--- title: "Motion Graphs: Gradients and Areas" description: "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." canonical: https://lightmysky.com/learn/mathematics/motion-graphs-gradients-and-areas-mt_KhJeCMYBSL source: https://lightmysky.com/learn/mathematics/motion-graphs-gradients-and-areas-mt_KhJeCMYBSL.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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. Subject: Mathematics · Area: Mechanics · Ages 16 to 17 Page: https://lightmysky.com/learn/mathematics/motion-graphs-gradients-and-areas-mt_KhJeCMYBSL ## Ready when they can - 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 ## Lesson: Reading motion from graphs 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. **Example.** 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. **Tip.** 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. **Recap.** Slope of position gives velocity, area of velocity gives displacement, and the shape tells the journey. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Modelling Motion in a Straight Line](https://lightmysky.com/learn/mathematics/modelling-motion-in-a-straight-line-mt_no78_6w1rS) ## Optional connections - [Plotting Linear Graphs](https://lightmysky.com/learn/mathematics/plotting-linear-graphs-mt_-3udyo6VyB) ## Opens up - [The Constant-Acceleration Equations](https://lightmysky.com/learn/mathematics/the-constant-acceleration-equations-mt_Nyc3BcikQ9)