Deriving the suvat Equations | CIE A Level Maths: Mechanics Revision Notes 2020

Deriving the suvat Formulae

suvat is an acronym for the five quantities used when modelling motion in a straight-line with constant acceleration
- s – displacement (from the starting point)
- u – initial velocity
- v – final velocity
- a – acceleration
- t – time
All except time are vector quantities and can be negative
- time is a scalar quantity

v = u + a t v^{2} = u^{2} + 2 a s s = \frac{1}{2} (u + v) t s = u t + \frac{1}{2} a t^{2} s = v t - \frac{1}{2} a t^{2}

The equations can only be used when the motion has constant acceleration
All equations connect four of the five quantities
- Knowing any three allows a fourth to be found
The first four equations are provided in the exam
- $s = v t - \frac{1}{2} a t^{2}$ is not given but it is easy to remember as it is very similar to $s = u t + \frac{1}{2} a t^{2}$

The four equations that involve time can be derived from a velocity-time graph
- The velocity-time graph will be a straight line as the acceleration is constant
- The fifth equation can be found by choosing any two of the equations and eliminating the t variable (see the worked example

qJL_I9LZ_2-3-1-deriving-the-suvat-formula-diagram-1_2

Two of the equations can also be derived using calculus
- Velocity is found by integrating acceleration
- Displacement is found by integrating velocity

2.3.1_WE_Deriving the suvat formulae_1

Use the constant acceleration equations

$s = \frac{1}{2} (u + v) t$ and $v = u + a t$

to show that

$v^{2} = u^{2} + 2 a s$ .

2-3-1-deriving-the-suvat-formula-example-solution

If you are asked to derive one of the formulae then the question will likely give you a hint as to which method to use. They may provide a velocity-time graph. Make sure you show each step and state any reasons such as the gradient of the graph being the acceleration.
If the question does not ask you to derive the formulae, then you can use them freely without proof.