---
title: "Newton's Second Law as a Differential Equation"
description: "Writing the second law as mass times the second derivative of position equals force turns a force rule into an equation whose solution is the whole future motion. The same statement covers a falling s"
canonical: https://lightmysky.com/learn/science/newtons-second-law-as-a-differential-equation-mt_LzSNSkEovH
source: https://lightmysky.com/learn/science/newtons-second-law-as-a-differential-equation-mt_LzSNSkEovH.md
retrieved: 2026-09-12
---

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# Newton's Second Law as a Differential Equation

Writing the second law as mass times the second derivative of position equals force turns a force rule into an equation whose solution is the whole future motion. The same statement covers a falling stone, a spring and a charged particle.

Subject: Science · Area: Forces & Motion · Ages 18 to 19
Page: https://lightmysky.com/learn/science/newtons-second-law-as-a-differential-equation-mt_LzSNSkEovH

## Ready when they can

- Writes the equation of motion for a stated force and identifies what has to be solved for
- Solves the equation for a constant force and for a force that depends only on time
- Explains why two initial conditions are needed before a single trajectory is picked out

## Lesson: Force rules become motion equations

Before any equation, you list the pushes and pulls. Gravity pulls down with m times g, a spring pulls back toward rest, and each force gets a direction. Only outside forces on your chosen system count, since inside forces cancel out.

Newton says net force equals mass times acceleration. Since acceleration is the second derivative of position, written x double prime, the law is really mass times x double prime equals force. Writing it out states exactly what you must solve for: the position function over time.

**Example.** A steady pull gives the familiar parabola, because integrating a constant twice makes position grow with t squared. A pull that fades with time needs the full integration machinery instead. Either way the law hands you the equation, never the answer itself.

A second slope equation always needs two starting values before one path is picked out. Take the starting position plus the starting velocity, and the future motion is pinned down. With fewer, many different trips still satisfy the same force rule.

**Recap.** List the forces, write mass times x double prime equals force, then solve with two starting values.

## Practice

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

## Needs first

- [Recovering Motion by Integrating Acceleration](https://lightmysky.com/learn/science/recovering-motion-by-integrating-acceleration-mt_CXo7wKOA46)
- [Newton's Second Law: Force, Mass and Acceleration](https://lightmysky.com/learn/mathematics/newtons-second-law-force-mass-and-acceleration-mt_SKO85AcKdb)

## Opens up

- [Simple Harmonic Motion as a Second-Order Equation](https://lightmysky.com/learn/science/simple-harmonic-motion-as-a-second-order-equation-mt_HHmR-uCvwO)
- [Drag and Terminal Speed from the Equation of Motion](https://lightmysky.com/learn/science/drag-and-terminal-speed-from-the-equation-of-motion-mt_In6VeoADzZ)
- [The Time-Independent Schrödinger Equation](https://lightmysky.com/learn/science/the-time-independent-schrodinger-equation-mt_zeBVPery5T)