Hello there, I'm Mr. Forbes and welcome to this lesson from the Measuring and Calculating Motion Unit.

This lesson's all about displacement-time graphs and how we can use them to find the velocity of a moving object.

By the end of this lesson, you're going to be able to look at displacement-time graphs and read off values for displacement and time, and also be able to calculate velocity of objects based upon those readings.

In addition, you'll be able to sketch displacement-time graphs when given the appropriate information.

The keyword you need to understand for this lesson are shown here, the first is displacement, and that's distance in a particular direction.

And then the main part of the lesson is all about displacement-time graphs, and these are graphs that show you what the displacement is for different times, and the final keyword is motion sensor.

And you'll be using a motion sensor in order to produce a displacement-time graph.

And here's a set of explanations for those keywords that you can return to at any point during the lesson.

This lesson's in three parts.

In the first part, we're going to look at what a displacement-time graph is and how we can read values of displacement and time from it to see where an object is.

In the second part, we're going to try and plot a displacement-time graph in real time by using a motion sensor to record our movement, and in the third part, we're gonna analyse the patterns in the graphs in more details to try and find out when an object's moving quickly, slowly, backwards, or forwards.

So when you're ready, we'll begin by looking at displacement-time graphs.

We'll start by revising the idea of displacement.

So the displacement of an object is how far away that object is in a straight line from a starting position, and we have to specify the particular direction it is as well, so we always need to describe that direction.

There are several different ways of doing that.

One of the most common is using compass direction, so I might walk a certain distance northeast, 10 kilometres in that direction, or I might walk southeast, and those displacement would be different.

So here I've just done the 10 kilometres northeast.

We could also use a simpler system of, perhaps, just left and right, so I might go three metres right or two metres left.

And of course, there's up and down as well.

We can move upwards and downwards if we're in an elevator.

Displacement are vector quantities, and when we add a vector together, we need to take into account the direction of movement.

So I'll give you an example of that.

If we follow these movement instructions, so let's say I've got a starting point and I walk five metres to the right, and then I move two metres upwards.

I've got to put that direction onto my diagram to make sure I know that it's not still in the same straight line as before.

Then I've got three metres left and two metres down.

Overall my final displacement is just two metres right, despite the fact I've walked all those additional distances.

So I have to to sum up my displacement, taking into account the directions of motion.

Right, let's check if you can work out a final displacement now.

I've got a pupil following the instructions shown in the box there, and I'd like you to work out what's the displacement from the starting point.

Is it one metre north, 27 metres south, seven metres north or 27 metres north? So pause the video, work that out, and then restart please.

Okay, welcome back.

Hopefully you came up with the solution of seven metres north.

If we draw a diagram of that here, we've got the starting point, we move the 10 metres, then the four metres, and then that second 10 metres is going backwards towards the start really, and then finally, the three metres, so overall we've moved four metres and three metres to the north.

That gives us seven metres to the north.

Well done if you've got that.

Instead of constantly writing out things like northwest, northeast, we can describe displacement along the straight line by just using positive and negative values, a little bit like this.

So I've got a number line here ranging from minus five to plus five, and I can draw a dis displacement as being plus two metres.

I can have another displacement further in that direction towards the right, plus four metres, or I could describe a negative displacement like this, as minus four metres.

And it's often much simpler to just have negative and positive numbers instead of always stating the direction completely.

So we're going to be using those throughout most of the rest of the lesson.

Now the main focus of this lesson is looking at graphs of displacement and time.

So we're gonna start by looking at this one here.

This is the displacement of a mobility scooter at different times.

You can see along the bottom, I've got a time axis running from naught to 120 seconds, and up on the Y axis, I've got a displacement, and that displacement is running from naught to 300 metres.

And we can look at the motion to find a position of an object at any time.

So I could look at, say, 30 seconds and see that going upwards and across, the displacement at 30 seconds is 100 metres, and I can do that for 60 seconds and see the displacement's 150 metres there, so we're just reading the values of the two axes here.

And finally, look at 90 seconds and go across, you can see there's a displacement of 200 metres there.

So it's fairly straightforward to be able to read displacement for particular times.

