Lesson video

Hello there, my name's Mr. Forbes and welcome to this lesson from the Measuring and Calculating Motion Unit.

In the lesson, we're going to be analysing different displacement-time graphs and using them to calculate speed and velocity.

By the end of this lesson, you're going to be able to look at displacement-time graphs and take values from those graphs in order to calculate speed or velocity of a range of different objects.

Here are the three keywords and phrases that you'll need to understand to get the most from this lesson.

The first is displacement-time graph, and a displacement-time graph shows you the displacement of an object over a period of time.

We're gonna be looking at a wide range of those in the lesson.

The second is gradient, and that's the steepness of a line on those graphs.

And third is instantaneous velocity, which is the velocity at a particular moment in time.

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

This lesson's in three parts, and in the first part, we're going to be reading information from displacement-time graphs and calculating the total distance travelled by different objects.

In the second part of the lesson, we're gonna be reading similar information, but this time, we're gonna be calculating velocity based on that information.

And in the third part of the lesson, we're going to be calculating average and instantaneous speed by looking at the gradients of lines on the graph.

So when you're ready, let's start with calculating distance travelled.

Displacement-time graphs can show positive and negative displacement.

So you can see I've got a graph here, showing the movement of an object and it's got both positive displacement in the first section there, negative displacement in the middle section, and then positive displacement again in the end section.

Those positive and negative numbers show opposite directions, so opposite displacement, and they can be things like north and south or east and west, or even left and right.

As long as those two words mean opposite things, then I'm okay.

So here's a nice easy question to start with.

This graph shows positive and negative displacement.

A positive displacement on a graph represents up, I'd just like you to decide what does the negative displacement represent? So is it sideways, down or further up? Pause the video, make your selection and restart.

And as I said, that was a nice easy one.

If the positive displacement is up, then obviously, the negative displacement is going to be down.

So well done, if you got that.

A displacement graph doesn't just show displacement.

It can also be used to work out total distance travelled.

So we'll go through an example of that here.

So I'm gonna start with a displacement-time graph like this.

It's empty at first, and a person at position zero with no displacement, and I've marked just the forwards direction on this number line.

So if I follow a set of instructions about movement, if I go forwards two metres in 10 seconds, you can see a line like that on a graph.

And that's actually showing constant speed during that motion there.

And in the second section, well, stationary for 10 seconds and you can see the little flat part of the line there means the displacement isn't changing over those 10 seconds.

So stationary for 10 seconds looks like that on a graph.

And a move again, this time moving forward's five metres in 20 seconds.

So my displacement's gone up now.

It's got a total displacement of seven metres, and in the final bit of movement, forward's two metres in 20 seconds.

So you can see there, the final displacement is nine metres on the graph after 60 seconds.

The distance travelled is also the sum of those distances.

So I've got two metres, five metres, and three metres.

So the total distance travelled is nine metres as well.

So the displacement and the distance travelled is the same, because I've not changed direction at all during that motion.

A quick check now, I'd like you to work at the total distance travelled by a car according to the graph here, is it, a, 200 metres, b, 400 metres, or c, 600 metres? Pause the video, make your selection, and restart, please.

Okay, welcome back.

Well, in this journey it's 600 metres.

The car travelled 400 metres and stopped for a while, and another 200 metres.

So the total distance travelled is 600 metres there.

Well done, if you've got that.

Now in the examples we've seen so far, the final distance travelled is not equal to the total displacement, but that's only because there's no changes in direction.

Let's have a look what happens when there is a change of direction in the motion.

So again, I'm gonna be starting at zero and I've got movements towards the right here as positive and I can show different movements on the graph.

So first of all, let's move right, eight metres.

So there's eight metres movement in 20 seconds, and you can see a constant speed on the graph there, a nice straight line.

And to move again, sorry, I stay stationary for 10 seconds, so there's no change in displacement there.

And then the third part of the movement is, I'm gonna move backwards.

I'm gonna move left five metres in 15 seconds.

And you can see, in that instance, the displacement's gone down.

The distance of travelled though has increased, so they're no longer the same value.

The third bit, stationary for five seconds, and then another movement towards the right there, three metres in 20 seconds.

You can see the final displacement is six metres according to the graph, but that's not the same as the distance I travelled.

I travelled eight metres at first, then another five metres, and then another three metres, and gives me a total distance of travelled of 16 metres.

