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