I'd like to check if you can read the displacement of a graph now please.

So what's the displacement at time equals 40 seconds for this windup toy? Pause the video, make your selection, and then restart.

Welcome back.

Well, that was six metres.

If you look at 40 seconds, looking upwards, and then across to the displacement axis, that gives us a displacement of six metres.

Well done if you've got that.

We can show movement in opposite directions on a displacement-time graph.

I'm gonna show you how to do that here.

So I've got a number line there showing my distance from the left to the right, and that's in metres, and I've got a graph on my position at the bottom as well.

And we're gonna try and walk through a few different movements, so first of all, I'm gonna start with a displacement of zero.

It's the easiest place to start, and then I'm gonna follow the instruction, walk four metres to the right in 10 seconds, and as you'd expect on the graph, I'd find at a time equals 10 seconds, my displacement would be four metres, and I've walked at a steady speed here, so I've got a nice straight line.

Follow the next instruction, walk five metres to the right in 20 seconds.

Well, my time will be now at 30 seconds, and my total distance towards the right will be nine metres.

So there's my second phase of movement.

As you'd imagine, moving backwards, so I'm going to walk six metres to the left in 20 seconds.

What will happen is my displacement's now gone down to three metres, so my graph will go back downwards, my final displacement there, three metres in 50 seconds.

And then if I just stand still for the rest of it, I'll have a constant displacement.

Okay, let's see if you can interpret a displacement-time graph based upon instructions about walking.

So we've got four sets of instructions here and I've got a matching graph.

All I'd like you to do is to work out the order of those four instructions please.

So pause the video, work out the order, and then restart.

Okay, welcome back.

Let's have a look at the first stage.

Well, the first stage, I've walked forwards four metres in 10 seconds, so that matches the first stage of the motion there.

In the second stage, you can see my displacement's not changing at all, so I've stopped moving for 10 seconds.

In the third phase, I've walked for 30 seconds and have got a increase in displacement of two metres, so that matches this one here.

And finally, I've walked three metres forward in 10 seconds.

Well done if you've got that order.

So, so far I've only looked at positive displacement at distance in front of my starting point.

We're gonna move on and look at some negative displacement now where we can have values where we are behind our starting point basically.

So again, I'm gonna start at position zero, displacement zero, and I'm going to follow a set of movements, first of those, walk forward five metres and 20 seconds, and that'll give me a line on the graph like this.

You can see I'm at five metres at 20 seconds there.

Then I'll walk forward another one metre in 10 seconds and that'll give me another movement here on the graph.

And then the final movement, I'm gonna walk backwards.

I'm gonna walk eight metres to the left in 30 seconds.

And as I'm only six metres in front of the starting point, I'm gonna end up at minus two metres, two metres behind my starting point, and I have a line that goes down like this.

So I've got a complete journey in three phases shown there.

So as before, I'd like you to work out the correct order for these movements.

So I've got a graph and I've got four different movements.

I'd like you to match each part of that graph to one of the phases of movement to get the correct order please.

So pause the video, work that out, and then restart.

Welcome back.

When looking at the four phases, in the first part, in the first 10 seconds, I've moved forwards three metres, so that's the one that matches.

Then I've moved backwards five metres in the next 10 seconds, so I'm ending up at minus two displacement.

Then I've walked forwards again for a 20-second period, so that must match this one here, walk six metres forwards.

And finally, I've walked backwards in 20 seconds, four metres.

Well done if you've got those four.

Okay, it's time for the first task now.

And what I'd like you to do is to sketch some displacement-time graphs for me.

I've got two pupils and I've got two different sets of movement here.

So I've got pupil A and there's five phases to the movement described there.

And I've got pupil B, and there's only four phases to the movement there.

What I'd like you to do is to sketch the displacement-time graph showing the movement for both of those pupils.

And you can use the same axis for both of them.

Just draw two separate lines.

So pause the video, sketch the graph, and then restart please.

Okay, welcome back.

And your graph should look something like this.

As you can see, pupil A I've drawn with a purple dashed line there and they go forwards for 10 seconds, then they stop for 10 seconds, forward a bit more, and then backwards quite a further distance.

