Motion Graphs: Gradients and Areas
Read the gradient of a displacement-time graph as velocity and the area under a velocity-time graph as displacement, and describe a whole journey from the shape of the graph alone.
What a learner can do afterwards
- Find a velocity from the gradient of a displacement-time graph
- Find a displacement from the area under a velocity-time graph, counting area below the axis as negative
- Describe in words the journey shown by a given velocity-time graph
1 · Read
A displacement time graph hides velocity in its slope. Divide rise by run: change in position over change in time. A straight line from (0 seconds, 0 metres) to (4 seconds, 24 metres) rises 24 over a run of 4, so the velocity is 6 metres per second. Steep means fast, flat means stopped, and downhill means you are heading back.
A velocity time graph hides displacement in its area. Flat at 3 metres per second for 8 seconds is a rectangle of 24 metres. Rising from 0 to 6 metres per second over 4 seconds is a triangle of half times 4 times 6, which is 12 metres. Area below the axis counts as negative, so a triangle of 15 metres above the axis with a rectangle of 6 metres below gives displacement 9 metres but distance 21 metres.
Read a journey left to right, one straight piece at a time. A straight slope up to 8 metres per second means speeding up steadily, then a flat line means steady cruising. A flat line at zero is the only flat line that means standing still.
Split every graph into straight sections before you measure anything. Find each gradient or area on its own, then add the pieces with their signs. A velocity time slope is acceleration: uphill means speeding up above the axis, slowing down below it.
Slope of position gives velocity, area of velocity gives displacement, and the shape tells the journey.
The rewritten readings were written for this dataset. Practice items adapted from WeBWorK Open Problem Library problems by Brandon Lostracco (Brock University); Paul Pearson (University of Rochester); Wynne Reichheld (Brock University) (CC BY-NC-SA 3.0).
2 · Watch
Take it off screen
Where it sits
Learn first
Optional connections
This opens up
14 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.