Gradient Detective

Gradient Detective

A distance-time graph hides an entire journey inside one line, and the whole of it can be read from the gradient, because on this graph the gradient IS the speed. That is worth stating carefully. Gradient is change in the vertical divided by change in the horizontal, and here the vertical is distance and the horizontal is time. Distance divided by time is speed. Nothing has been invented; it is the same gradient you already know. And it tells you the units for free: kilometres up, hours across, so the gradient comes out in kilometres per hour without your having to remember anything.

Reading the axes

Everything here follows from what the two axes are measuring:

What is each line telling you?

Find the fastest section

This distance-time graph shows one car journey in four sections. Tap the section where the car is travelling fastest.

Two graphs that look identical

Distance-time and speed-time graphs are drawn the same way and mean completely different things. The lines below are the same shapes on each:

A flat line

A graph shows a horizontal line running across the page for several minutes. What is the object doing?

• It is impossible to say without looking at the vertical axis. On a distance-time graph a flat line means stopped; on a speed-time graph it means travelling at a constant speed, which could be very fast indeed
• Stopped. A flat line always means nothing is happening
• Travelling at a constant speed, since the line is not changing
• Slowing down, because the line is no longer climbing

One car journey

The graph you have been looking at, as a table. Distance in kilometres from home, time in hours: • At 0 h: 0 km • At 1 h: 60 km • At 2 h: 60 km • At 3 h: 180 km • At 5 h: 0 km One section worked, as the model. Between 0 and 1 hour the distance changes from 0 to 60 km, so the change in distance is 60 km and the change in time is 1 hour. The gradient, and therefore the speed, is 60 ÷ 1 = 60 km/h. Work out the other three for yourself. Note that between 3 and 5 hours the distance is falling, which is the drive home, and that a speed is still a positive number however the line slopes: it is the direction of travel that has changed, not the speed.

The fast stretch

Between 2 and 3 hours, the car's distance from home goes from 60 km to 180 km. What is its speed over that section, in km/h?

The whole journey

Over the full five hours, the car drove 60 km out, stopped, drove another 120 km out, then drove all 180 km home. What was its average speed for the whole journey, in km/h?

Why not just average them?

The four sections have speeds of 60, 0, 120 and 90 km/h. Averaging those four gives 67.5 km/h, but the average speed for the journey is 72 km/h. Why do the two disagree?

• Because the sections last for different lengths of time, and a plain mean treats a two-hour stretch as equal to a one-hour one. Average speed is always the total distance travelled divided by the total time taken, never the mean of the individual speeds
• Because the stopped section should be left out of the average altogether
• Because 67.5 has been rounded somewhere along the way
• Because the return journey should count as negative, since the car is going backwards

Four things to check

Read the vertical axis before the line. Distance or speed changes the meaning of every shape on the page. Distance from the start is not distance travelled. A car that drives 180 km out and 180 km home finishes at zero on the graph, having covered 360 km. Questions ask for both, and they are different numbers. Average speed is total distance ÷ total time. Never the average of the section speeds, unless every section happens to last the same length of time. A curve means the speed is changing. A straight line means a steady speed, so anything bending is telling you the speed is not steady: a curve getting steeper is speeding up, and a curve flattening off is slowing down.

A curving line

A distance-time graph starts almost flat and curves upwards, getting steeper and steeper. What is happening?

• The object is speeding up. The gradient is the speed, and a curve that gets steeper has an increasing gradient, so the speed is increasing throughout
• The object is getting further away, which is all the curve shows
• The object is slowing down, since a curve means it cannot maintain a steady speed
• It started stationary and then stopped again at the top of the curve

Find a section's speed

Put the steps for finding the speed of one straight section of a distance-time graph into order.

• Check the vertical axis really is distance, and note the units on both axes
• Read the distance and the time at the start of the section
• Read the distance and the time at the end of the section
• Subtract to get the change in distance and the change in time
• Divide the change in distance by the change in time, and give the answer in the units from the axes

The detective rules

On a distance-time graph the gradient of the line gives the _____, and the units come straight from the two axes. A flat section means the object is _____, while on a speed-time graph the same flat line would mean a constant speed instead, which is why the axis label has to be read first. A curved line means the speed is _____. And the average speed for a whole journey is always the total distance divided by the total _____, not the mean of the separate section speeds.