Deriving the equations of motion

5 minAS Level2.1.6

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can derive the four equations of uniformly accelerated motion, starting from the definitions of velocity and acceleration, and state when they apply.

Four equations, two of them free

The suvat equations solve almost every straight-line motion problem at AS. Your data sheet gives you two of them. The other two you can rebuild in ten seconds flat, which is exactly what "derive" questions test.

Symbol	Meaning	Unit
$s$	displacement	m
$u$	initial velocity	$\text{m s}^{-1}$
$v$	final velocity	$\text{m s}^{-1}$
$a$	acceleration (constant)	$\text{m s}^{-2}$
$t$	time taken	s

The fine print: they only work for uniform (constant) acceleration in a straight line.

Equation 1: straight from the definition of acceleration

Acceleration is the rate of change of velocity. With constant acceleration,

$a = \frac{v - u}{t} \quad\Longrightarrow\quad v = u + at$

Equation 2: average velocity

Velocity is the rate of change of displacement, so displacement = average velocity × time. With uniform acceleration the velocity changes steadily, so its average is exactly halfway between $u$ and $v$ :

$s = \left(\frac{u + v}{2}\right)t$

On the velocity–time graph, that's the area of the trapezium under the line.

The triangle's height is v − u, which equation 1 tells you is at.

Quick check

Multiple choiceQuestion 1 · Recall

Answer the 2 questions above to continue.