So they've ended up with negative displacement after about 45 seconds there, they've got negative displacement, and they stay with that negative displacement for a bit until the end.

Pupil B, well, they start by going backwards, so you can see the line goes downwards, a green solid line there, downwards to minus three metres, then you stop for a while, then you go forward quite a bit, and then backwards a bit more.

So well done if you've got a graph like that.

Now it's time for the second part of the lesson.

And in this part, we're going to be trying to plot a displacement-time graph, but instead of doing that manually, we're gonna try and get a motion sensor to help plot that graph.

So let's have a look.

A motion sensor's a device that we connect to a data logger and to a computer, and it allows us to measure the position of an object compared to the sensor, and from that, we can plot graphs.

So I've drawn a motion sensor here as a little blue box and connected it to my laptop, and I've got an object and I'm gonna measure the distance to it, and the way it works is it sends out a sound pulse and this is an ultrasound pulse so you can't hear it, very high frequency, and it sends out that sound pulse towards the object.

And when it reaches the object, it will reflect off it, so the sound hits the object and reflects back towards the sensor.

So we have a reflected sound pulse there as well.

And what the sensor can do is it can measure the amount of time it takes for that sound pulse to be sent to the object and back again, so it's measuring the time it takes.

It also knows the speed of sound 'cause it's travelling through the air, and so it can do a calculation to work out what distance that sound pulse has travelled to the object and back again.

So we can then measure that distance and that data is collected by the computer, and that allows us to plot a displacement against time graph.

So I've got a stationary object there, but if it was moving backwards and forwards, the sound pulse would travel for different amounts of time and the computer could work out what distances that object is away from the sensor.

Okay, let's see if you can remember how to do calculations based on speed, distance, and time.

With this question here, I've got the speed of sound in here, it's about 330 metres per second.

How long will it take a sound wave to travel a distance of 40 centimetres? Is it A, 132 seconds, B, 1.

3 seconds, C, 0.

01 seconds, or D, 0.

001 seconds? So pause the video, try and work that out, and then restart please.

Welcome back.

Hopefully you worked out it was answer D, 0.

001 And to work that out, we can do the maths here.

The distance was 40 centimetres and that's 0.

40 metres, and it's important we convert that to metres for the calculation.

And there's the equation, speed equals distance divided by time, V equals X divided by T.

We've rearranged that, put the values in, and that gives us an answer of 0.

001 seconds, which is one millisecond.

Well done if you got that.

Okay.

So what you're going to be doing is you're going to be using some apparatus to produce a graph of a trolley moving in real time.

So we're gonna use a dynamics trolley and try and move that across the surface of a flat desk or a bench like that.

We're gonna position the sensor, usually it's best to try and raise the sensor up slightly so we don't get too many reflections off the desk.

And we're gonna position a trolley a few centimetres away from it, perhaps 30 or 40 centimetres to start.

And on that trolley, to make sure we get a nice reflection back towards the sensor, we're gonna position a piece of cardboard, so there's a big sheet of cardboard there to allow the reflection of the sound wave to go backwards and forwards there.

And you're gonna move that trolley throughout a movement range, perhaps from about 10 centimetres closest to the sensor up to a distance of about a metre is best.

So moving the trolley backwards and forwards will allow the sensor to measure distance for different times.

And from that, we'll produce a graph.

When you're moving the trolley away from the sensor, what's gonna happen there is the displacement's going to increase over time, so you're gonna get a graph that looks a little bit like this.

If you move the trolley very precisely and you're getting a constant velocity, you're going to get a straight line on your graph.

And so you can see in this one, the displacement's increased from 10 centimetres to 100 centimetres across about 4.

5 seconds there.

But more likely, you're going to have some variations in the velocity.

And in this example, I've got the velocity increasing.

So imagine you've just tapped the trolley and it's moving, but friction's slowing it down, and so you'll get a gradual decrease in velocity.

So you're gonna get a slightly curved shape like that one.

Similarly, if you're moving the trolley towards the sensor, what's gonna happen is the displacement is going to decrease over time.

So if you've got a constant velocity, you're going to get a line like this.

