Distance-Time Graphs

Two people walk the same route. One walks steadily. The other sprints, stops for a rest, then walks slowly. Sketch what each graph of distance against time would look like, and say which single feature of the second graph tells you when the rest happened.

On a distance-time graph the gradient is the speed. Steeper means faster, horizontal means stationary, and a section that falls means the distance from the reference point is decreasing: the object is returning.

On a distance-time graph, each visual feature has a fixed meaning.

Steepness is speed. The gradient is distance divided by time, which is a speed. A steeper section is a faster one.

Horizontal means stationary. Zero gradient is zero speed. The object is not moving, though time continues, which is why the section has a length.

Rising means moving away from the reference point, and falling means returning towards it.

Straight means a constant speed, and curved means the speed is changing.

The starting and finishing points matter too. The height at the left edge is where the object began, which need not be zero: a graph starting at $(0, 15)$ describes something that was already $15$ km from the reference point when timing started. The height at the right edge is where it finished.

A falling section never means a negative distance. The vertical axis usually measures distance from a particular place, and that measurement genuinely gets smaller as the object comes home.

For a straight section, the speed is the gradient, calculated from the two endpoints and quoted with units.

A section from $(0,0)$ to $(3, 60)$, with kilometres up and hours across:

$$\text{speed} = \frac{60 - 0}{3 - 0} = 20 \ \text{km/h}$$

For a falling section from $(4.5, 60)$ to $(7, 0)$, the gradient is

$$\frac{0 - 60}{7 - 4.5} = \frac{-60}{2.5} = -24$$

and the natural description is "returning at $24$ km/h". The negative sign records the direction of travel; quoting a speed as a negative number is unusual, so let the word do the work.

Average speed is a different quantity and is calculated differently:

$$\text{average speed} = \frac{\text{total distance travelled}}{\text{total time}}$$

For the whole journey above, the object travelled $60$ km out and $60$ km back, so $120$ km in $7$ hours, giving about $17.1$ km/h. That is not the average of $20$ and $24$, because the two phases lasted different lengths of time, and the stationary hour and a half counts towards the total time as well.

A curved section means the speed is not constant, and the shape of the curve says which way it is changing.

A curve getting steeper is speeding up. Each successive time interval covers more distance than the last, so the graph rises more sharply as it goes.

A curve flattening out is slowing down. Each interval covers less distance than the one before.

The same applies to a falling section: a fall that steepens is a return that is speeding up, which is what the diagram above shows.

What you cannot do is quote a single speed for a curved section, because there is no single speed to quote. Two honest responses are available:

describe it qualitatively, as "speeding up throughout" or "slowing down towards the end";
calculate the average speed over the section, using its two endpoints, and say that is what it is.

The average over a curved section is the gradient of the straight line joining its endpoints, which is a real and useful number. Calling it "the speed" is the error; calling it "the average speed over that interval" is correct.

Two different graphs describe the same journey, and reading the axis label is the only way to tell them apart.

Distance from a fixed point. This is the common one. It rises on the way out, is flat during a stop, and falls on the way home, finishing at zero if the object returns to where it started.

Total distance travelled. This can never fall, because distance already travelled cannot be undone. It rises on the way out, is flat during a stop, and rises again on the way home. It finishes at the total journey length.

So for the journey in the diagram, the first version ends at $0$ km and the second ends at $120$ km. Same journey, entirely different graph.

A quick check when reading an unfamiliar graph: if any section falls, the axis must be measuring distance from somewhere, not distance travelled. If nothing ever falls, either could apply and the label decides.

Speed-time graphs are a third kind again, with speed rather than distance on the vertical axis, and their gradients mean something different once more. Those belong to senior courses; the point here is simply to read the label before interpreting anything.

Five items on distance-time graphs. Work each one, then reveal the answer.

1. 1 What does a horizontal section of a distance-time graph mean?

   Show answer
   The gradient, and so the speed, is zero.The object is stationary

2. 2 A section runs from $(0,0)$ to $(4, 100)$, in km and hours. Find the speed.

   Show answer
   Gradient is $\tfrac{100}{4}$.$25$ km/h

3. 3 What does a falling section mean, on a graph of distance from home?

   Show answer
   The measured distance is getting smaller.Returning towards home

4. 4 A section curves and gets steeper. What is happening?

   Show answer
   Steeper means faster.Speeding up

5. 5 Someone travels $30$ km in $1$ h, rests $1$ h, then returns in $1$ h. Find the average speed.

   Show answer
   Total distance over total time: $\tfrac{60}{3}$.$20$ km/h