But as I've said, the final displacement is just six metres to the right.

So when there's a change in direction, the final displacement and the distance travelled are not the same.

Okay, I've got a graph of motion for Sam here.

I'd like you to work at the total distance travelled by Sam according to the graph, bearing in mind, it changes the direction.

So I've got, is it, a, two metres, b, six metres, c, 10 metres, or d, 12 metres? Pause the video, make your selection, and then restart, please.

Okay, welcome back.

And the answer to that was 10 metres.

Although, Sam ended up with a final displacement of just two metres, they moved six metres forward at first according to the graph, and then another four metres backwards.

You can see the displacement is decreased by four metres there, but the total distance that Sam moved is six metres, plus the four metres, and that's 10 metres.

Well done, if you've got that.

As you've seen in some earlier examples, displacements can be negative as well as positive.

So we've got a graph here showing a negative displacement as well as positive displacement, but we can still work out the total distance travelled by analysing the movement.

So I've got a movement here.

If I look at the first section of the graph, I've moved, well, I've decided to call the positive displacement north here, so north 600 metres in 100 seconds.

And then stayed still for a bit, stationary for a 100 seconds.

Then I moved south.

So I'm getting back to zero displacement here.

Moved south 600 metres in 100 seconds.

Then further south, and that's what gives me negative displacement.

I'm south of my starting point now, so south 400 metres in 100 seconds, then I stopped again for a 100 seconds and then moved north 400 metres in a 100 seconds.

So the total distance I've travelled there is the sum of all those separate distances in each phase of movement.

So the distance travelled is 2,000 metres there.

But my final displacement is zero, so definitely not the same as the distance travelled.

So let's see if you can find another example of total distance travelled.

I've got bicycles riding here, and I'd like you to find the total distance travelled by it over those 180 seconds.

Is it 100 metres, 200 metres, 300 metres or 500 metres? Pause the video, make a selection and restart, please.

Okay, welcome back.

Hopefully, you chose 500 metres.

As you can see, there are three separate phases of movement here where the distance, or, sorry, the displacement's changing, I move forwards, 200 metres, backwards, 100 metres, then forwards again, 200 metres.

That gives me a total distance travelled of 500 metres.

The sum of those values.

Well done, if you've got that.

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

I'd like you to have a look at the information shown in the box there and draw me a displacement-time graph based upon those movement instructions.

I'd like you to also state the final displacement after the journey, and you'll see that from the graph.

The third of all, calculate the total distance travelled during that journey, please.

So pause the video, draw that graph and answer the other two questions and then restart, please.

Welcome back, and here's the graph you should have drawn.

As you can see, there's quite a few different phases of movement there.

I've got movement forward, stopped, backwards, forwards, stopped, and then backwards a little bit again.

So well done, if you've drawn a graph that looks that shape.

The final displacement if you drew the graph accurately should be 700 metres and you can see that on my graph as well.

And finally, the total distance travelled could add together all of the movements of each phase and that gives me a total sum of 1,300 metres.

Well done, if you've got that.

Now it's time for the second part of the lesson, and in it, we're going to be looking at how to calculate velocity by reading information from displacement-time graphs.

Let's have a look at how we calculate velocity.

The average velocity can be found using this equation, which you may have seen before.

Average velocity is change in displacement divided by time.

So we can write that as a set of symbols as v equals s, divided by t.

And the change in displacement is s, measured in metres, velocity is measured in metres per second.

That's symbol v, and time, t, is measured in seconds.

From that, you can see that both displacement and time can be read from a displacement-time graph, and that's what's going to allow us to calculate average velocity.

Let's start by finding out how you can find the average velocity for a complete journey.

As you saw in the equation, we need to have a displacement and a time.

So all we really need to do is to find the displacement from the graph.

So looking at this graph for the complete journey, after 500 seconds, the displacement is 600 metres.

So we get s equals 600 metres.

I can also look at the end of the graph and find the total time for the journey, and that was 500 seconds.

So a, t, is 500 seconds.

I can then calculate the average velocity for the complete journey by writing out the equation, v equal s, divided by t.

Substituting those two values I've taken from the graph and then calculating the final answer, which gives me 1.

2 metres per second.

Okay, let's check if you can find the average velocity for a journey.

I've got a journey here that took 40 seconds.

I'd like you to find the average velocity for it, please.

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

Okay, welcome back.

