Lesson video

Hello, thank you for joining us for today's algebra lesson.

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

There's lots of ideas today that you're gonna want to think about and give yourself plenty of time to think for yourself before I give you some hints and help you out.

With that in mind, make sure you pause the video to give yourself that time to think.

You might also want to make sure you've got a pen and a piece of paper, and maybe a pencil and a ruler for any of the graph work that you're going to need to do.

Let's get started then.

Welcome to today's lesson on speed-time graphs.

By the end of this lesson, you'll be able to calculate time intervals and acceleration of sections of a speed-time graph.

So speed is the rate at which something is moving.

It is measured as the distance travelled per unit of time.

There's two keywords today that we're gonna be using a lot, so you might want to make some notes on these.

Acceleration is the rate of change of speed with respect to time.

Deceleration is the rate at which the speed of an object is decreasing, and, again, that is with respect to time.

So we're gonna start by looking at drawing and interpreting speed-time graphs.

Speed-time graphs are a type of graph which modelled the speed and object travels with respect to time.

So because we're modelling the speed of the object as time progresses, we plot time on the x-axis.

Now that's exactly the same as distance-time or displacement-time graphs, or maybe other types of time graphs you've looked at before.

Speed is then plotted on the y-axis.

there are lots of different units for speed.

Here I've used metres per second, and this is quite a common one when we are using a speed-time graph.

However, we could have miles per hour, or kilometres per hour, or any other measure of speed.

Now, metres per second can be written in a few different ways.

They're shown on your screen now.

You might wanna pause the video and just familiarise yourself with those three ways of writing speed.

So here is a speed-time graph for a 30-second period of a car travelling along a road.

Pauses the video.

What is the speed of the car after five seconds? what does the graph tell you? So we need to look at five seconds on the x-axis, and we can see the speed of the car is 13 metres per second.

Don't forget your units.

Okay, what is the speed of the car after 20 seconds? That's right, it's 13 metres per second again.

Now what is the initial speed of the car? Right, of course, it's 30 metres per second.

It's the speed when time is zero.

So think about what you've just looked at.

What do you think a horizontal line on a speed-time graph represents? Right, a horizontal line on a speed-time graph represents constant speed, and in this case it's a speed of 30 metres per second.

But if we drew that horizontal line somewhere else, that would represent a different constant speed.

Right, so here's a speed-time graph for a race that Sofia has run.

The graph starts at the origin.

That's because she's stationary when the race starts.

So her speed is zero when time is zero.

Right, how fast is she running after three seconds? Use the graph.

What do you notice? Right, well after three seconds, she's going at four metres per second.

A couple of different ways to write that I've written on the screen.

Okay, so after three seconds, she's running at four metres per second.

How fast is she running after six seconds? Right, eight metres per second.

So this diagonal line is showing that Sofia's speed is increasing.

She started stationary.

Then she was running at four metres per second.

Then eight metres per second.

So her speed is increasing.

We call this acceleration because this is a straight line.

The rate of acceleration is constant so her speed is increasing, but at a constant rate.

Right, so here is a speed-time graph for a car travelling between two sets of traffic lights.

What is happening to the car between zero and 20 seconds? What's that telling you? Right, we've got a straight diagonal line.

It's got positive gradient.

It's telling you that the car is accelerating, it's gone from zero metres per second to 10 metres per second.

What happened then after 20 seconds? Right, so that Line means we're maintaining a constant speed.

So we've got a constant speed of 10 metres per second for 15 seconds.

What do you think that final Line segment represents then? So on a speed-time graph, line segments with negative gradient represent the object slowing down or decelerating.

So you can see that the speed is decreasing over time.

So in this case, the car has gone from travelling at 10 metres per second to zero metres per second, and that's taken at 15 seconds.

Sofia uses an application on her phone to track her speed when skiing, and she's showing it on this speed-time graph.

Sofia says, "That must be when I stopped at the top to take photos." Do you agree? Well done if you said no.

The horizontal line is showing a constant speed of five metres per second, not Sofia being stationary.

You've gotta make sure you're not muddling your distance-time graphs and your speed-time graphs.

Sofia says, "Okay, that would be when I sat on the ski lift then." That sounds more sensible.

So let's have a look at the rest of this graph.

So here we've got rapid acceleration.

So that's probably when she got on the lift.

Then she was moving at a constant speed, and then we've got very quick deceleration.

