Lesson video

Welcome.

It's really nice to see you for today's algebra lesson.

My name is Ms. Davies, and I'm gonna be helping you as you work your way through this video.

There's lots of exciting things that we're gonna talk about today.

I hope you're really, really looking forward to it.

Make sure you've got everything you need, and let's get started.

Welcome to this lesson on non-linear distance time graphs.

By the end of this lesson, you'll be able to interpret non-linear distance time graphs.

It's a few keywords that we're gonna use today.

If you want to recap these, feel free to pause the video and have a look over them now.

So we're gonna start by looking at acceleration on a distance time graph.

In the second part of this lesson, we're gonna look at different graphs for the same journey.

So here is a distance time graph of a race Sofia ran last year.

Sofia says, "I ran a hundred metres in 16 seconds, so I drew this graph." Sofia has drawn a straight line.

What does this mean about the race? What do you think? So it's always good to check what sort of graph we're looking at.

We're looking at a distance time graph.

A straight line on a distance time graph means running at a constant speed.

So we could calculate the speed, if we wished, by finding the gradient of the line.

The gradient will be the same anywhere along this line, and that's showing us then that we've got this constant speed 'cause we've got this constant gradient.

So just have a think.

Why might a straight line be an unrealistic way to model Sofia's race? Well, it's unlikely that she's gonna run at a consistent speed throughout.

If she's starting from stationary, then she'll need to accelerate to get up to a constant speed.

So it's likely she's gonna accelerate at the start.

Might also be possible that she decelerates at the end.

Or if she sprints at the end, she might accelerate before decelerating.

It's very unlikely that it's going to be a constant speed throughout the entire race.

So let's think about what this will look like on a distance time graph.

So if she's accelerating, that means her speed is increasing.

The distance she can cover per second is therefore increasing.

So we think about after one second, she's covered a certain distance.

After two seconds, she can cover more distance.

After three seconds, she's going faster, so she can cover more distance in that second.

And so on.

While she's accelerating, her distance time graph will be a curve.

If she then maintained a constant speed, for example of 10 metres per second, what would that look like on the graph? So constant speed is represented by a straight line.

If her speed is 10 metres per second, then the gradient will be 10.

If she then decelerated at a constant rate, what do you think that's gonna look like on the graph? Pause the video, have a think before I show you.

Right, Lucas says, "If she's decelerating, I think the graph will curve downwards.

But Sofia says, "I still ran a hundred metres in 16 seconds.

so the graph must finish at that point." What do you think to those two statements? Does that fit with what you thought deceleration would look like? Because this graph is showing total distance, the distance travelled cannot decrease.

However, as she is decelerating, her speed is decreasing.

The amount of distance she can cover each second is decreasing.

So you'll get a curve like this.

What do you think of this one? True or false, distance time graphs have to be made up of straight line segments.

Think about a justification for your answer.

Well done if you said false.

Acceleration is represented by a curve on a distance time graph.

So which of these shows an increase of speed on a distance time graph? What do you think? So increasing speed or positive acceleration looks like this.

The first graph is showing a stationary object.

Then we've got one travelling at a constant speed.

This is speed increasing over time.

And the final one will be speed decreasing over time.

So Lucas drops a football out of his window.

"My window is four metres high, and it took two seconds for the ball to reach the ground.

So I've drawn this sketch." Alex says, "The ball is falling downwards, so your graph should have a negative gradient." What do you think? Alex is incorrect.

A distance time graph does not tell us the direction of the journey, just the distance the object has travelled over time.

So the distance the ball has travelled is four metres, and that is shown correctly on our graph.

Do you think Lucas's sketch is an accurate depiction of the distance the ball travelled over time? Pause the video, what do you think? So it's a really short journey, so it'd be hard to plot this any more accurately.

However, gravity causes objects to accelerate.

So if he's dropping the object out of the window, it's starting from stationary, so with no speed.

And then as it's dropped it would pick up speed.

Lucas says, "Now I've learned that gravity causes objects to accelerate, I've tried again." And that's probably more realistic now, isn't it? It's starting from zero speed and it's gathering speed as it falls to the ground.

If we're being really realistic, the ball is likely to bounce once it reaches the ground.

And that means more distance will be travelled just in an upwards direction now.

How much distance will depend how high it bounces, and also depend how many times it bounces.

So it might look something like this as a sketch.

Accelerating till it hits the ground.

