Distance-Time Graphs
Read and analyse distance–time graphs for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-05). A distance–time graph, or travel graph, shows how far something is from its start as time passes, so you can see at a glance when it moves, stops and comes back.
You will read distances and times off the graph, find the speed from the steepness of each segment, spot a stop or a return trip, compare two travellers, and calculate average speed for a whole journey — useful for any problem about speed, distance and time.
Theory
A distance-time graph (or travel graph) shows how far something is from its start as time passes. This Year 12 Standard 1 (NSW) guide shows how to read the graph, find the speed from the steepness of each segment, spot stops and return trips, and calculate average speed.
A distance-time graph has time \(t\) across the bottom and distance \(d\) from the start up the side. To read it, go up from a time to the graph and across to the distance axis, or the other way to find a time.
The speed of a straight segment is \(\dfrac{\text{rise}}{\text{run}}=\dfrac{\text{distance travelled}}{\text{time taken}}\). A steeper segment means faster. A flat segment means stationary (stopped), and a segment sloping down means returning towards the start. Where two lines cross, the travellers are at the same place at the same time.
The average speed \(=\) total distance travelled \(\div\) total time taken. On a return trip, count the distance out and back. This is a core Year 12 Standard 1 (NSW) Ratios and rates skill.
The speed of a straight segment is its rise over run:
Using two points on the segment, \((t_1, d_1)\) and \((t_2, d_2)\):
For the whole trip:
How to answer a distance-time graph question
- Read the axes: check the units of time and distance.
- Pick the segment the question is about and read its start and end points.
- Calculate the speed as distance gained \(\div\) time taken for that segment.
- For average speed, add all the distance travelled (out and back) and divide by the total time, stops included.
Go up from \(t=10\) to the line, then across. For the speed use the whole line, from \((0,0)\) to \((25,150)\).
| \(\text{(i) } d\) | \(=\) | \(60\text{ m}\) |
| \(\text{(ii) speed}\) | \(=\) | \(\dfrac{150}{25}\) |
| \(\therefore\ \text{speed}\) | \(=\) | \(6\text{ m/s}\) |
(i) \(60\text{ m}\); (ii) \(6\text{ m/s}\).
The flat part runs from \(t=1\) to \(t=3\). The last segment goes from \((3,20)\) to \((4,45)\).
| \(\text{(i) stop}\) | \(=\) | \(3 - 1 = 2\text{ h}\) |
| \(\text{(ii) speed}\) | \(=\) | \(\dfrac{45 - 20}{4 - 3}\) |
| \(\therefore\ \text{speed}\) | \(=\) | \(25\text{ km/h}\) |
(i) Stationary for \(2\) hours; (ii) \(25\text{ km/h}\).
Lena's line is straight from \((0,0)\) to \((4,60)\); the lines first cross at \((2,30)\).
| \(\text{(i) Lena}\) | \(=\) | \(\dfrac{60}{4} = 15\text{ km/h}\) |
| \(\text{(ii) cross at } t\) | \(=\) | \(2\text{ h}\) |
| \(\text{(iii) Omar}\) | \(=\) | \(\dfrac{30}{1} = 30\text{ km/h} \gt 15\) |
(i) \(15\text{ km/h}\); (ii) after \(2\) hours; (iii) Omar, because his line is steeper.
Count the distance out and back, and use the whole time, including the wait.
| \(\text{total distance}\) | \(=\) | \(45 + 45 = 90\text{ km}\) |
| \(\text{total time}\) | \(=\) | \(6\text{ h}\) |
| \(\text{average speed}\) | \(=\) | \(\dfrac{90}{6} = 15\text{ km/h}\) |
The bus travels \(90\text{ km}\) at an average of \(15\text{ km/h}\).
Common pitfalls
Frequently asked questions
How do you find speed from a distance-time graph?
Pick a straight segment and divide the distance travelled in that segment by the time it took. This is the rise over the run. A segment from (3 h, 20 km) to (4 h, 45 km) has a speed of 25 divided by 1, which is 25 km/h.
What does a flat line on a distance-time graph mean?
A flat, horizontal segment means the object is stopped. The distance from the start is not changing. The length of the flat part tells you how long it stopped.
What does a steeper line mean on a travel graph?
A steeper line means a faster speed, because more distance is covered in the same time. A line sloping down means the object is coming back towards the start.
How do you find average speed from a distance-time graph?
Divide the total distance travelled by the total time taken, including any stops. On a trip out and back, count the distance both ways. A bus that goes 45 km out and 45 km back in 6 hours averages 90 divided by 6, which is 15 km/h.
What does it mean when two lines cross on a travel graph?
Where two lines cross, both travellers are at the same distance from the start at the same time, so one has caught up with the other.