That horizontal line on the x-axis means a constant speed of zero, so that is when she's stationary.

And then we're accelerating again to 15 metres per second.

And then decelerating.

You can see that we're decelerating a lot quicker 'cause that line's a lot steeper.

That last section you can see that she gets back down to zero metres per second, so she's stationary again.

And then she accelerates not to the same speed as before.

She's only gone to 30 metres per second that time.

Then she's decelerating again, and then she's come to a final stop.

So now we know what speed-time graphs look like, we can draw them from key information.

So we've got sprinter accelerates from zero metres per second to 10 metres per second in six seconds.

Right, then they maintained that constant speed for further six seconds.

But Laura says, "After they've crossed the finish line they will stop, so I've added this to the graph." What could be the problem with Laura's new line segment? Pause the video.

What do you think? So this would suggest that the runner went from 10 metres per second to zero metres per second in no time.

Now this is impossible.

It's impossible for a runner to stop instantaneously.

All right, time for check then.

Jacob starts his cycle ride by accelerating to 4.

5 metres per second.

He maintains his speed as he cycles uphill.

He then accelerates down the other side before coming to a stop.

Which of those sketches could represent Jacob's journey? What do you think? Well, if you said A, the diagonal line with positive gradient is showing our acceleration.

It's maintaining that speed, so we have a horizontal line.

He accelerates again, so we need another diagonal line with positive gradient.

And then in order to come to a stop we need to decelerate.

He won't be able to instantaneously come to a stop, but you might be able to come to a stop quite quickly, which is why we've got a very steep line at the end.

Right, time for practise then.

So this graph shows the speed of a cart between two points on the motorway.

Have a look at the graph and see if you can answer those questions.

Once you're happy with your answers, come back for the next one.

Well done.

This time it's your go to draw your own.

So Sofia is practising her skiing.

I would like you to draw a speed-time graph for her journey.

It's up to you to come up with your own scales for the axes.

So read through the question first and decide what a suitable scale would be.

Come back when you're happy with your answer.

And finally, this graph Shows the speed of a race car in its first 20 seconds of a race.

First of all, I'd like to know what was the race car's fastest speed, and then see if you can describe the first 20 seconds of this race.

This try to reference each part of the journey.

Talk about whether we're accelerating, decelerating, constant speed, whether we come to a stop at any point.

See if you can describe everything that is happening in that time.

Give it a go and then come back to the answers.

Let's have a look then.

So the initial speed of the car, so that's the speed when time is zero, and that's 10 metres per second.

Ake sure you've got your units.

How long was the car accelerating for? Well we're looking at that first section of line, and that's two seconds.

From zero to two seconds, the car was accelerating.

The horizontal line segment represents the car travelling at a constant speed, and it was a speed of 22 metres per second if you added that information.

And then what happened after five seconds or the car started to decelerate? Well done if you used that key word right? Right, And then Sofia's graph.

Now I looked through and saw the highest speed was 12 metres per second, so steps of two make the most sense on the y-axis.

The total time was 55 seconds.

So steps of 10 makes the most sense on the x-axis.

If you use the same scales as me, your graph should look like mine.

Pause the video and look at each of those stages.

And finally, so for the race car's fastest speed, we're looking at the largest y value, so that was 40 metres per second.

And then some of the things you might have thought about in your description.

So the car accelerated at a constant rate for four seconds, reaching a speed of 32 metres per second.

See if you mentioned any of those bits in your description.

Then it decelerated for two seconds, and then travelled a constant speed of 24 metres per second for further four seconds.

Again, see if you've included that in your description.

The car then accelerated for four seconds until it reached a top speed of 40 metres per second, and then that top speed was maintained for a further six seconds.

Pause the video if you need to check your answer against mine.

Fantastic.

So now we're gonna have a look at Calculating rates of acceleration.

So here's a distance-time graph.

On a distance-time graph, a diagonal line segment represents constant speed.

So we can calculate the speed of the different sections of the journey by calculating the gradients.

Now you may have done this before on a distance-time graph.

The reason that the gradient is speed is because speed is the change in distance over time.

So here we can see that, in one second, the distance has increased by 10.

so this speed would be 10 metres per second.

Now let's return to our speed-time graphs.

So on a speed-time graph, the speed is easy to see.

We can just read it off the y-axis.

So this time we can see the speed by reading the Y coordinates.

