Displacement

Displacement is a fancy way of saying distance. In physics, displacement is measured in meters, or m.

Displacement doesn’t exactly have a formula, because you could measure small distances with a ruler, and large distances with, well…a large ruler.

However, if you know the average velocity and time taken, you can place those values into this formula to find the displacement:

Displacement should not be confused with distance. The difference is that displacement is the distance “as the crow flies” from the start; and distance is the path traveled. To show the difference, let’s have an example.

A car takes a small trip around town. Shown below is the path it travelled.

So what is its distance travelled?

To find the distance, we add up all the measurements together.

600+900+600+900 = 3000m

Well that was easy, now let’s find the displacement.

To find the displacement, we measure the same distance, but in vectors. A vector is a distance with a direction attached; for example a vector could be 500m to the right, or even 3cm upwards. When are calculating the displacement, some of the vectors will cancel out.

In order from start to finish, the vectors are 600m right, 900m down, 600m right, and 900m up.
We can then cancel out the 900m down and 900m up, because they are both the same distance; and opposite directions. This then leaves the two 600m right vectors.

Because the direction is the same, we can add the two measurements together, giving us 1200m right. This is the displacement of the car. We can then check our answer by measuring the distance from start to finish, and we find that it also equals 1200m.

Note: If the path traveled is just a straight line, there won’t be any vectors to cancel out. Therefore the distance would be exactly the same as the displacement.

Here’s an example using the formula. If a car is travelling in a straight line at 5ms^-1 for 1 minute, what will be its displacement?

For starters, we need to change the amount of minutes into secondsYou should know about the SI unit basics by now, but if you need to brush up on them they are under the Before We Begin section..

1 minute = 60 seconds

Now, we put the measurements into the formula.

d = 5×60
d = 300m (1 s.f.Don't know what this is? Significant figures are explained in the Before We Begin section.)

We already know that if the path is a straight line, then the distance is the same as the displacement. Therefore the displacement is 300m.

Okay, now it's time to take a another trip around town again, but for figuring out the displacement we are going to use a tool we like to call... TRIGONOMETRY!

Yes, that's right. Even though this is a physics website, we're going to use some maths.

Let's see where we travelled this time, and figure out the displacement with a bearing. Don't panic, it's easier than it sounds.

There are a few things to realise with this picture. First, this picture is not to scale, so you can't use a ruler and guess the answer.

Secondly, that white symbol in the top-left part of the triangle is the greek letter, theta. It tells us that the top-left corner of the triangle is the angle we will find out shortly.

Thirdly, the cream coloured arrow is the displacement. Remember that displacement is the direct distance from the start to the finish.

To begin solving this problem, we need to find the displacement shown by the cream arrow. To figure this out, we use the Pythagorean theoremRemember this from Year 11? It's a²+b²=c²..

a²+b²=c²
500²+1200²=c²
250000+1440000=c²
1690000=c²
c=1300

Ok, so we know that the displacement is 1300 meters, but we also want to know the bearingThis is part of Year 11 trigonometry too. It just replaces a compass direction with a number, telling us how many degrees clockwise from North we are travelling in..

To do this, we first need to calculate the angle, theta. Do you remember SOHCAHTOA?

The right angle triangle requires an angle, but we know the opposite and adjacent angles; which means we need to use the function "tan".

So if you have a wonderful knowledge of trigonometry, you should be able to tell us that...

angle = tan^-1(12÷5)
angle = tan^-1(2·4)
angle = 67° (2 s.f.)

Great! But that's not the final answer...

Let's look at the North, South, East, West diagram and change it slightly.

This is the diagram from which bearings are made. Making the bearing is really simple, but explaining how to do it is another story.

To make a bearing, you look at the diagram and find the line that is travelling in the North, South, East or West direction. Then, you add or subtract the angle depending if it is clockwise from the line, or anticlockwise.

In this case, the angle is clockwise from East, so the bearing is 90+67 which is 157°.

So putting the bearing and the measurement together our answer is 1300m at a bearing of 157°.

There's the answer! Well done if you got that answer.

A quick question before we move on:

If you wake up and fall asleep in the same bed, what is your displacement at the end of the day?

Answer: 0!

Why? Because displacement is the direct distance from start to finish. Because the start and finish are in the same place, the distance between them is zero.

Once you think you’re ready to move on, try looking at Velocity.