Equations of Motion

These are another very useful tool for tackling a certain type of problem. Problems where an object is accelerating uniformly.

There are 5 quantities which, if we know some of, we can calculate the others using one or more of the equations.

s	Displacement	m
u	Initial velocity	m/s
v	Final velocity	m/s
a	Acceleration	m/s2
t	Time	s

The equations are;

v = u + at        s = ut + 1/2 at²       v² - u² = 2as        average velocity = s / t  = (v + u) / 2

 

Here are a few examples

1. A car accelerates from rest at 2 m/s². How fast will it be travelling after 5 seconds? How far will it have travelled?

To find the velocity use v = u + at.     Putting in the numbers   v = 0 + 2 x 5 = 10 m/s

To find the distance we could use s = ut + 1/2 a t².     s = 0 + 1/2 x 2 x 10²  =  100 m

2. A boy throws a stone up into the air. It reaches a height of 11 m. What speed did he throw it up at? How long will it spend in the air?

We need to define which direction is positive, upwards or downwards. I suggest downwards so that the initial velocity of the stone is negative and its acceleration is +9.8 m/s². (as long as we are consistent it doesn't actually matter which direction is positive)

To calculate the initial velocity use v² - u² = 2as for the journey up.   As v = 0 this  becomes u² = 2as  or  u  =  √ ( 2 x 9.8 x 11) = 14.7 m/s

The time in the air will be twice the time it takes to reach a height of 11 m

if v = u + at    then     t = (v - u) / a   =  ( 0 - - 14.7) / 9.8 = 1.5 s        The total time in the air is therefore 3 s.