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 the speed or the 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 displacements.

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 0 with no displacement.

And I've marked just the forwards direction on this number line.

So if I follow a set of instructions for that movement, if I go forwards 2 metres in 10 seconds, you can see I get a line like that on the 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 I move again, this time moving forwards 5 metres in 20 seconds.

So my displacement's gone up now.

It's got a total displacement of 7 metres.

And in the final bit of movement, forwards 2 metres in 20 seconds.

So you can see there, the final displacement is 9 metres on the graph after 60 seconds.

The distance travelled is also the sum of those distances.

So I've gone 2 metres, 5 metres, and 3 metres, so the total distance travelled is 9 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 out 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, then stopped for a while, and another 200 metres.

So the total distance travelled is 600 metres there.

Well done if you got that.

Now, in the examples we've seen so far, the final distance travelled is equal to the total displacement, but that's only because there's no changes in direction.

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

So again, I'm gonna be starting at 0, and I've got movement towards the right here as positive.

And I can show different movements on the graph.

So first of all, let's move right 8 metres.

So there's 8 metres movement in 20 seconds, and you can see a constant speed on the graph there, a nice straight line.

And I 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 5 metres in 15 seconds.

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

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

The third bit, stationary for 5 seconds, and then another movement towards the right there, 3 metres in 20 seconds.

You can see the final displacement is 6 metres according to the graph, but that's not the same as the distance I travelled.

I travelled 8 metres at first, then another 5 metres, and then another 3 metres.

That gives me a total distance I've travelled of 16 metres.

But, as I've said, the final displacement is just 6 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 out the total distance travelled by Sam according to the graph, bearing in mind changes of direction.

So I've got, is it A, 2 metres, B, 6 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 2 metres, they moved 6 metres forward at first according to the graph, and then another 4 metres backwards.

You can see the displacement has decreased by 4 metres there.

But the total distance that Sam moved is 6 metres plus the 4 metres, and that's 10 metres.

Well done if you 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 displacements.

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 100 seconds.

Then I moved south, so I'm getting back to 0 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 100 seconds and then moved north 400 metres in 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 0, 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 a bicycle 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 your 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, 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 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.

And 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 forwards, 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, I've gotta add together all of the movements of each phase, and that gives me a total sum of 1,300 metres.

Well done if you 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 of an object can be found using information about its change in displacement and time.

And using the equation here, which you may have seen before, or variations of it, average velocity is change in displacement divided by time.

If you write that in symbols, it's usually written as v equals x divided by t.

And change in displacement is represented by x, which is measured in metres.

Velocity is v, measured in metres per second.

And time, t, is measured in seconds.

Now, if you look carefully at the equation, you should notice that displacement and time, the two values that we need to calculate velocity, can both be found on displacement-time graphs, and that's exactly what we're going to do to find velocity.

So let's find the average velocity for a complete journey first.

So I've got a graph here showing a complete journey of an object, and we need to read two values from it.

We need to read displacement and time.

So starting with displacement, you can just look at the end of the graph.

The final point, looking across, that gives us a final displacement of 600 metres, so x is 600 metres.

Similarly, we need to find the time it took for that journey.

And looking down from the final point in the graph, that's 500 seconds, so t is 500 seconds.

And then we can calculate the average velocity quite simply.

Write out the equation using symbols to save a bit of space, v equals x divided by t.

Substitute in those two values, and then complete the calculation, 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 0.

05 metres per second.

If you look at the graph and find the endpoint, you'll find the final displacement is 2 metres, looking across there.

And looking down, you find the final time is 40 seconds.

We substitute those into the equation, v equals x divided by t, and that gives us a final answer of 0.

05 metres per second.

Well done if you got that.

We can also find the average velocity for part of a journey by looking at the change in displacement and the change in time.

So let's try that here.

We can look at just this section of the journey between 300 seconds and 500 seconds, and we find the change in displacement first.

So looking at that section, you can see the displacement's gone up from 400 metres to 600 metres, and we can calculate that's a 200 metre change.

