Distance-Time Graphs
In 2007, iPhone's GPS drew Australia's first real-time distance-time graph in every driver's pocket — a flat line means you stopped for coffee.
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If you walked 100 metres in 2 minutes, then stood still for 1 minute, then walked another 100 metres in 2 minutes, how would you sketch a distance-time graph of this journey?
What does a very steep line on a distance-time graph tell you about the object's motion? What about a flat horizontal line?
● Know
- A distance-time graph shows how distance changes over time.
- The gradient (slope) of a distance-time graph represents speed.
● Understand
- A steeper gradient means faster speed; a horizontal line means the object is stationary.
● Can do
- Interpret distance-time graphs and calculate speed from the gradient.
Look at a table of 20 GPS positions recorded every 5 seconds and you will struggle to see whether the car was speeding or slowing. Plot those same numbers on a distance-time graph and a picture jumps out: a steep section means fast, a flat section means stopped, a bend means the speed is changing. The graph reveals patterns that the raw numbers hide, and the key to reading it is understanding what the slope represents.
The slope (gradient) of a distance-time graph equals the object speed. A steep slope means high speed; a gentle slope means low speed; a flat line (zero slope) means zero speed - the object is stationary.
A straight line indicates constant speed. The object covers equal distances in equal time intervals. A curved line indicates changing speed. If the curve gets steeper, the object is speeding up. If it gets flatter, the object is slowing down.
The y-intercept shows the starting position. A positive y-intercept means the object started ahead of the reference point. A negative y-intercept (if used) means it started behind.
A student walks 100 metres to school in 2 minutes, waits 1 minute at the gate for a friend, then walks another 200 metres to the classroom in 3 minutes. The distance-time graph shows: a straight line from (0,0) to (2,100) with slope 50 m/min; a flat horizontal line from (2,100) to (3,100) showing no movement; then a straight line from (3,100) to (6,300) with slope about 67 m/min. The second walking segment is steeper, showing faster speed.
Australian transport planning: Transport for NSW uses distance-time graphs and similar analyses to optimise Sydney train timetables. By plotting train positions over time, planners identify where delays occur, where express trains can overtake slow services, and how to minimise journey times. The Sydney Metro system uses automated trains that maintain precisely calculated speed profiles to maximise throughput on the line.
A downward sloping distance-time graph means the object is moving backward in time. This is false. Time always moves forward on the horizontal axis. A downward slope means the object distance from the starting point is decreasing - it is returning toward the start. If you define distance as total path length, the graph never goes down. If you define it as displacement from origin, negative values or downward slopes simply indicate direction toward the origin.
- Straight line going up
- Horizontal flat line
- Curved line getting steeper
- Straight line going down
- Moving at constant speed away from start
- Speeding up - acceleration
- Stationary - not moving
- Moving back toward start at constant speed
Calculating speed from a distance-time graph uses the same method as calculating any slope: rise over run. The rise is the change in distance (vertical axis), and the run is the change in time (horizontal axis).
Speed = change in distance / change in time = rise / run
This calculation works for any straight-line segment. For curved segments, you can calculate average speed over an interval using the same formula, or estimate instantaneous speed by drawing a tangent line at a point and calculating its slope.
When comparing multiple objects on the same graph, the steepest line indicates the fastest object. Lines that are parallel have the same speed. If two lines cross, the objects are at the same position at that time - but they are not necessarily moving at the same speed.
Two cyclists race on the same straight road. Cyclist A graph is a straight line from (0,0) to (10, 50) - speed = 5 m/s. Cyclist B starts 20 metres ahead but cycles slower: line from (0,20) to (10, 60) - speed = 4 m/s. The lines cross at t = 20 seconds, when both are at 100 metres. Cyclist A catches up because of higher speed, despite starting behind. After the crossing, Cyclist A leads because the faster cyclist eventually overtakes.
Australian athletics: Athletics Australia uses high-speed video and laser tracking to generate distance-time graphs for elite sprinters. Coaches analyse these graphs to identify where athletes accelerate, where they maintain top speed, and where they decelerate. Even world-class sprinters like Cathy Freeman had characteristic graph shapes that coaches used to refine training. Modern systems track athletes at 100 Hz, producing exquisitely detailed motion graphs.
The highest point on a distance-time graph is where the object is fastest. This is false. The highest point simply means the object is farthest from the start. Speed depends on slope, not height. An object could be stationary at its maximum distance (flat line at the top) or moving very slowly. Conversely, an object passing through the origin could be moving extremely fast with a steep slope.
