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 the velocity of objects based upon those readings.

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

The keywords you need to understand for this lesson as 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.

The lesson's in three parts.

And in the first part, we're going to look at the structure of a displacement-time graph and how we can read values off it.

In the second part, we're going to try and plot a displacement-time graph using a motion sensor.

And in the third and final part, we're going to try and look at displacement-time graphs and analyse the motion, seeing when an object's moving fast, slow, or if it stopped.

So when you're ready, we'll begin with the first part, 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 clearly, there are several different ways of doing that.

One of the most common is using a compass direction, so I might walk a certain distance northeast, 10 kilometres in that direction, or I might walk southeast, and those displacements 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 3 metres right or 2 metres left.

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

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

Displacements of 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, and if we follow these movement instructions, so let's say I've got a starting point and I walk 5 metres to the right, and then I'll 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 got 3 metres left and 2 metres down.

Overall, my final displacement is just 2 metres right, despite the fact I've walked all those additional distances.

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

Right, let's check if you can work out our 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 1 metre north, 27 metres south, 7 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 a solution of 7 metres north.

If we draw a diagram of that here, we've got the starting point, we move the 10 metres, then the 4 metres, and then that second 10 metres is going backwards towards the start, really.

And then finally, the 3 metres.

So overall, we've moved 4 metres and 3 metres to the north, and that gives us 7 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 a 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 5 to plus 5, and I can draw a displacement as being plus 2 metres.

I can have another displacement further in that direction towards the right, that's plus 4 metres, or I could describe a negative displacement like this as minus 4 metres.

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

So we're gonna 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 6 metres.

If you look at 40 seconds, looking upwards, and then across to the displacement axis, that gives us a displacement of 6 metres.

Well done if you've got that.

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

And 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 of 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 4 metres to the right in 10 seconds.

And as you'd expect on the graph, I'd find our time equals 10 seconds, my displacement would be 4 metres.

And I've walked at a steady speed here, so I've got a nice straight line.

Follow the next instruction, walk 5 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 9 metres.

So there's my second phase of movement.

As you'd imagine, moving backwards, so I'm going to walk 6 metres to the left in 20 seconds, what will happen is, my displacement's now gone down to 3 metres, so my graph will go back downwards, my final displacement there, 3 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 the 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 4 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 I've got an increase in displacement of 2 metres, so that matches this one here.

And finally, I've walked 3 metres forwards in 10 seconds.

Well done if you've got that order.

So far, I've only looked at positive displacement, a 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're behind our starting point, basically.

So again, I'm gonna start at position zero, displacement zero, and I'm going to follow our set of movements.

The first of those, walk forwards 5 metres in 20 seconds, and that'll give me a line on the graph like this, you can see I'm at 5 metres at 20 seconds there.

Then I'll walk forward another 1 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 8 metres to the left in 30 seconds.

And as I'm only 6 metres in front of the starting point, I'm gonna end up at minus 2 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, we're looking at the four phases.

In the first part, in the first 10 seconds, I've moved forwards 3 metres, so that's the one that matches.

And then, I've moved backwards 5 metres in the next 10 seconds, so I'm ending up at minus 2 displacement.

Then I've walked forwards, again, for a 20-second period, so that must match this one here, I walked 6 metres forwards.

And finally, I've walked backwards in 20 seconds 4 metres.

Well done if you've got those four.

Okay, it's time for the first of the tasks now.

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

So we've got two pupils, and they're given these sets of movement instructions here.

We've got Pupil A and they've got those five sets of instructions there.

And we've got Pupil B, and they've got a different five instructions.

And what I'd like you to do is to sketch displacement-tine graphs for both those pupils, please.

And to make this a little bit easier, you can use the same axes for both pupils, just draw two separate lines.

So pause the video, sketch the graphs of that movement, and then restart, please.

Okay, welcome back, and hopefully your graphs look something like these.

I've got Pupil A here drawn in the purple dash lines, and you can see they go forward, they stop for a while, forward again, and then backwards in two different phases, so that's that line.

And the second one, Pupil B, they don't do anything for the first 10 seconds, then they move forwards and then backwards, then they stop, and then forwards again.

Well done if your graph looks anything like this.

Now it's time for the second part of the lesson, and in this part, we're going to be looking at plotting a displacement-time graph.

But instead of doing that manually by hand, we're going to try and use a motion sensor to help us with it.