Decelerating after the bounce, until it reaches its highest points.

It's gonna slow down until it gets to the top of its bounce, then it'll accelerate towards the ground again.

And then it might bounce again.

And then we've got a horizontal line showing it finally stopping.

Okay, have a go at this one.

Give a reason why this might not be an accurate sketch for a sprinter completing a hundred metre race.

See if you can put this into words.

What do you think? So this graph does not show any deceleration before coming to a stop.

It's gonna be impossible for a sprinter to accelerate, run at a constant speed, and then come to an immediate stop.

Time for a practise.

Here is a distance time graph for a car travelling between two sets of traffic lights.

See if you can answer those questions.

Off you go.

For question two, you have a distance time graph for a ball rolled along a carpet.

I'd like you to have a go at filling in the labels for the different sections of the journey.

Think about what is happening, see if you can explain why.

Once you've given that one a go, come back for the next bit.

Sofia tries to run the same route every day.

She uses her smartwatch to track her distance over time.

You've got three scenarios and three graphs.

See if you can match them up.

Off you go.

And finally, match each description to the most appropriate graph.

See if you can explain your reasoning.

Give this one a go.

Let's have a look then.

So we need to find 10 seconds on the graph, and it looks to be approximately 25 metres.

Acceleration then is shown by that first curve, so that's taken up 20 seconds.

The constant speed goes from 20 seconds to 35, so that's 15 seconds.

After 35 seconds, the car started to decelerate.

And then how far the car travelled in 50 seconds.

So if you find 50 seconds on the X axis, that looks to be 325 metres.

Well done if you got that.

You've shown that you can interpret these distance time graphs.

So for question two.

So the start of our graph is showing the ball is accelerating after it is released.

If you said increasing in speed, that's the same thing.

Then we've got a straight line showing the ball travelling at a constant speed.

Then we've got another curve, where the ball is decelerating.

That's that bottom right label.

Now, I wonder if you thoughts about why it might decelerate.

Well, if you've rolled a ball along the ground, it will eventually decelerate to a stop, won't it? So you might have said something like friction between the ball and the carpet is slowing it down.

Or you might have said it reached its fastest speed and then it's gonna slow to a stop.

That final line segment then is showing the ball coming to a stop.

A horizontal line on a distance time graph shows the object is stationary.

For question three, A is gonna match with graph F.

The graph is showing acceleration at the beginning of the run.

So if she starts from stationary and then begins to run, she's gonna need to accelerate before she gets to a constant speed.

B is gonna match up with graph D.

Because she starts with a constant speed, that must mean she's already started tracking her run when she's at a constant speed.

C is gonna match with graph E.

And you can see that there's that moment of acceleration towards the end of the journey before she then decelerates to a stop.

And question four.

So a dropped rugby ball is graph B.

You can see how it's accelerating due to gravity, and then you've got that bounce at the end.

A dropped water balloon is gonna be D.

It's got the acceleration at the beginning, but then it comes to a sudden stop.

If you imagine it breaking on the ground so it doesn't travel any further, it's gonna stop very suddenly.

Different to the ball that will keep on travelling as it bounces.

The car between two junctions on a motorway is graph A, 'cause it's maintaining a constant speed.

A ball thrown upwards and then caught again.

This one was a bit interesting.

If you throw a ball upwards, the speed is gonna slow down.

Then when it gets to its highest point, it'll start accelerating again as it falls back to the ground.

That's quite a difficult concept to get your head round, so well done if you've got that one.

You might have found that one through process of elimination of the others.

And finally, E could be a car approaching a busy roundabout, because it would need to decelerate before it got to the roundabout.

So now we're gonna have a look at different graphs of the same journey.

So it's important to be aware of what type of graph you are interpreting or drawing, 'cause we'll look at distance time graphs, speed time graphs.

You might have looked at displacement time graphs before as well.

So this is a speed time graph.

What is it showing? Pause the video, see if you can put it into words.

So this is showing an object maintaining a constant speed of five metres per second for 10 seconds.

What would that look like on a distance time graph? So this is gonna be a diagonal line on a distance time graph.

Let's have a go at plotting it.

So we're gonna need a different set of axis.

We still need time to go up to 10 seconds, but we're gonna need distance on the Y axis now.

So after zero seconds, zero metres have been travelled.

So the graph must start at the origin.

And if we're going at five metres per second, after one second would've travelled five metres.

This is constant speed so we'll get a linear graph.

