---
title: "Velocity and Acceleration as Derivatives"
description: "Velocity is the time derivative of the position vector and acceleration the derivative of velocity, taken one component at a time. Motion whose acceleration changes needs no new formulas once this is "
canonical: https://lightmysky.com/learn/science/velocity-and-acceleration-as-derivatives-mt_RdhbiJ6CTk
source: https://lightmysky.com/learn/science/velocity-and-acceleration-as-derivatives-mt_RdhbiJ6CTk.md
retrieved: 2026-09-12
---

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# Velocity and Acceleration as Derivatives

Velocity is the time derivative of the position vector and acceleration the derivative of velocity, taken one component at a time. Motion whose acceleration changes needs no new formulas once this is the definition.

Subject: Science · Area: Forces & Motion · Ages 18 to 19
Page: https://lightmysky.com/learn/science/velocity-and-acceleration-as-derivatives-mt_RdhbiJ6CTk

## Ready when they can

- Differentiates a position function to get velocity and acceleration at any instant
- Handles two-dimensional motion by differentiating each component separately
- Says why the constant-acceleration equations stop applying as soon as acceleration depends on time

## Lesson: Velocity and acceleration as derivatives

A speedometer needs more than distance: it needs to know how fast that distance is changing right now. That rate of change is velocity, the time derivative of position, and acceleration is the derivative of velocity in turn, making it the second derivative of position. Take s(t) equal to t cubed plus 2t. Then v(t) equals 3t squared plus 2, and a(t) equals 6t. At t equal to 2, velocity is 14 and acceleration is 12.

**Example.** In two dimensions position is a vector with one component per axis. Differentiate each component. For r(t) equal to the pair t squared and 3t, velocity is the pair 2t and 3, and acceleration is the pair 2 and 0. At t equal to 2 velocity is the pair 4 and 3, whose length is 5. Speed is that length, always nonnegative, while velocity keeps its direction.

Velocity is the limit of displacement over time as the window shrinks to zero, which is what a speedometer reads. The sign of velocity carries the direction along the axis: positive one way, negative the other.

The constant acceleration equations stop applying the moment acceleration depends on time. With a(t) equal to 6t the acceleration keeps changing, so those formulas stay on the shelf. Differentiate or integrate instead: velocity is the derivative of position and acceleration the derivative of velocity, one component at a time.

**Recap.** Differentiate position for velocity and velocity for acceleration, work axis by axis, and drop the constant acceleration formulas once acceleration varies.

## Practice

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

## Needs first

- [Kinematics with Calculus](https://lightmysky.com/learn/mathematics/kinematics-with-calculus-mt_2kNspOOoDw)
- [Resolving Vectors into Components](https://lightmysky.com/learn/science/resolving-vectors-into-components-mt_tc698unGJG)

## Opens up

- [Recovering Motion by Integrating Acceleration](https://lightmysky.com/learn/science/recovering-motion-by-integrating-acceleration-mt_CXo7wKOA46)