So let's have a look at that.

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, it's very high frequency, and it sends out that sound pulse towards the object.

And when it reaches that 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 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 air, 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 seconds.

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

The distance was 40 centimetres and that's north 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 = 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 our 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 the 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 1/2 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 is 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 5 seconds there, at a 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 a gradual slowing down of the trolley due to friction, and you can see it's a little bit of a curved 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 the 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 naught centimetres away, and then it will gradually increase in 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 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 showed you.

But if you've got something like this, well done.

Okay, it's time for the third part of the lesson now.

So far we've looked at drawing displacement-time graphs and reading simple values off them.

What we're going to do now is to try and look at what the graphs tell us about the movement, about how fast an object's moving or how slow.

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, and 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 is increased every second.

And what that means is the 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 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 objects 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, and 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 backward, but it's got a shallow gradient, so its speed is slow compared to the other two.

Well done if you 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 first section, we've got a forwards constant velocity.

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

In the second phase of motion, and 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, in the positive direction.

In this section here, all of the values 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 lower 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 2 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, hopefully, you selected negative velocity, and 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.

As you've seen, the motion of an object can have different phases, there's different parts to it, where the velocity changes.

So I've got a graph here, and as you can see, there's four separate parts of the motion you can see, an upward slope, a flat bit, another upward slope, and a flat bit again.

And I've labelled those A, B, C, and D.

We can describe each of those sections separately, and we can analyse what's going on in each section separately as well.

So in part A, we're moving forwards, the displacement's increasing as time goes on, and in part B, we're stationary, there's no change in displacement during the time, 200 to 300 seconds, so we're stationary.

In part C, we're moving forwards, and in this instance, we're going forwards a bit more slowly than we were in part A, because the gradients less steep.

And in part D, again, we're stationary, so we can analyse each separate part of the motion just like that.

So let's see if you can interpret this graph correctly.

The graph shows the movement of an underground train moving between three stations.

I'd like you to identify, which type of motion is represented in each section? So we've got four words there, faster stationary, slower, and stationary, again, I'd like you to just match those to those letters there, please.

So pause the video, match up the words and letters, and then restart.

Okay, welcome back.

Well, in the first section, that's when it's moving it's faster, so we'll label that faster.

In Section B, that was stationary, In Section C, that's moving slower, and in Section D, where stationary again.

So that was the order there.

Well done if you've got that.

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, 1 metres per second, or is it C, 2 metres per second? So pause the video, make your selection.

and restart, please.

Okay, welcome back, and hopefully you selected 1 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, 1 metre 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 and 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, we've reached the final task of the lesson, and what I'd like you to do is to analyse the motion of a robot in a warehouse.

So I've got a graph here, and I'd like you to describe the motion of the robot during each of the marked stages there, A, B, C, and D.

Then I'd like you to try and explain why that graph is unrealistic.

And finally, I'd like you to try and sketch a more realistic version of that graph of motion.

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

Welcome back, and here's my descriptions of each of those phases.

So I've got Phase A, it's moving forwards, it's got a constant positive velocity, and its displacement's positive.

In Phase B, I'm moving backwards, constant negative velocity, and positive displacement, again.

C, it's stationary, so we've got a stationary robot there with a positive displacement, and D, the robot there is moving forwards, it's got a constant positive velocity, and positive displacement, well done if you got that.

For the second part of the task, I asked you to explain why it's unrealistic.

Well, it'd be unrealistic because there's very sudden changes in velocity, it instantly changes from forwards to backwards motion, and that's just not possible.

So it should have more gradual changes.

So when I asked you to sketch a more realistic version, what you could have done is rounded off those connection points, as I show here in the diagram.

So well done if you did that.

We've reached the end of the lesson now, and here's what you should have learned from it.

Displacement-time graphs show the movement of objects.

So we've got time on the X axis, and we've got displacement on the Y axis, and they show the position or the displacement of an object over time.

The steeper the gradient of the line on those graphs, the greater the velocity is.

And you can see a steep gradient there in the purple dots, that's fast, and a shallower gradient in the black line, and that's a lower forward velocity.

Positive and negative gradients show the different directions.

So you can see backwards and forwards motion on it.

If the line's straight, you've got constant velocities, and if you've got curves, it shows a change in velocity.

So that's the end of the lesson.

Well done for reaching it.

I'll see you on the next one.