Anyone who has studied mechanics / dynamics will have come across Newton’s equations of motion (not to be confused with his *laws* of motion). The ones I get my students to use are

where is the initial velocity, is the velocity at time , is the *displacement* and is the acceleration. Note, these equations are only true for constant acceleration, but that actually covers quite a lot of situations. They can all be derived from the definition of acceleration.

## Derivation of Equation 1

We start off with out definition of acceleration, which is the rate of change of velocity. Writing that mathematically,

This is an example of a first order differential equation. To solve it we integrate. So we have

When we integrate without limits, we have to include a constant term, so we can write

where is our constant. To determine the value of the constant we need to put in some conditions, such as (but not necessarily) initial conditions. When we have defined that , so we can write

## Derivation of Equation 2

To derive equation two, which we notice involves the displacement (the vector equivalent of distance), we do the following

When so we can write

## Derivation of Equation 3

To derive equation 3 we use the trick of writing the acceleration in terms of the velocity and the displacement . To do this we write

So, writing

Again, we can work out by remembering that so

and so

on 11/11/2014 at 07:35 |Newton’s equations of motion – revisited | thecuriousastronomer[…] week, I showed how one could derive 3 of Newton’s equations of motion. As a colleague of mine pointed out to me on FaceBook, the […]

on 11/12/2014 at 07:30 |Newton’s 2nd law of motion, force and acceleration | thecuriousastronomer[…] this equation also allows us to derive the three equations of motion, equations like and , as I did in this blog. It tells us that it is more difficult to accelerate a more massive object than it is a less […]

on 11/02/2015 at 11:00 |Time dilation in General Relativity | thecuriousastronomer[…] just comes from Newton’s 2nd equation of motion , see my blog here which derives those […]

on 11/09/2015 at 16:41 |abhishekThise is a very most impartant of studen.

on 02/09/2016 at 04:09 |ChristySimple and catchy,is ur lectures. Thnk u

on 02/12/2016 at 15:02 |DeepakThank you. This was very useful to me

on 05/12/2016 at 12:09 |Shikha MishraThank you it was very helpful for me.

on 07/02/2017 at 21:43 |Princeton UniversityIt’s surprising to find on wordpress.com a resource so precious about equations.

We will note your page as a benchmark for Derivation of Newton’s equations of motion .

We also invite you to link and other web resources for equations like http://equation-solver.org/ or https://en.wikipedia.org/wiki/Equation.

Thank you ang good luck!

on 25/09/2017 at 17:23 |AryamanThanks

on 16/02/2018 at 18:06 |SAMINUGood iwas benefits alot

on 24/02/2018 at 06:25 |DeepikaTqu u it was very useful to me……

on 13/10/2019 at 10:25 |MantarajHow many ways to find out these equation

on 24/11/2019 at 07:58 |AfolabiakoredePls can you explain more further how you got equation 2