So what does the fact that we've got a diagonal line represent on a speed-time graph? We've just looked at this.

This is acceleration at a constant rate.

Acceleration is the change in speed over time.

Let's have a think about the units of acceleration then.

So we can measure acceleration in metres per second per second because it's the change in speed, which is measured in metres per second, per second.

So if the units of speed are metres per second, how do you think we would write the units for acceleration? What do you think? So let's think about this metres per second per second.

So that means metres per second divided by seconds.

It's the speed divided by the time.

Well, let's write that with a division, rather than with a fraction.

So metres per second divided by seconds.

And then we know that dividing by a value is the same as multiplying by its reciprocal.

And simplifying that, we've got M over S squared.

So metres per second per second can be written as metres per second squared.

Now there's a few other ways to write that.

You can see it with a forward slash, or you can see it as a fraction, or you can see it using negative indices.

So ms to the negative two.

Right, well, let's see if we can calculate acceleration now.

So how much has the speed changed in the first 10 seconds of the journey? Well, let's have a look at the graph.

We've gone from zero to eight metres per second.

So that's an increase of eight metres per second, and that's taken us 10 seconds.

So if we use a ratio table in order to calculate acceleration, we need to calculate the change in speed over time.

Therefore, if we divide both by 10, we get the unit ratio of one to 0.

8.

So that means, for every second, the speed is increasing by 0.

8.

So the object is accelerating at a rate of 0.

8 metres per second squared.

Now this is just the gradient of the line.

So on a speed-time graph, the gradient of the line is acceleration.

Just to remind us, on a distance-time graph, the gradient is speed.

But on a speed-time graph, the gradient is acceleration.

So let's calculate the different rates of acceleration for this journey.

Well, here we can see that in the first 20 seconds, the speed is increased by 10 metres per second.

So we've got a acceleration rate of 0.

5 metres per second squared.

Now what's the rate of acceleration between 20 and 35 seconds? Pause the video, come up with your answer.

Remember we said that a horizontal line means constant speed, so that means the acceleration is zero.

The speed is not changing.

Okay, let's look at this final part of the journey.

So the object is decelerating 'cause speed is decreasing.

We can see, in 15 seconds, the speed is decreased by 10 metres per second.

We calculate the gradient of that line then.

It's approximately 0.

67.

Now I've written negative 0.

67 'cause the gradient is negative.

So it is accelerating at a rate of negative 0.

67 metres per second squared, or we can say it's decelerating at a rate of 0.

67 metres per second squared.

So just notice the difference.

If we've got a negative acceleration, that means it's decelerating.

If we use the word deceleration, we can just use the positive value 'cause we already know that means it's decreasing in speed.

Okay, which of these are possible units for acceleration? What do you think? Of course it's metres per second squared.

The other three are all units for speed.

Which of these is the correct acceleration rate for this object? What do you think? So the speed has increased by two metres per second in four seconds.

So the acceleration rate is 0.

5 metres per second squared.

Have a look at this speed-time graph.

Jun says the object is decelerating at a rate of negative two metres per second squared.

What is wrong with his statement? Right, he's mostly correct.

The only problem is because he said it's decelerating, he doesn't need the negative sign in his answer.

So he's got the correct rate.

What he should have said is that the object is decelerating at two metres per second squared, or accelerating at negative two metres per second squared.

They mean the same thing.

Okay, so with enough information, we can draw speed time graphs given the rate of acceleration.

So car starts stationary and accelerates a rate of two metres per second squared for four seconds.

Let's look at how we draw this.

Now remember the acceleration rate is the gradient of the line.

And because the scale on these axes is in the ratio one to one, we can easily draw a line with gradient two.

We need that to last four seconds.

So keep drawing with a gradient of two until we get to four seconds.

Right, it then immediately decelerates at a rate of three metres per second squared until it is travelling at two metres per second.

It then maintains that speed for a further two seconds.

Let's have a look at this together.

So because the car is decelerating, the gradient will be negative, and we want to stop when we get to two metres per second.

So I've shown you that on the Y-axis, and then we need our gradient of three, and we just stop when we get to two metres per second.

Then we've got a constant speed, so we can use a horizontal line for the next two seconds.

Pause video if you want to check my drawing against the description.

Now, we do need to be careful with the scales on the axes 'cause they're not always gonna be one to one.

Let's try this one together.

A car accelerates from rest at a rate of 3.

