Deriving the suvat Formulae (Cambridge International AS Mathematics)

Deriving the suvat Formulae

What is suvat?

• 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

What are the suvat (constant acceleration) equations?

• The five equations for motion in a straight line are:

$v=u+at\,v^{2}=u^{2}+2as\,s=\frac{1}{2}(u+v)t\,s=ut+\frac{1}{2}at^{2}\,s=vt-\frac{1}{2}at^{2}$

• 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

  • $s=vt-\frac{1}{2}at^{2}$ is not given but it is easy to remember as it is very similar to $s=ut+\frac{1}{2}at^{2}$

How do I derive the suvat equations?

• 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

• Two of the equations can also be derived using calculus

  • Velocity is found by integrating acceleration
  • Displacement is found by integrating velocity

Worked Example Use the constant acceleration equations

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

to show that

$v^{2}=u^{2}+2as$.

Answer:

Exam Hint

• 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.