After 10 seconds, we will have travelled 50 metres.

The gradient of the distance time graph is going to be the speed.

So it should be five metres per second, 'cause that was the speed on the speed time graph.

So you can just check that you've got your gradient of five.

Right, will a horizontal line segment on a speed time graph always be a diagonal line on the distance time graph? Well, let's try another one.

Let's try an object moving at three metres per second.

On the speed time graph, that's gonna look like a horizontal line, but at three metres per second.

Let's look at it on the distance time graph.

So after one second, we've travelled three metres.

It's constant speed, so we'll have a linear graph.

After five seconds, we would've travelled 15 metres.

And so on.

So at the moment we've tried two cases where a horizontal line on a speed time graph is a diagonal line on a distance time graph.

Jun says, "I think there's a scenario which would be a horizontal line on the speed time graph, but not a diagonal line on the distance time graph." Do you agree with him? So there is one case.

The only constant speed that will not produce a diagonal line is a constant speed of zero.

So you can see on our speed time graph, a horizontal line on the X axis is a constant speed of zero.

So the line on the distance time graph will also be horizontal.

Where the horizontal line is will depend if this is at the beginning of the journey or if some distance has already been travelled.

So if it's at the beginning of the journey, then it'll be a horizontal line along the X axis.

If it's after some distance has been travelled, it may be somewhere else on the graph.

So this works the other way around.

We can draw a speed time graph from a distance time graph.

Well, here's the distance time graph showing constant speed.

We need to work out what the speed is.

Make sure you check your axis, 'cause the X axis has steps of one, but the Y axis has steps of two.

So the gradient of our line is two.

That means the speed is two metres per second.

On our speed time graph then, we can have a horizontal line at two metres per second.

We need to make sure this covers the right number of seconds.

So it was seven seconds on the distance time graph, so it should be seven seconds on the speed time graph.

Right, let's see if we've got that.

So which of these could be a speed time graph of the journey shown by this distance time graph? What do you think? Well done if you said C.

We need the gradient of the line to find the speed.

The speed is three metres per second, so we need a horizontal line at three on the speed time graph.

So here is another speed time graph.

This time the line is diagonal rather than horizontal.

What does a diagonal line mean on a speed time graph? Can you remember? Right, this means constant acceleration.

So every second the speed is increasing by one metre per second.

This means the object is accelerating at one metre per second squared.

This is shown in the gradient of the line.

So because the gradient of the line is one, the acceleration is one metre per second squared.

What does constant acceleration look like on a distance time graph? Roughly, what is this gonna look like? Right, this is gonna be a curve.

Acceleration is a curve on a distance time graph.

Now, it is mathematically possible to calculate the distance travelled at any point from a speed time graph.

This may be something you look at in the future.

And this would mean we could plot a distance time graph accurately.

However, for now we're just gonna appreciate the general shape of the graphs.

So our speed time graph is a diagonal line showing constant acceleration.

So our distance time graph is a curve showing acceleration.

I've plotted a few points on this curve just to show you what is happening.

And you might be able to see that our distance is increasing, but it's not increasing at a linear rate.

It's increasing by north 0.

5, then 1.

5, then 2.

5.

We can see from the table of values that the distance travelled per second is increasing.

This shows that the speed is increasing.

There's more distances travelled in the same amount of time.

We can also see that the acceleration is one metre per second squared.

If you have a look at how those values are increasing, the amount added on is increasing by one each time.

That is our acceleration rate.

This is not something you need to use at the moment, but if you look at this in more detail in the future, you'll explore these structures a little bit more.

So here's a sketch of a speed time graph.

What does this graph show? This is showing constant deceleration, 'cause the speed is decreasing over time.

Let's remind ourselves what deceleration looks like on a distance time graph.

So the distance is still increasing, but the rate at which the distance is increasing is decreasing over time.

So you get this curve.

At zero seconds, no distance has been moved.

And then we've got the fastest part of the journey at the beginning, so the most distance has travelled then.

Then as the object slows down, the distance travelled each second is decreasing.

When the object reaches a speed of zero, the curve will flatten into a horizontal line.

Read through these statements.

Can you fill in the blanks? Off you go.

Okay, a horizontal line on a distance time graph means the object is stationary.

Well done if you got that the right way round.

A horizontal line on a speed time graph means the object is travelling at a constant speed.

So recap on gradient then.