We've got moving from 100 centimetres away to 10 centimetres away in five seconds there at constant velocity.

Or again, if there's some sort of frictional force, you might get it slowing down slightly.

So in this example, I've got gradual slowing down of the trolley due to friction.

You can see it's a little bit of a curve line there.

So let's check if you understand what you'll see on the graph of motion when you use a movement sensor.

I've got a dynamics trolley, it's pushed towards the motion sensor, it hits the sensor, bounces off it, and moves away.

Which of those three graphs would've been produced by that motion please? So pause video, make your selection, and restart.

Okay, welcome back.

Well, you should have chosen A.

You can work that out fairly easily because as the trolley's moving towards the sensor, the displacement's decreasing, and it hits the sensor so it's not centimetres away and then it will gradually increase its displacement.

So you need a downward slope, followed by an upward slope for that to work.

C isn't right because that's clearly not got close enough to the centre, it's still at 50 centimetres, so it just didn't bounce off it.

Well done if you got A.

Okay, it's almost time for you to carry out the experiment now, and you'll need to set up the equipment like this with the motion sensor, the trolley with the reflecting cardboard.

You'll also need a stopwatch to allow you to try and match some timings.

I'm gonna show you a graph that I'm gonna ask you to try and replicate by moving the trolley backwards and forwards in that space in a moment, but you'll need a stopwatch in it order to get some timings.

So you're gonna set your equipment up like that, and I'd like you to then try and produce these three different shapes for me using three different movements of the trolley.

So I've got three different lines there and I'd like you to try and replicate those as best you can.

So pause the video, try and replicate those using the equipment, and then restart.

Okay, welcome back, and hopefully, your experiment went well.

As you can see, I've got a set of results like these here and they're not perfectly matching the original graphs I gave you to try and replicate.

I don't expect anybody could perfectly replicate them.

You need some sort of robotic movements to get smooth displacement graphs like I should do, but if you've got something like this, well done.

Okay, it's now time for the third part of the lesson, and in it we're going to look in detail at using displacement-time graphs.

We're going to try and compare the motion of objects when it changes direction, stops and starts and so on.

So let's go on with that.

Okay, let's start by comparing the motion of two objects.

So I've drawn the motion of two different objects, the displacement against time on this graph here.

I've got a pink dashed line and a blue dotted line.

What you need to know about them is that the gradient of those lines tells us the change in displacement per second, how much the displacement has increased every second.

What that means is a gradient must be equal to the velocity of the object 'cause velocities change in displacement per second, the rate of change of displacement.

The steeper the lines are, the greater the velocity of the object must be.

So my pink dashed line has got a higher velocity than the blue dotted line.

What I'd like you to do is look carefully at these three graphs.

They've all got the same set of axes, they've got the same displacement scale and the same timescale, and what I'd like you to do is to decide which of those is the highest velocity, please.

So pause the video, make your decision, and then restart.

Okay, welcome back.

Hopefully you chose B.

And the reason it's B is B's got the steepest gradient there, so as long as those axes are all the same on each of those, that's clearly the highest velocity well done if you chose that.

Now, we've seen that the gradient shows us the velocity, but it also can tell us the direction of movement.

So I've got two lines here.

These two objects are moving in opposite directions.

The pink dashed line has got a displacement that's decreasing over time, and the green dotted line has got a displacement that's increasing over time, so that they must be going in opposite directions.

It might be up or down.

If we choose forwards as a positive, then this one is moving forwards and this one is moving backwards.

So we can describe the opposite directions of motion on the graph like that.

Okay, now what I'd like you to do is to decide which of these object is moving slowest.

So which has the lowest speed? Again, each of those three graphs has got the same displacement scale and the same time scale for easy comparison.

So just decide which of those is moving slowest please.

Pause the video and then restart.

Okay, welcome back.

You should have chosen C for that one.

And it's moving slowest because the gradient is shallowest.

It's got a negative velocity, it's moving backwards, but it's got a shallow gradient, so its speed is slow compared to the other two.

Well done if you've got that.

Okay, we're going to look at some more complex motion now, where we've got changing velocities and speeds.

So I've got a graph here, and as you can see, there's three different parts of the motion here.

