---
title: "The Constant-Acceleration Equations"
description: "Derive the five constant-acceleration equations from a straight-line velocity-time graph, then choose the one that uses the three quantities you know and gives the one you want."
canonical: https://lightmysky.com/learn/mathematics/the-constant-acceleration-equations-mt_Nyc3BcikQ9
source: https://lightmysky.com/learn/mathematics/the-constant-acceleration-equations-mt_Nyc3BcikQ9.md
retrieved: 2026-09-12
---

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# The Constant-Acceleration Equations

Derive the five constant-acceleration equations from a straight-line velocity-time graph, then choose the one that uses the three quantities you know and gives the one you want.

Subject: Mathematics · Area: Mechanics · Ages 16 to 17
Page: https://lightmysky.com/learn/mathematics/the-constant-acceleration-equations-mt_Nyc3BcikQ9

## Ready when they can

- Derive v = u + at and s = (u + v)t / 2 from the velocity-time graph
- List the five quantities in a problem and pick the equation that leaves out the unwanted one
- Solve for time when the equation is quadratic in t and say what both roots mean

## Lesson: Picking the right motion equation

Five letters run the whole show: s for displacement, u for starting velocity, v for final velocity, a for acceleration, t for time. On a velocity time graph the slope gives v equals u plus a t, and the area of the trapezium gives s equals (u plus v) times t over 2. A car starting at 5 metres per second with acceleration 2 reaches 13 metres per second after 4 seconds, since 5 plus 2 times 4 is 13.

List what you know, name what you want, and pick the equation that never mentions the leftover letter. When time is unknown and unwanted, use v squared equals u squared plus 2 a s: a car starting from rest with acceleration 5 over 10 metres reaches 10 metres per second, since v squared is 100. When you know u, a and t and want s, use s equals u t plus half a t squared: with u 5, a 2 and t 6, s is 30 plus 36, which is 66 metres.

**Example.** A quadratic in t gives two roots, and you keep the one that fits the story. Starting from rest with acceleration 2, reaching 36 metres gives t squared equals 36, so t is 6 or negative 6, and the journey takes 6 seconds. A root of t equals 0 marks the moment the clock starts.

**Tip.** Write the five letters on paper every time, tick the knowns, circle the want, and cross out the leftover before you choose. Then check the sign of each number against your direction.

**Recap.** List the five letters, pick the equation that skips the leftover, and read both roots against the story.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [The Quadratic Formula](https://lightmysky.com/learn/mathematics/the-quadratic-formula-mt_C-4brnRYHY)
- [Motion Graphs: Gradients and Areas](https://lightmysky.com/learn/mathematics/motion-graphs-gradients-and-areas-mt_KhJeCMYBSL)

## Opens up

- [Vertical Motion Under Gravity](https://lightmysky.com/learn/mathematics/vertical-motion-under-gravity-mt_IgNj9bn14O)