Deriving (SUVAT) equations of motion
They are called ‘SUVAT’ equations because the quantities s, u, v, a, and t are used in the equations where;
- s = displacement (s)
- u = initial velocity (ms-1)
- v = final velocity (ms-1)
- a = acceleration (ms-2)
- t = time (s)
A list of the SUVAT equations of motion that you will need to memorise has been given below;
- v = u + at
- s = ½(u + v)t
- s = ut + ½at²
- v² = u² + 2as
Deriving the (SUVAT) equations of motion
v = u + at
Acceleration is defined as the rate of change of velocity over time. This involves final velocity – initial velocity in a given period of time. We can write this as;
We can rearrange the expression to get the first SUVAT equation;
s = ½(u + v)t
Velocity is defined as the rate of change of displacement over time. We can find average velocity by adding the initial and final velocities and dividing by 2; this can be written as;
We also know the distance or displacement is equal to velocity x time; (s = vt). So we can replace v with average velocity to get;
s = ut + ½at²
From the previous and first equations we derived we know that;
We can express displacement in terms of u, a and t rather than u, v, and t. We can substitute v = u + at in the previous equation to get;
…which simplifies to give;
v² = u² + 2as
This equation was created by ‘Evangelista Torricelli’ to find the velocity of an object moving with a constant acceleration without knowing the time interval. It is therefore known as Torricelli’s equation.
To derive the Torricelli’s equation we shall use the following equations that we have derived above;
We can make t the subject in the first equation to get;
So now we can replace t in the displacement formulae to get;
Using calculus
We know that velocity is the rate of change of displacement over time;
That must mean that;
Integrating a with time should give us an expression for velocity v, this is because acceleration is the rate of change of velocity with time;
Integrating velocity with time should provide an expression for displacement, this is because velocity is the first derivative of displacement modelled as a function of time;
…therefore;
We can use algebraic manipulation to find the other equations;
