Position-Time and Distance-Time Graphs
A position-time graph plots an object’s position on the vertical axis against time on the horizontal axis, and a distance-time graph plots the total distance traveled against time; in both, the slope (gradient) of the line gives the speed or velocity. In brief: a horizontal line means the object is at rest, a straight sloping line means constant speed, and a curve means changing speed. On a position-time graph, a negative slope means movement in the negative direction, but a distance-time graph can never slope downward.
Graphs show the whole motion of an object at a glance. This guide explains how the graphs are drawn, how to read them, how to calculate speed and velocity from the slope, compares the shapes of lines in a table, works through an example with a cyclist and corrects common mistakes.
How are motion graphs drawn?
Time is the independent variable and is placed on the horizontal (x) axis. Distance or position is plotted on the vertical (y) axis, with a scale and a unit on each axis. Points from a table of data are plotted and joined with a straight line or a smooth curve. A good graph has a title, labeled axes with units and a scale that uses most of the paper.
- Distance-time graph: shows how far the object has gone in total. It starts at zero and can only increase or stay constant.
- Position-time graph: shows where the object is relative to the origin. The values can be negative and can decrease when the object moves backward.
What does the slope show?
The slope of a line is rise ÷ run = Δy ÷ Δx. On a distance-time graph, this is Δd ÷ Δt, which is the speed. On a position-time graph, it is Δx ÷ Δt, the velocity (with a sign for direction).
- A horizontal line has zero slope: the object is at rest.
- A straight line with a constant slope shows constant speed. The steeper the line, the greater the speed.
- A curve that gets steeper shows increasing speed (acceleration), and a curve that flattens shows decreasing speed.
- For a curve, the speed at one instant is the slope of the tangent to the curve at that point.
- The average speed over an interval is the slope of the straight line joining the two end points (the chord).
Graph shapes compared
| Shape of the line | Distance-time graph | Position-time graph |
|---|---|---|
| Horizontal | At rest | At rest |
| Straight, upward | Constant speed | Constant positive velocity |
| Straight, downward | Not possible | Constant negative velocity |
| Curve getting steeper | Speeding up | Speeding up (in the direction of motion) |
| Curve flattening | Slowing down | Slowing down |
Worked example: reading a distance-time graph
A cyclist’s journey has these points on a distance-time graph: (0 s, 0 m), (4 s, 20 m), (6 s, 20 m), (10 s, 32 m).
- 0 to 4 s: slope = (20 − 0) ÷ (4 − 0) = 5 m/s. The line is straight and rises, so the speed is constant.
- 4 to 6 s: the distance does not change (20 m), the line is horizontal, so the cyclist is at rest. Speed = 0.
- 6 to 10 s: slope = (32 − 20) ÷ (10 − 6) = 12 ÷ 4 = 3 m/s. The line is less steep than the first part, so the speed is lower.
- Average speed for the whole trip: 32 m ÷ 10 s = 3.2 m/s.
If this were a position-time graph and the final point were (10 s, 8 m) instead of (10 s, 32 m), the slope from 6 to 10 s would be (8 − 20) ÷ 4 = −3 m/s: the cyclist would move back toward the origin at 3 m/s. The displacement for the whole trip would be 8 m, but the distance traveled would be 20 + 12 = 32 m.
How did graphing motion develop?
Nicole Oresme drew graphs of changing quantities in the fourteenth century, and René Descartes’ coordinate geometry in the 1600s made it possible to plot equations. Galileo analyzed motion using geometric diagrams. Today, motion sensors and data loggers plot position-time graphs on a screen in real time during experiments.
Where are motion graphs used?
Graphs are used in physics experiments, in the analysis of sports performance, in transport planning (timetable graphs of trains), and in checking the data from GPS devices.
Which mistakes are common?
Mistake 1: Reading the graph as a picture of the path
A rising line does not mean that the object is going uphill. The axes are distance and time, not the shape of the trajectory.
Mistake 2: Saying that a horizontal line means constant speed
On a distance-time or position-time graph, a horizontal line means that the object is at rest.
Mistake 3: Using the height of the line instead of the slope for speed
The speed is the slope, not the y-value.
Mistake 4: Confusing the intersection with a collision
Where two lines cross on a position-time graph, the objects are at the same position at the same time. They do not need to collide.