---
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: 2026-09-12
---

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# 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 (5 seconds, 20 metres) rises 20 over a run of 5, so the velocity is 4 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 6 metres per second for 5 seconds is a rectangle of 30 metres. Rising from 0 to 4 metres per second over 10 seconds is a triangle of half times 10 times 4, which is 20 metres. Area below the axis counts as negative, so a triangle of 12 metres above the axis with a rectangle of 8 metres below gives displacement 4 metres but distance 20 metres.

**Example.** Read a journey left to right, one straight piece at a time. Flat at 5 metres per second means cruising at a steady speed, then a straight slope down to zero means slowing steadily to a stop. 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. The slope of a velocity time graph is acceleration, so an uphill slope means speeding up.

**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

- [Plotting Linear Graphs](https://lightmysky.com/learn/mathematics/plotting-linear-graphs-mt_-3udyo6VyB)
- [Modelling Motion in a Straight Line](https://lightmysky.com/learn/mathematics/modelling-motion-in-a-straight-line-mt_no78_6w1rS)

## Opens up

- [The Constant-Acceleration Equations](https://lightmysky.com/learn/mathematics/the-constant-acceleration-equations-mt_Nyc3BcikQ9)