Real-world motion is rarely as simple as constant speed. Objects speed up, slow down, stop, and change direction. Distance-time graphs capture these complexities through their shapes.
Speeding up (acceleration): The graph curves upward, getting steeper over time. Each successive time interval covers more distance than the previous one.
Slowing down (deceleration): The graph curves downward, getting flatter over time. Each successive time interval covers less distance.
Stationary: The graph is horizontal. Distance does not change.
Returning to start: The graph slopes downward toward the time axis. The object distance from the origin decreases.
When interpreting graphs, always check the axes first. A distance-time graph is different from a speed-time graph or a displacement-time graph. The same shape can mean completely different things on different axes.
A car journey might produce this distance-time graph: steep slope leaving home (accelerating onto the highway), moderate constant slope (cruising at 100 km/h), flat section (stopped for petrol), moderate slope again (back on the highway), and finally shallow slope (slowing down in suburban streets). Each feature tells a story about what happened during the journey. A transport engineer could use this graph to analyse traffic flow, identify congestion points, and estimate fuel consumption.
Australian road safety research: The Centre for Accident Research and Road Safety Queensland (CARRS-Q) analyses vehicle motion data from GPS trackers, dashcams, and telemetry systems. Distance-time graphs derived from this data help researchers understand driver behaviour, identify risky patterns like hard braking and rapid acceleration, and design safer roads. Insurance companies increasingly use telematics devices that record detailed motion graphs to set premiums based on actual driving behaviour.
A curved distance-time graph means the object is moving along a curved path. This is false. The graph is a mathematical representation, not a map. A curved graph simply means speed is changing. The actual path could be perfectly straight. Conversely, an object moving along a curved path at constant speed would produce a straight distance-time graph (if distance is measured along the path). The graph shape reflects how speed changes, not the geometry of the path.
Find the error in this graph interpretation.
- The graph shows distance increasing over time.
- The slope gets steeper, then flatter, then steeper again.
- Therefore, the object is always speeding up.
- The total distance travelled is the final y-value.
1. Describe what each of these features on a distance-time graph represents: a steep straight line, a shallow straight line, and a horizontal line.
2. A distance-time graph shows a straight line passing through (0,0) and (5 s, 25 m). What is the object's speed? Explain how you calculated it.
- Confusing the height of the graph with speed. — The height shows distance from the start. Speed is shown by the gradient (steepness) of the line.
- Calculating gradient as run/rise instead of rise/run. — Gradient = change in y (distance) / change in x (time). Getting this backwards gives the wrong speed.
📓 Copy Into Your Books
▼Distance-Time Graph Basics
Vertical axis = distance, horizontal axis = time. The line shows how far an object is from the start at each moment.
Interpreting the Line
Upward straight line = constant speed. Horizontal line = stationary. Steeper line = faster speed.
Calculating Speed
Speed = gradient = change in distance / change in time.
Curved Lines
A curved line on a distance-time graph means the speed is changing (acceleration or deceleration).
You learned that distance-time graphs show how distance changes over time, and the gradient represents speed.
Two cars travel along the same road. Car A's distance-time graph is steeper than Car B's. What does this tell you about their speeds?
The hook pointed out that every GPS tracking app draws a distance-time graph in real time — a steep line means fast movement, a flat line means you've stopped for a coffee.
Now that you can read and sketch distance-time graphs yourself, how would you describe what your own movement looked like today as a graph? Does the GPS example make more sense now that you can interpret what the slope actually means?
1. On a distance-time graph, what does a horizontal line represent?
2. How is speed calculated from a distance-time graph?
3. An object travels 30 m in 6 s. What is its speed?
4. A steeper line on a distance-time graph means:
5. What does a curved line on a distance-time graph indicate?
Describe the motion of an object whose distance-time graph shows a steep straight line for 3 seconds, then a horizontal line for 2 seconds, then a shallow straight line for 4 seconds. (3 marks)
Hint: Consider what each section of the graph tells you about speed and movement.
A car travels 60 m in the first 10 seconds, stops for 5 seconds, then travels another 30 m in the next 10 seconds. Sketch the distance-time graph and calculate the average speed for the whole journey. (3 marks)
Hint: Total distance divided by total time gives average speed.
Explain why the gradient of a distance-time graph represents speed, using the formula for speed in your explanation. (3 marks)
Hint: Link gradient (rise/run) to the speed formula (distance/time).