The next thing we do is find the change in time for the section of the graph.

And you can see again, we've got a change in time of 200 seconds, up from 300 seconds to 500 seconds.

And then we substitute those two values, those two readings from the graph, into the equation v equals x divided by t.

Put them in, and we get an average velocity for just that section of the graph of 1 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 a section between 120 seconds and 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, welcome back.

Let's have a look at the answer to that, and it was 6.

7 metres per second.

And the way we found that answer was looking at the graph, we can see there's a change in time of 30 seconds.

It's gone up from 120 to 150 seconds.

And there's a change in displacement of 200 metres, up from 100 metres to 300 metres.

We substitute those two changes into the equation, and we get a velocity of 6.

7 metres per second.

Well done if you got that.

Sometimes we've got situations where the displacement is decreasing, and that's gonna give us a negative velocity.

We're moving backwards towards the origin point.

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

We've got a section here of 100 seconds between 200 and 300, and you can see that the displacement has gone down during that section.

It was 600 metres, and it decreases down to 0 metres back to the start.

So, we can find the change in displacement by looking carefully and seeing it's a negative value, it's -600 metres, and we've gotta represent that by using the minus sign.

We then can find the change in time.

The change in time here is 100 seconds.

And then we can just put those two values into the equation, remembering to include that minus sign for the displacement or the change in displacement.

So putting those values in, that gives us a value of -6 metres per second.

So we have got a negative velocity here.

The displacement is decreasing during that time.

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 what you should have selected was -0.

2 metres per second.

If you look carefully at the graph, you can see there's a change in time of 20 seconds and a change in displacement of -4 metres.

So substituting those into the equation gives us a velocity of -0.

2 metres per second.

Well done if you 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.

Okay, welcome back.

Let's have a look at the complete journey first.

Well, we can see that there's a total displacement at the end of 6 metres, a total time of 60 seconds, and that gives us this calculation giving a velocity of 0.

1 metres per second.

Between 20 and 30 seconds, this section highlighted here, we've got a change in time of 10 seconds and a change in displacement of 3 metres.

So we can put those into the equation and get 0.

3 metres per second.

And in the final section that I asked about, between 40 and 50 seconds, a change in time of 10 seconds and a change in displacement of -2 metres, giving a velocity of -0.

2 metres per second.

Well done if you 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 we've got a graph here, and there's several different sections to it.

And in each of the sections, the object's moving at a 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 100 seconds of time passed, and the object's moved 600 metres, and I can get a velocity of 6 metres per second there.

So the velocity is constant at 6 metres per second.

So at any time between 0 and 100 seconds, the velocity's going to be 6 metres per second.

In this section here, again, I can find the velocity is a constant -4 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 instantaneous speeds there of 6 metres per second and 4 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.

Welcome back.

The answer was 0.

3 metres per second.

To find the instantaneous speed, we look at this section of the graph where the 30 seconds is.

So 30 seconds is right in that section of that line.

And we can find that there's a change in time of 20 seconds and a change in displacement of 6 metres.

And we can use those to find an instantaneous speed of 0.

3 metres per second.

And it was okay to use 6 metres instead of -6 metres because I just wanted the speed there, so the direction wasn't important.

Well done if you got that.

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

But on many graphs, there are curved 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 changing 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's 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's 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 0 at the end 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's 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's 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, so 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.

Welcome back, and here's the solution to the first question.

The total distance travelled here was 1,300 metres.

And because I asked for the average speed, I need the total distance travelled in this calculation.

So I've got that, 1,300 metres, and then I've divided it by the total time, and that gives me an average speed of 1.

4 metres per second.

Well done if you got that.

And here are the solutions to part two.

For each of those, I found the change in time.

In each case, it was 100 seconds.

And I found the change in displacement for each of those, which is equal to the change in distance as well.

And that's given me a set of answers like this.

Well done if you got these.

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 but a final displacement of only 3 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.