---
title: "Recovering Motion by Integrating Acceleration"
description: "Given acceleration as a function of time, integration returns velocity and then position, with the constants of integration fixed by the initial conditions. The constant-acceleration equations reappea"
canonical: https://lightmysky.com/learn/science/recovering-motion-by-integrating-acceleration-mt_CXo7wKOA46
source: https://lightmysky.com/learn/science/recovering-motion-by-integrating-acceleration-mt_CXo7wKOA46.md
retrieved: 2026-09-12
---

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# Recovering Motion by Integrating Acceleration

Given acceleration as a function of time, integration returns velocity and then position, with the constants of integration fixed by the initial conditions. The constant-acceleration equations reappear as the case of a constant integrand.

Subject: Science · Area: Forces & Motion · Ages 18 to 19
Page: https://lightmysky.com/learn/science/recovering-motion-by-integrating-acceleration-mt_CXo7wKOA46

## Ready when they can

- Integrates a given acceleration twice and uses initial position and velocity to fix both constants
- Derives the constant-acceleration equations by integrating a constant
- Reads a change in velocity as the area under an acceleration-time graph

## Lesson: From acceleration back to motion

You know the forward chain. The slope of position is velocity, and the slope of velocity is acceleration. Integration runs that chain backward. One integral turns acceleration into velocity, and a second integral turns velocity into position.

Each integral brings one unknown constant, so you need two starting values. The velocity at time zero fixes the first constant, and the position at time zero fixes the second. The change in velocity over any stretch of time is the area under the acceleration curve there.

**Example.** A motorboat glides at 5.0 meters per second, then slows against its motion. One integral gives a velocity that reaches zero after about 6.3 seconds. A second integral gives a stopping distance of about 21.1 meters. Past that moment the math gives negative velocity, so the boat would drift back.

**Tip.** When acceleration stays constant, the two integrals hand back old friends. Velocity is starting velocity plus a times t, and position is starting position plus starting velocity times t plus half a times t squared. Constant acceleration is simply the case of a constant integrand.

**Recap.** Integrate acceleration twice and fix each constant with a starting value.

## Practice

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

## Needs first

- [Integration as the Reverse of Differentiation](https://lightmysky.com/learn/mathematics/integration-as-the-reverse-of-differentiation-mt_edtk3ArxRk)
- [Velocity and Acceleration as Derivatives](https://lightmysky.com/learn/science/velocity-and-acceleration-as-derivatives-mt_RdhbiJ6CTk)

## Opens up

- [Newton's Second Law as a Differential Equation](https://lightmysky.com/learn/science/newtons-second-law-as-a-differential-equation-mt_LzSNSkEovH)