The gradient of a distance time graph represents the speed, whereas the gradient on a speed time graph represents the acceleration.

If you want to make some notes on that, just pause the video now.

Sofia says, "Is it possible to have a curve on a speed time graph?" We've seen curves on distance time graphs, that's acceleration and deceleration.

But what about on a speed time graph? Jun says, "I think so.

If a diagonal line represents constant acceleration, then a curve would mean the acceleration is not constant." And that's absolutely possible.

You can accelerate, but at a rate that's not constant.

It's quite hard to wrap your head around that.

So here the speed is increasing, but not at a constant rate.

The amount the speed is increasing by is increasing every second.

This shows that the rate of acceleration is actually increasing.

Right, what about if Sofia drew this curve? Pause the video.

See if you can wrap your head around what's going on here.

Well, the speed is still increasing, 'cause the Y values are increasing as time increases, but the amount the speed is increasing by is slowly decreasing.

At some point, this curve will become horizontal and the speed would be constant.

So you can see that the rate of acceleration is slowing down.

Alright, Jun says, "I think all these graphs are showing deceleration.

What do you think? Right, Jun is correct.

Let's look at them together.

The first one's showing constant deceleration.

Then we've got speed decreasing.

And the rate the speed is decreasing is actually increasing all the time.

So it starts off where it's only decreasing a little bit, and then it gets faster and faster and faster.

So the rate of deceleration is increasing.

And the last one, you've got the rate of deceleration decreasing.

So you're decelerating really fast to start with, and then that's slowing down.

Right, that is a really weird concept, as I say, to wrap your head around.

You might wanna spend some time just having a look at those graphs and playing around.

Quick check then.

It is possible to have curves on a distance time graph.

What do you think? Of course that's true.

Which is the correct justification for your answer? Right, curves on a distance time graph show acceleration.

How about this one? True or false, it is possible to have curves on a speed time graph.

Yes, it is, we've just looked at these.

Which is the correct justification for your answer? Right, curves on a speed time graph show acceleration which is not constant Time for a practise.

Here is a speed time graph for a journey.

Have a go at drawing a distance time graph of the same journey.

Don't forget to label your axes.

And now the other way around.

Here is a distance time graph.

Can you draw a speed time graph of the same journey? This time you need to label your axes, and you need to write on an appropriate scale.

Well done.

Now I'd like you to match the descriptions to the correct graph.

All these graphs are distance time graphs.

Make sure you're thinking about the right thing.

For question four, you've got some distance time graphs and some speed time graphs.

I'd like you to match the distance time graph to the speed time graph of the same journey.

Give this one a go.

And finally, can you match the descriptions to the correct graphs? These are all speed time graphs.

Take your time to think about what is happening with each.

When you think you've got this, come back and we'll look at the answers.

So for question one, you should have this graph.

It should have a gradient of two.

Notice that the Y axis is increasing in steps of two.

We need to label it with the distance in metres and the time in seconds.

For question two, we need to find the gradient first.

So the gradient is five, so the speed is five metres per second.

You've got choices on your scale this time.

The X axis should increase in twos, just like the distance time graph.

Make sure you've labelled it time in seconds.

The Y axis, you could have gone up in steps of one, in which case your line will be at the very top where five is.

Or you can do what I've done and go up in steps of two.

And then your horizontal line will be at five, halfway between four and six.

So just check that you are happy with that, and make sure your graph stops at 10 seconds and doesn't go any further.

For question three, A is gonna match up with H, the object is stationary.

B matches with G, the object is moving at a constant speed.

C matches with F, the object is decelerating.

And D matches with E, the object is accelerating.

Speed is increasing over time.

And for question four, A matches with D 'cause both are showing constant speed.

B matches with F, both are showing constant deceleration.

And C matches with E, both are showing constant acceleration, speed increasing over time.

And this was the one to wrap your head around.

So A, we've got constant acceleration, straight line on a speed time graph.

B, we've got the rate of acceleration decreasing over time.

C, we've got the rate of deceleration increasing over time.

For D, we've got constant deceleration.

For E, we're decelerating, but the rate of deceleration is decreasing over time.

And finally F, the rate of acceleration is increasing over time.

Well done if you've got some or all of those correct.

Fantastic, you've worked really hard today.

Thank you very much for joining me.

If you want to pause the video and just check through all the things we've covered, feel free to do that now.

Otherwise, I really look forward to you joining us for another lesson in the future.