Hopefully, you selected naught 0.

5 metres per second.

And you should see that the displacement there was two metres at the end of the journey and the time taken was 40 seconds.

So if we substitute those into the equation and we get an output of naught five metres per second.

Well done, if you found that.

You can also find the average velocity for part of a journey instead of the complete journey just by looking at the changes in the values for the displacement and the change in the value for the time.

So we're gonna do that here with an example.

We're gonna look at this part of the journey from 300 seconds to 500 seconds.

So we start by finding the change in the displacement during that time.

And as you can see, the displacement was 400 metres and then went up to 600 metres.

That gives us a change in the displacement of 200 metres and it might be on the graph.

You can find a change in time, and as I mentioned, we've got from 500 seconds in 300 seconds.

That gives us a change in time of 200 seconds there.

And then we substitute those values into the equation just like we do for the other calculations.

So write out the equation, put the values in, and it gives us an average velocity for that part of the journey of one metre per second.

Okay, let's see if you can find an average velocity for a section of movement.

I've got a graph here and I've highlighted the section between a 120 seconds and a 150 seconds.

I'd like you to find the average velocity for just that section of movement, please.

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

Okay, let's have a look at the solution to that and it's 6.

7 metres per second, and we can find that by taking readings from the graph.

There's a change in time of 30 seconds there and a change in displacement of 200 metres.

And so we can substitute those two values into the equation, giving a value of 6.

7 metres per second.

Well done, if you've got that.

Sometimes displacement is decreasing and that gives us negative velocities and we have to be very careful with doing calculations involving negative velocities beause we've got negative numbers.

So let's have a look at an example of that.

We've got a section of a graph here, and you can see clearly that the velocity is negative, because the displacement is decreasing between 200 and 300 seconds there.

So we can find the change in displacement, and this time we've gotta be very careful because it's at negative value.

We've gone down by 600 metres, so we'd represent that as minus 600 metres.

And then we can look at the change in time, and the change in time is a 100 seconds, times always positive there.

So then we can calculate the velocity by substituting the values into the equation as we've done before.

But being very careful, because we've got to use that value of minus 600 metres.

So we've got minus 600 metres, divided by a 100 seconds, that gives us minus six metres per second.

And the minus is telling us, we've got negative velocity with moving back to the origin point.

So I'd like you to find the average velocity for the highlighted section of the graph here between 20 seconds and 40 seconds.

So pause the video, find the velocity for that section, the average velocity, and then restart, please.

Okay, welcome back, and you should have chosen minus not 0.

2 metres per second.

You can see there's a change in time of 20 seconds and a change in displacement of minus four metres there.

We substitute those into the equation and that gives us minus 9.

2 metres per second.

Well done, if you've got that.

And now it's time for the second task of the lesson and I've got a graph showing the motion of a robotic arm here, and you can see it's a displacement-time graph.

What I'd like you to do is to calculate the average velocity for the complete 60 seconds of action there.

Then I'd like you to calculate the average velocity between 20 and 30 seconds.

And finally, the average velocity between 40 and 50 seconds.

So pause the video, work out those average velocities and restart, please.

Welcome back, and let's have a look at the complete journey first.

So for the complete journey, the total displacement is six metres and 60 seconds.

So we can substitute those into the equation.

We'll get a velocity of 0.

1 metres per second.

For the next section of the graph between 20 seconds and 30 seconds, just this section here, we can see there's a change in time of 10 seconds and a change in displacement of three metres.

So we get a velocity of naught 0.

3 metres per second, and for this section of the graph here, we get a change in time of 10 seconds, and a change in displacement of minus two metres.

So that gives us a velocity of minus naught 0.

2 metres per second.

Well done, if you've got those.

And now it's time for the final part of the lesson, and in it, we're going to be looking at how we can use the gradient of a displacement-time graph to find the instantaneous speed.

The instantaneous velocity of something is how fast it's going at a particular moment in time and in what direction.

And we can find that from the gradient at a specific time on a displacement-time graph.

So I've got a graph here and there's several different sections to it.

And in each of the sections, the object's moving at constant velocity.

So it's giving me a straight line there.

And the instantaneous velocity is going to be the same as the average velocity for that section.

So if I can find the average velocity, that will give me the instantaneous velocity as well.

So for example, we can look at this first section here and we can see that there's a 100 seconds of time passed and the objects move 600 metres, and I can get a velocity of six metres per second there.

