. The vertical change is 16 − 0, giving sixteen metres. Subtracting two from six gives a horizontal change of four seconds. Divide sixteen by four to estimate a speed of four metres per second. Your answer is an estimate because drawing the tangent and reading its coordinates involve some uncertainty.

6:09Choosing two points on the curve instead would usually give an average speed between those times. It would not necessarily give the speed at four seconds. A larger triangle on the tangent reduces the effect of small reading errors. This graphical method does not require differentiation, which is covered separately in this series.

Speed-time graphs, acceleration and area

6:26On a speed time graph, the vertical axis shows speed directly. A horizontal section above zero therefore represents constant speed, not a stop. A line along the time axis represents rest because its speed is zero. Rising and falling sections represent acceleration and deceleration respectively, with straight sections showing constant rates of change. The gradient of a speed time graph gives acceleration. Its vertical change is the change in speed, and its horizontal change is the time taken.

6:54For these simple journeys in one direction, acceleration measures how quickly speed changes. Metres per second squared means the speed changes by a certain number of metres per second during each second. For the first section, subtract two from ten to obtain a speed increase of eight metres per second. The time taken is four seconds. Dividing eight by four gives an acceleration of two metres per second squared. This means the speed increases by two metres per second during each second of that section.

7:20During the final section, the speed change is 0 − 10, giving negative ten metres per second. The elapsed time is 15 − 10, which is five seconds. Dividing negative ten by five gives negative two metres per second squared. If asked for the deceleration, state its magnitude as two metres per second squared. The area under a speed time graph gives distance travelled. For constant speed, the region is a rectangle whose height is speed and whose width is time.

7:47Multiplying them gives speed multiplied by time, which is distance. At six metres per second for five seconds, the area represents thirty metres travelled. When speed changes uniformly, the region can be a triangle or a trapezium. A triangular region is half the corresponding rectangle because speed changes steadily between zero and the maximum value. For a trapezium, the average of the starting and ending speeds gives its effective height.

8:11We only need areas involving linear sections for this topic. An exam question asks for the total distance travelled during the fifteen seconds shown. The first region is a trapezium because the starting speed is two, not zero. Add the parallel vertical sides, two and ten, to obtain twelve. Half of twelve is six, and multiplying by the width of four gives twenty-four metres. The middle interval lasts 10 − 4, giving six seconds.

8:35Its rectangular area is 10 × 6, giving sixty metres. The final triangle spans the last five seconds, from ten to fifteen. Half of five, times a height of ten, gives twenty-five metres. Add twenty-four, sixty and twenty-five to obtain a total distance of one hundred and nine metres. The answer is in metres, not square metres, because speed multiplied by time produces distance. Calling the first region a triangle would miss the distance from its starting speed of two metres per second.

9:02Splitting it into a rectangle and a triangle would also work. These graphical methods must work without a calculator on Paper two, as well as with a calculator on Paper four. Use exact arithmetic where possible, and keep full values during calculations. Unless instructed otherwise, give calculated answers that are not exact to three significant figures. Graph readings should reflect the scale and the precision the question permits, rather than suggesting unrealistic accuracy.

Exam technique and recap

9:28In an examination, use the command word to decide what your answer needs. Describe the middle section by stating that the object travels at a constant speed of ten metres per second. Calculate acceleration using a gradient, with subtraction shown before division. Calculate distance travelled using the area under the speed time graph, showing each region separately. The main mistakes come from interpreting the shape before reading the axes. A horizontal distance time section means rest, but a horizontal speed time section above zero means constant speed.

9:58Gradient gives speed on a distance time graph and acceleration on a speed time graph. Area gives distance only for the speed time graphs considered here. A practical graph connects quantities, so its meaning comes from its labels, units and scale. Conversion graphs connect equivalent measurements, while travel graphs describe how motion develops over time. Straight-section gradients give constant rates, and tangent gradients estimate rates at particular points on curves.

10:24Reading those relationships carefully lets you explain the situation instead of simply memorising graph shapes.