We've got three different straight lines showing that we've got three different velocities, because the gradient shows the velocity.

In the the first section, we've got a forward constant velocity.

It's a straight line, so that means it's constant velocity.

In the second phase of motion, then we're moving backwards.

You can see the displacement's decreasing in this section.

So we've got backwards at constant velocity.

And in the third phase of motion here, we've got forwards constant velocity again.

If you compare the gradients, you can see this final forwards constant velocity.

That's not very fast compared to the other sections where the gradients much steeper.

We can look at the graph in a little bit more detail and describe the displacement in each section as well as the velocity.

So we can look at the Y axis to try and identify if the displacement's positive or negative.

So looking at this first section here, in all of that section, we've got positive displacement, whatever the object is, it's ahead of the starting point, the positive direction.

In this section here, all of the value for the displacement are negative.

So between those two times in that section, we've got negative displacement.

And again, just towards the end of the motion, we've got some positive displacement again.

So displacement can be both positive and negative.

We have to treat displacement and velocity separately.

Velocity can be positive even if displacement is negative, and the other way around as well.

So in this first section of the graph, we've got negative velocity and positive displacement.

You can see that the object is going backwards, its displacement is decreasing, and it's got positive displacement in all that time.

In this low section of the graph, we've got negative velocity still.

We're moving towards the negative, but we've already reached negative displacement.

In this section, we've got a stationary object and it's stationary because you can see that the gradient is zero there, it's at minus two metres.

There's no change in that over the time.

And in this section of the graph, we've got positive velocity.

It's moving forwards again, but it's still got negative displacement.

And in the final section of the graph here, we've got positive velocity and positive displacement.

It's moving forwards and it's already got a positive displacement, so we have to be very careful when describing the different sections of the motion.

Okay, let's check if you understood that.

I'd like you to describe the movement between time equals zero seconds and time equals 10 seconds as shown by the graph here.

And I've highlighted that section for you to make it easier to see.

So I'd like you to choose the two correct options from this list.

Is it positive velocity or negative velocity? Is it positive displacement or negative displacement? So choose two of those.

So pause the video, make your selections, and then restart.

Okay, welcome back.

You should have selected negative velocity because the displacement is decreasing, it's moving towards negative values, and positive displacement because all of displacement values there are still positive.

Well done if you selected those two.

Okay, another check of understanding of graphs here.

What I'd like you to do is to describe the movement in the highlighted section shown in green there between time 22 seconds and 28 seconds.

Is it positive velocity, negative velocity, positive displacement, or negative displacement? I'd like you to select the correct two please.

So pause the video, make the selection, and restart.

Okay, welcome back.

Hope you selected negative velocity.

You've got a gradient that's going downwards again, so that's negative velocity, and it's also got negative displacement in that section.

Well done if you selected that.

And another quick check here, I'd like you to describe the movement between 32 seconds and 38 seconds.

So decide which velocity it's got and which type of displacement it's got please.

So pause the video, make your two selections.

and restart.

Welcome back.

In this instance, it was positive velocity.

You can see the displacement is starting to increase, to go towards positive values again and will go on an upward slope, and it's still got negative displacement in that section.

So well done if you selected those two.

And another check here, and in this one, I'd like you to work out the average speed for the journey of this miniature train.

And you've got to use the speed equation that you should already know.

So is it A, naught metres per second, B, one metres per second, or is it C, two metres per second? So pause video, make your selection, and restart please.

Okay, welcome back, and hopefully, you selected one metre per second.

As you can see, the distance travelled was 600 metres, and if we put that into our speed equation, speed equals distance divided by time, we can get an average speed for the journey of 600 metres divided by 600 seconds, one metre per second.

Well done if you got that.

Displacement-time graphs can be used to show different phases of motion.

We've already seen a few examples of that.

I've got a more complex one here, and as you can see, there's changes in the gradient of this graph several times throughout the journey, and that means the velocity is changing.

So we can look at each section in turn.

In this first section, we're moving forward at six metres per second.

We can calculate that from the distance travelled order displacement change, that's 600 metres in 100 seconds, and that will give us six metres per second.

