Equations of Motion

There can be a huge variety of problems that need to be solved depending on the variables that are given in the question. Variables that can be given, or, that need to be determined include – time, displacement, initial and final velocity and acceleration. 

The equations of motion are three neat equations that can be used in isolation to solve a problem if enough information is given, or, they can be used together to solve more difficult problems by using a series of intermediate steps. The individual equations can also be manipulated to make a different variable the subject of the equation.

There is a skill in being able to solve motion problems confidently, which involves being able to analyse a problem to correctly identify the variables given, determine the variable that needs to be calculated and using the correct equation/s to solve the problem. With motion problems, it is very much a case of practice makes perfect!

Equations of Motion

These equations will be on your data sheet so you don’t have to remember them. You do however, need to become confident at identifying variables and rearranging the equations:

 s=ut+\cfrac { 1 }{ 2 } a{ t }^{ 2 }


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


 u= the initial velocity (in m/s)

 v= the final velocity (in m/s)

 t= the time (in s)

s= the displacement (in m)

 a= the acceleration (in m/s2)

Example 1:

A car at rest accelerates at 1.4 m/s for 5.4 seconds. How far does the car travel in this time?


 u=0 m/s

 t=5.4 s

 a=1.4 m/s2

s=? m

With the information given and the variable that we need to find, the equation:  s=ut+\cfrac { 1 }{ 2 } a{ t }^{ 2 } can be used:

s=(0\times 5.4)+\cfrac { 1 }{ 2 } (1.4){ (5.4) }^{ 2 }


s=20.4 m

Example 2:

A child accelerates from rest down a water slide reaching a final speed of 4 m/s and entering the water. The water slide was 10 metres long. How long does it take the child to travel down the slide?


 u=0 m/s

 v=4 m/s

 s=10 m


With the information given and the variable we need to find, we would first need to determine acceleration, as there is no equation that has only u, v, s and t.

using:  { v }^{ 2 }={ u }^{ 2 }+2as and rearranging to give: a=\cfrac { { v }^{ 2 }-{ u }^{ 2 } }{ 2s }  

a=\cfrac { { 4 }^{ 2 }-{ 0 }^{ 2 } }{ 2(10) } 

a=0.8 m/s2

now substituting into:  v=u+at and rearranging to give: t=\cfrac { v-u }{ a }

t=\cfrac { 4-0 }{ 0.8 }

t=5 seconds