So the velocity is constant at six metres per second.

So at any time between north and a 100 seconds, the velocity is going to be six metres per second.

In this section here, again I can find the velocity is a constant minus four metres per second.

So we've got the velocity there.

The instantaneous speed is going to be the same as the instantaneous velocity, but without the direction.

So I've got an instantaneous speeds there of six metres per second and four metres per second.

So I'd like you to find the instantaneous speed at 30 seconds on this graph, please.

So find the time, 30 seconds, and work out what the instantaneous speed would be there.

Pause the video and then restart when you're done.

Okay, welcome back.

And you should have found the instantaneous speed there is naught 0.

3 metres per second.

If we look at this section of the graph, where the 30 seconds is, in the middle of that section there, then we can find that the time in that section is 20 seconds.

The change in displacement is six metres.

So we've got 20 seconds and six metres there, so we can find the instantaneous speed.

Remember, that doesn't have direction, which is why I'm muting six metres and I get an instantaneous speed of naught 0.

3 metres per second.

Well done, if you've got that.

In the graphs we've looked at so far, we've just looked at constant velocities or constant speed, and that gives us straight line sections on a graph.

But on many graphs, there are curve lines showing that the velocity's actually changing.

So in this graph, I've got a series of curves, and in this first section, you can see the gradient is increasing in those first 200 seconds, the object is speeding up, its speed is increasing.

And in this section of the graph and the object is actually slowing down, the gradient is decreasing.

So the speed and the velocity is decreasing there.

Objects can speed up as they're moving towards you as well.

And in this graph, I've got an object change in its speed as it's moving towards you.

So in this section of the graph, the gradient of the graph is becoming more steep, and so the object speed is increasing, the velocity is becoming more negative.

We've got negative velocity here, because the object is moving towards you.

In this section of the graph, the object is slowing down, the gradient is becoming shallower, and the speed is decreasing there.

And as you can see towards the end, it's becoming almost flat.

The velocity is approaching zero in there.

Okay, let's see if you understand what a curve on a displacement-time graph represents.

So I've got a graph here of a roller skater, and what I'd like you to do is to identify in which of those four parts, the speed of the roller skater is increasing, not the velocity, the speed of the roller skater.

So pause the video, select whichever options you think are correct, and then restart, please.

And welcome back, you should have selected, a.

In that section, you can see the graph is becoming more steep, so they're speeding up.

The velocity's actually increasing there.

And also, c, because at that point, the velocity is in the opposite direction.

But again, we've got the graph becoming steeper in a downwards direction, so we've got an increase in speed there as well.

So well done, if you selected those two.

And now it's time for the final task of the lesson.

And what I'd like you to do is to have a look at this graph here, which shows the vertical movement of a drone.

And I'd like you to calculate the average speed for the complete journey, the full 500 seconds of movement there.

Then I'd like you to calculate the instantaneous speeds at these points, t equals 50 seconds, or time equals 50 seconds, t equals 250 seconds, and t equals 450 seconds.

So pause the video, work out the answers to those, and then restart, please.

Okay, welcome back, and here's the solution to the first part of that, the average speed for the complete journey.

So I'm looking for average speed here, so I need to work out the total distance travelled.

As we learned earlier in the lesson, we add up all or small distances, and that gives us a total distance travelled of 1,300 metres, and we divide that by the time, which was 500 seconds.

And that gives us an average speed of 1.

4 metres per second.

Well done, if you've got that.

And for question two, we have the solutions here.

And as you can see for each section, I found the time and the change in displacement and have calculated our speed for that section.

So the average speed for each of those sections, because that's going to be equal to the instantaneous speed as well.

So well done, if you've got these three.

Now we've reached the end of the lesson, and here's a quick summary of all the information we've covered.

So displacement-time graphs can show positive and negative displacement.

The total distance travelled in any journey is equal to all of the changes in displacement added together.

Total distance is usually different from the final displacement, because the direction of travel can change.

As you can see in this graph, the direction of travel is changing several times.

I've got a distance travelled of 23 metres for a final displacement of only three metres.

Instantaneous velocity is equal to the gradient of a displacement-time graph at a point, and instantaneous speed is equal to the size of that gradient.

So we've calculated several gradients during the lesson.

Well done for reaching the end of the lesson.

I'll see you in the next one.