The second phase of motion here is simpler to describe.

It's stationary, there's no change in the displacement at all during those 100 seconds, so it must be stationary.

In this next phase of motion, we're moving backwards at six metres per second.

And then, a little bit more complex.

We've got this phase of motion here, we're still moving backwards, our displacement's decreasing, so we've got a backwards motion and the gradients not quite as steep there.

We're moving at four metres per second if we calculate that.

Another stationary phase here.

And finally, we've got a forward motion again at the end.

So even though our displacement's negative, we're moving forward back towards the starting point.

So let's check if you can understand more complicated motion.

What I'd like you to do is look at this graph here.

The graph shows the movement of an airport shuttle bus, and it's moving between three different car parks.

I'd like you to find out the average velocity and the average speed for the bus for one complete journey from the start to the end, as shown on the graph there.

So pause the video, try and work that out, and then restart please.

Okay, welcome back.

Well, the average velocity was actually quite easy to find because the final displacement is zero, so when I try and calculate an average velocity, I'm always going to get zero.

I've not actually achieved any overall displacement.

To find the average speed, what I needed to do was to add together all of the different phases of movement, so I moved forward 600 metres, then backwards 600 metres, then I moved backwards another 400 metres, and then forwards again 400 metres.

That gives me a total distance of 2,000 metres travelled, and to calculate the distance travelled then, I use the speed equation there, and I get an average speed of 3.

3 metres per second.

Well done if you've got that.

In the graphs we've drawn so far, we've drawn sharp changes in motion and that's not actually very realistic.

So if I've got a displacement-time graph like this and I've got a sudden change in direction like that, a sudden change in velocity, that's not very realistic at all.

In fact, when objects are changing velocities, they have to accelerate, and there's a gradual change in velocity.

So it would be much more realistic if I tried to draw my diagram like this, where I've drawn a curve connecting those two points.

Now, often we don't do that because we just want simple graphs, but a curve on a graph shows that the object is actually slowing down and then stopping and starting to speed up in the opposite direction.

So a realistic displacement-time graph would have curves connecting the straight lines.

Let's see if you can spot a realistic displacement-time graph.

I've got three separate graphs here.

Which of those is the most realistic in the way it shows changes in velocity? So pause the video, make your selection, and restart.

Hello again, and you should have chosen C, and that's the more realistic one because there's a gradual change in velocity.

There's a curve during the change there.

So well done if you selected that.

Okay, time for the final task of the lesson, and I've got a graph of the motion of a robot in a warehouse here.

And what I'd like you to do is to look carefully at that graph and try and describe what's happening in each of those marked phases of motion.

So I've marked them A to E for you.

Then I'd like you to explain why that graph is actually a bit unrealistic.

And the final part of the task is to try and sketch a more realistic version of that motion.

So pause the video, try those three questions, and then restart please.

Okay, welcome back.

Well, I've got all my descriptions of motion here.

In phase A, I'm moving forwards, constant positive velocity with positive displacement.

For part B, I'm moving backwards with a constant negative velocity and I've still got positive displacement.

In section C, still moving backwards with a constant negative velocity and my displacement there is negative, or the robot's displacement, I should say.

Part D, moving forwards again because my displacement's increasing again, positive velocity, negative displacement.

And part E, moving forwards with constant positive velocity and my displacement's positive there.

So well done if you've got those.

For the second part of the task, you are asked to explain why the graph was unrealistic.

And the reason for that is the sudden changes in velocity, the sharp points on the graph are quite unrealistic because the velocity would have to change gradually.

So what you should have are some curves like that.

So your sketch version of it should have curves where those points are.

So well done if you did that.

So we've reached the end of the lesson now, and here's a summary of everything you should have learned during it.

Displacement-time graphs show the movement of objects, and I've got a graph here showing quite a variety of movements.

The steeper the gradient, the greater the velocity.

So the steep purple dotted line is a fast moving object, and the shallower black lines are lower velocities.

Positive and negative gradients show different directions, straight lines show constant velocities, and curves show change in velocities.

So you can see we've got a decreasing velocity for the red dashed line there.

Well done for reaching the end of the lesson.

I'll see you on the next one.