5 metres per second squared until it reaches a speed of 14 metres per second.

So we know, in one second, we can go from Zero to 3.

5.

in two seconds, we can go from zero to seven.

In three seconds, we can get up to 10.

5.

And four seconds, we can get up to 14.

So it would take four seconds to reach 14 metres per second.

So we can draw that on.

Jun says, "Or we can use the formula, acceleration is the change in speed over time." If you want to put that in your notes, feel free to pause the video now.

Okay, let's use that then.

So if the Acceleration rate is 3.

5, that is gonna equal the changing speed.

Well, the speed has gone from zero to 14, so that's 14 takeaway zero, divided by the time.

Now, it's the time we're trying to work out.

So you can multiply both sides by T and divide both sides by 3.

5.

So you get T is 14 divided by 3.

5.

That gives you T is four, and of course that's seconds.

Right, the car then maintains that speed for eight seconds before decelerating for eight seconds at a rate of 1.

5 metres per second squared.

Well, let's look at that together then.

So we want the constant speed for eight seconds, then decelerating at 1.

5 metres per second squared.

Well, in eight seconds, if you do 1.

5 times eight, in eight seconds, the speed will decrease by 12 metres per second.

If it started at 14 and it decreased by 12, we'd end up at two metres per second.

Again, pause the video if you just want to check through what we've done.

Alright, quick check then.

A car accelerates at a rate of two metres per second squared, starting at four metres per second, and reaching 12 metres per second.

Which of these could be a speed-time graph for that journey? What do you think? Well done if you went for that second one.

Our line needs to start at four metres per second, end at 12 metres per second, and we should have a gradient of two.

Time to put that all into practise then.

See if we can calculate the rate of acceleration for each of these one stage journeys.

Give that a go.

For question two, you've got a speed-time graph of a runner in a 100-meter race.

I'd like you to describe what happens in this race.

Even better answers will include rates of acceleration at each stage.

Once you are happy that you've got as much information possible in your answer, come back for the next bit.

Okay, you've got a scenario on the right-hand side, and this time I'd like you to draw a speed-time graph for that journey.

Before you start, you might wanna consider the scale on your axes to make sure you've got the right gradients at the right points.

Give that one a go.

And finally, we've got a speed-time graph for a car accelerating when it leaves a set of traffic lights.

So I'd like you to work out the rate of acceleration for each of those three stages so far.

Then I want the average acceleration for the first two stages combined.

And then, after 35 seconds, the car decelerates at a rate of 1.

25 metres per second squared, and then it comes to a stop.

So I'd like you to draw that last stage on the graph.

When you're happy with your answer, come back and we'll look at it together.

So for the first one, you should have 1.

25 metres per second squared.

Well done if you remember to put your units.

For the second one, you should have 0.

3 metres per second squared.

For C, you could have written this as negative 10 metres per second squared as an acceleration rate, or you could say it was decelerating at 10 metres per second squared.

For question two, there's lots of things you could have written about.

So some things you may have said.

The runner started from stationary and accelerated at a rate of two metres per second squared for four seconds.

They then accelerated at a slower rate of 0.

5 metres per second squared for four seconds.

They reached a top speed of 10 metres per second and maintained this for four seconds.

Finally, they decelerate at a rate of 2.

5 metres per second squared until coming to a stop.

Pause video and just check your answer is similar to mine.

So for question three, your speed-time graph should look like this.

Check you've got the two horizontal lines in the right place to show the constant speed, and our line should finish on the coordinate 10,4.

Pause the video and compare your graph to mine.

And finally, so the acceleration rate for the first stage is 0.

4 metres per second squared.

The second stage is 0.

8 metres per second squared.

And that third stage we're travelling at a constant speed, so acceleration is zero.

So the average acceleration for the first two stages, we've gone from zero to 10 in 20 seconds.

So that'll be 0.

5 metres per second squared.

You need to look at the overall speed increase and the time.

You can't just average out the two values because the first rate of acceleration was maintained for longer.

And finally, your line should look like that.

It should reach the x-axis at 43,0.

Good work today, guys.

We've looked at of speed-time graphs now, so we should be getting really familiar with what they're showing.

We've talked about the different sections, what a horizontal line looks like, what accelerating looks like, and what decelerating looks like.

And we've started to work out those rates of acceleration.

It's fantastic to learn with you today and I really hope you decide to join us again.