Deriving the equations of motion

You are only expected to know how to use the four suvat equations, and you don't need to know how to derive them from first principles for the exam. However, seeing where they come from can help you understand them fully, and remember them more easily.

The term first principles means starting from the most basic assumptions we can make about a physical system — in this case, a physical law such as Newton's laws of motion, and the basic definitions of velocity and acceleration:

Equation 1

This is the simplest to derive, and comes straight from the definition of acceleration:

$\Delta v$ can be written as $(v-u)$, the final velocity minus the initial velocity, so:

Multiplying both sides by $t$:

Adding $u$ to both sides gives the final equation:

Equation 2

If an object travels from point a to point b, its velocity may vary along the way. Its average velocity between the two points can be found by adding the instantaneous velocity at a ($u$) to the instantaneous velocity at b ($v$), and dividing by two:

Combining this average velocity with $s=vt$ (using this average in place of a single velocity) gives:

Equation 3

We know that on a velocity-time graph, displacement is represented by the area underneath the line.

The area under the bottom (rectangular) section of the graph is the initial velocity, $u$, multiplied by the time, $t$:

The area under the top (triangular) section is found by subtracting the initial velocity from the final velocity, and multiplying by the time:

So the total displacement is the sum of the two areas:

The term $(v-u)$ also appears in the equation for acceleration, where $(v-u)=at$, so it can be substituted in directly, giving:

Equation 4

This is the hardest to derive, but often the most useful of the four. Starting with equation 2, $s=\frac{(u+v)}{2}t$, and substituting $t=\frac{(v-u)}{a}$ (rearranged from the definition of acceleration) gives:

This can be rearranged to:

Multiplying out the brackets:

Which finally gives: