Lesson video

Hello there, I'm Mr. Forbes, and welcome to this lesson from the measuring and calculating motion unit.

The lesson's all about velocity-time graphs, and we're going to have a detailed look at those graphs, so that we can work out velocity at certain times, calculate acceleration, and also calculate distance travelled.

By the end of this lesson you'll be able to describe velocity-time graphs in detail.

You'll be able to take readings from those graphs to find the velocity at any time, and you'll be able to compare the acceleration of objects by looking at the gradient of the graphs.

You'll also be able to calculate the acceleration, and find the distance an object travels when it's moving at constant velocity or accelerating uniformly.

These are the keywords you'll need to understand to get the most of the lesson.

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

And remember, the displacement is how far an object is from its starting point in a particular direction.

A velocity-time graph shows how the velocity varies over time.

Acceleration means a change in velocity, so acceleration is the rate of change in velocity.

And deceleration is used to describe an object that is slowing down, where its velocity is decreasing over time.

And here are those keywords again with the descriptions.

You can return to this slide at any point during the lesson.

This lesson's in three parts.

In the first part, we're going to be looking at velocity-time graphs and how they're different than displacement-time graphs.

We're gonna be using them to find the velocity at certain times and some changes in velocity.

In the second part of the lesson, we're going to move on to looking at acceleration and how we can tell an object accelerating from a velocity-time graph, and also calculate some accelerations.

In the third part of the lesson, we're going to be using velocity-time graphs to calculate the distance travelled by objects that are going at constant velocity or the velocity is changing uniformly, it's increasing at the same rate.

So when you're ready, let's start with velocity-time graphs.

Okay, let's start this lesson by looking at a displacement-time graph, which is something you should have seen before.

A displacement-time graph shows the displacement of an object over a period of time.

And the displacement is how far the object is from a starting point.

So we can have a graph like this showing the movement of an object, and as you can see, displacement changes over time.

The displacement's shown on the y-axis here, and it's usually measured in things like metres, but it could be measured in other distance units.

And the time is shown on the bottom axis here.

In this graph, you can see in the first five seconds displacement's increasing, so the object's moving away.

Then in the next 10 seconds, the object's displacement isn't changing, so we have a constant position.

And in the final 15 seconds, the object is moving away again.

This time the displacement's not going up as fast, so it's going a bit slower.

And here's a velocity graph, and straight away you can see it looks very similar to a displacement-time graph, but this graph is showing how the velocity changes over time, not the displacement.

So even though it looks similar, it gives different information.

So time again is shown on the bottom axis, the x-axis there, but this time the velocity is shown on the y-axis.

So in the first 20 seconds here, you can see the velocity is increasing.

Then in the next 20 seconds, the velocity is constant, and then the velocity is increasing again for the final 20 seconds.

As I've said, velocity-time graphs and displacement-time graphs may look very similar, but they show different information.

So we've got a displacement-time graph here and a velocity-time graph here.

This one shows displacement.

This one shows velocity.

It's important to check those axes very carefully, so you know what type of graph you're trying to analyse to find information from, because they show very different information, as we'll see during this lesson.

Okay, it's time for the first check now, and what I'd like you to do is to identify which two of these graphs are velocity-time graphs, please.

So pause the video, check the graphs carefully, select the correct two and then restart, please.

Okay, welcome back.

And you should have selected graph A and graph C.

And the reason for that is, well, we can examine the y-axis here, and in the first one for graph A, it says velocity and gives a unit of metres per second, so that's obviously a velocity-time graph.

Graph B, that's got displacement, so that's not.

And graph C, although it just has a letter, the Y, it's got a unit of kilometres per hour, and kilometres per hour is a velocity or a speed, so that must be a velocity-time graph.

Well done if you selected those two.

Well, the first thing we can do with a velocity-time graph is we can find the velocity of an object at any particular moment.

So I've got a graph here that shows the movement of a bicycle, and as you can see, the velocity's shown on the y-axis there going from 0 to 6 metres per second, and the time's across the bottom there on the x-axis.

So if we wanted to find out the velocity of the bicycle at time equals 20 seconds, what we'd do is find 20 seconds on the axis, and then look upwards until we find the line of motion.

And then we could look across and read off the velocity, and it's two metres per second there.

If you wanted to do the same thing at 35 seconds, we could look up and then across and find the bicycle's moving at four metres per second there.

And we can do the opposite as well.

We can find a time for a certain velocity.

So when does the bicycle reach five metres per second? Well, we find five metres per second on the y-axis, and then we look downwards, we find it's 40 seconds, so it took 40 seconds to reach five metres per second.

So let's check that you can read values off the graph.

I've got a velocity-time graph here for a lorry, and I'd like to know what's the velocity of the lorry at time equals 20 minutes.

So pause the video, find that, and then restart, please.

Okay, welcome back, and hopefully you selected 35 kilometres an hour.

If you look up from the time 20 minutes to find the line and then across, you'll find it's halfway between 30 and 40 there, 35 kilometres per hour.

Well done if you got that.

The slope or gradient on a velocity-time graph shows when the velocity is changing.

So I've got a velocity-time graph here with changing velocity in three different sections.

So if we look at the first section here, you can see that the velocity is increasing during those first 20 seconds.

So from 0 to 20 seconds, we've got an increasing velocity, and we can say that the object is accelerating, accelerating meaning an increase in velocity there.

In this second section of the graph, these 20 seconds, you can see that the velocity isn't changing, it's constant at four metres per second throughout that, so the object isn't accelerating at all.

There's a constant velocity between 20 and 40 seconds.

And in the final section of the graph, you can see the velocity here is decreasing.

So there's a decrease in velocity, and we can describe that as decelerating.

The object is decelerating or slowing down.

Okay, let's check if you can understand motion from a velocity-time graph.

I've got a graph here and it's in four sections, A, B, C, and D.

And I'd like you to decide whether the remote control car is accelerating, decelerating, or moving at content velocity for each phase of motion.

So pause the video, make your decision for each, and then restart, please.

And welcome back.

For section A, you can see there that the velocity is not changing throughout that first section of motion, so we've got constant velocity there.

In section B, you should see that the velocity's increasing, so that's an acceleration.

Section C, again, the velocity is constant, so constant velocity.

And for section D, the velocity is decreasing, so that's decelerating.

So well done if you got that.

So far we've only seen velocity-time graphs that show positive velocity, but the graphs can show positive and negative velocity as well.

So here we've got that movement of a goods train, a really large train, and it's got positive and negative velocity shown on the graph.

So this positive velocity will indicate direction in one motion, and this section at the bottom will indicate motion in the opposite direction.

Those directions might be anything, such as north and south, but they could be, if this was a different object, it might be up and down, or it could be left and right.

But it just shows opposite motion in the top half and the bottom half of this graph.

So it's time for a check to see if you can understand information from a velocity-time graph.

So I've got a graph here showing the movement of an elevator, and I'd like you to describe how can the motion be described for time 20 seconds to 25 seconds, the section I've highlighted on the graph there.

So choose two of the options on the left, please.

Pause the video, make that selection, and then restart.

Welcome back.

Well, you should have selected moving downwards.

You can see that the velocity is negative in that section, and that means it's moving downwards, and its speed's increasing.

The velocity is going down to minus four metres per second, and that's faster than minus one metre per second or zero metres per second.

So well done if you selected those two.

Time for the first task of the lesson now.

And I've got motion showing three remote control cars in an eight second long race.

And what I'd like you to do is to identify which car stops at the end of the race, identify which car reached the highest velocity and when that happened, state the velocity for each of the three cars at time equals four seconds, and describe the movement of just car B between two seconds and five seconds.

So pause the video, answer those four questions and restart, please.

Okay, welcome back.

And first of all, let's identify the car which stops.

Well, that's car C.

It's velocity is zero at end of the race.

The other two cars A and B are still moving.

Identify which car's reached the highest velocity and when.

Well, we can see that the highest point's that blue line there and that's car B, and it reached the highest velocity at that point, which is 1.

5 seconds.

State the velocity of each car time equals four seconds.

Well, we have to look carefully up from the four seconds mark here, and we should be able to see that car A is going at one metres per second, and so is car B, and car C is travelling at 1.

5 metres per second.

Well done if you got those three.

And the final part of the task was to describe the movement of car B between two seconds and five seconds.

And as you can see, car B is decreasing its velocity.

It's gone down by 1.

5 metres per second during a period of three seconds.

So well done if you identified that information.

Okay, it's time for the second part of the lesson now, and in it we're going to be using velocity-time graphs to find the acceleration of objects.

Let's start this part of the lesson with a look at the acceleration equation that you should already be familiar with.

So the acceleration of an object is the rate of change of velocity, and that means how much the velocity is changing every second.

So an acceleration of four metres per second would mean that the object's velocity is changing by four metres every second.

Acceleration can be calculated using this equation.

Acceleration is change in velocity divided by time taken, or written in symbols, A equals delta V over T, where A is the acceleration and that's measured in metres per second squared.

The change in velocity is represented by delta V, and that's measured in metres per second.

And the time is measured in seconds, T.

To calculate acceleration then, we need the initial velocity and the final velocity in order to calculate change in velocity, and both of those can be read from a velocity-time graph, and we can use that then to calculate the average acceleration for a section of motion.

So we've got a simple velocity-time graph here, and we're going to try and find the acceleration during the first five seconds.

And to do that we identify the initial velocity, and that's zero metres per second there at the bottom, and then the final velocity after those five seconds, and that's four metres per second there.

And we can then get a change in velocity, which is four metres per second, the difference between those two values.

Then we can substitute into the equation for acceleration, and that gives us a calculation like this.

We do the sums and that gives us an acceleration of 0.

8 metres per second squared for that five seconds of motion.

To check if you can do that, I'd like you to find the average acceleration during the first eight seconds of this ball that's rolling across a desk.

So you can see I've got a graph there.

What I'd like you to do is to calculate the acceleration during those first eight seconds, please.

Pause the video, do your calculation and then restart.

Welcome back.

Well, we write out the initial velocity and final velocity by identifying those two points on the graph there.

Got initial velocity of zero, final velocity of four, so I've got a change in velocity of delta V of four metres per second.

Then I substitute it into the equation, putting in the value for delta V and the change in time, and that gives me an acceleration of 0.

5 metres per second squared.

Well done if you've got that.

It's important to realise that we need to use the change in velocity to calculate acceleration, not an absolute value.

So looking at this graph, we can see there's several different key velocities there.

Starts at one metres per second, goes up to five metres per second, and then back down to two metres per second.

And if I want to work out the acceleration for the last part of the graph from 6 to 10 seconds, then I need to make sure I use the change in velocity.

So I identify the initial velocity, five metres per second there, and then the final velocity which is much less, it's two metres per second, so this object's slowing down.

And then I've got a change in velocity of minus three metres per second.

And then I can substitute that value into my calculation using the change in velocity and the change in time.

That's minus three metres per second divided by four seconds, and that gives me an acceleration of minus 0.

75 metres per second squared.

Okay, it's time for you to try and find an acceleration.

I'd like you to find the average acceleration between two seconds and seven seconds for this rolling ball.

So I'll take the information from the graph and calculate the average acceleration, please.

Pause the video, do that calculation and then restart.

Welcome back.

And what you should have done is identify the initial velocity and the final velocity, and that gives you a change in velocity of three metres per second.

And then you substitute that into the equation, putting in the change in time as well from two to seven seconds, there's a five second interval, and that gives an acceleration of 0.

6 metres per second squared.

Well done if you've got that.

So far we've calculated accelerations using just positive velocities, but as you've seen already, velocities can be both positive and negative.

So we've gotta take extra care when we're finding a change in velocity to take that into account.

So I've got a graph here for an object where the velocity is positive first and ends up negative later on.

What we're gonna do is try and find the acceleration for that sloping part of the graph there.

So we're gonna find the acceleration between 5 seconds and 19 seconds.

First thing we do is we find the initial velocity, and it started off at four metres per second there, and then we find the final velocity, that's minus three metres per second.

And next, we find the change in velocity, and this is where we've gotta be careful.

The change in velocity is minus three metres per second minus four metres per second.

That gives us a value of minus seven metres per second, which is what you can see in the graph.

It's gone down by seven metres per second.

So once we've got that, the rest is just the same, we find the change in time there, and we substitute those into the equation for acceleration.

That gives those minus seven metres per second divided by 14 seconds, and then we'll get a negative answer, an acceleration of minus 0.

5 metres per second squared.

So let's see if you can calculate an acceleration that involves both positive and negative velocities.

I'd like you to find the acceleration between 0.

1 seconds and 0.

2 seconds for this ball that's bouncing off a wall.

So pause the video, carry out the calculation, and then restart, please.

Okay, welcome back, and hopefully your calculation looks something like this.

The initial velocity there as I've marked is four metres per second, and the final velocity is minus two metres per second.

So the change in velocity is minus six metres per second.

We then find the change in time, and that's 0.

1 seconds, and substitute those values into the equation, and that gives us a final acceleration of minus 60 metres per second squared.

Well done if you got that.

When I've got motion that only involves constant accelerations, I can find the instantaneous acceleration at any point by looking at the gradient of the straight line sections of the graph.

So in this section of the graph, a straight line, I've got constant acceleration, and so the instantaneous acceleration is the gradient of that line.

And as you can see there's a four seconds and a change in velocity of four metres per second.

That gives us an acceleration of one metres per second squared for any point along that section of the line.

Between four and six seconds, there's no change in velocity, so there's no acceleration.

But then I can look at the instantaneous acceleration of this section of the graph and I find it's 0.

, sorry, it's minus 0.

5 metres per second squared between 6 and 10 seconds.

So for example, at eight seconds, the instantaneous acceleration is minus 0.

5 metres per second squared.

Okay, I'd like you to find an instantaneous acceleration for me using that technique.

I'd like to know what's the instantaneous acceleration for this object at time equals five seconds? So pause the video, work out the acceleration, and then restart, please.

Okay, welcome back.

You should have found the instantaneous acceleration is minus 0.

75 metres per second squared.

If we look at the gradient of this section, we've got a time of four seconds, and we've got a change in velocity of minus three metres per second.

So we'll get the acceleration by finding the change in velocity divided by the change in time, and that gives me minus 0.

75 metres per second squared.

Well done if you got that.

Acceleration isn't always constant, and so we end up with graphs that aren't just made up of straight lines.

So changing acceleration is shown by a changing gradient, and that gives us a curve on a graph.

So in this graph, we've got velocity and time, and the velocity is not changing uniformly, the acceleration is not constant.

The steeper the gradient, the greater the acceleration is.

So in this first few seconds of the graph, we've got high acceleration, because we've got a steep gradient, and then towards the end of the graph, we've got lower acceleration because the gradient is lower there.

So the gradient indicates the acceleration.

Okay, let's check if you understand the relationship between acceleration and gradient.

I've got the movement of a drone here, and I'd like you to decide at which point is the magnitude, the size of the acceleration greatest for this drone.

Is A, B, C, or D? So pause the video, make your selection and restart, please.

Welcome back.

You should have selected B for this.

That's where the gradient of that curve is steepest.

So the magnitude of the acceleration is greatest at point B.

Well done if you've got that.

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

I've got a graph here showing the motion of a robot that works in a warehouse moving things around, and I'd like you to look carefully at that graph.

I'd like you to then state the velocity of the robot at time equals 15 seconds, describe the movement of the robot between 40 seconds and 60 seconds, identify when the robot has the greatest acceleration, and finally, find the acceleration of the robot between 20 and 30 seconds.

So pause the video, work out your answers to that, and restart, please.

Welcome back.

Well, to state the velocity of the robot at time, 15 seconds, we look carefully at the graph, find 15 seconds and look across, find the velocity is two metres per second.

Describe the movement of the robot between 40 seconds and 60 seconds.

While the robot was decelerating, you can see the velocity's going down there from five metres per second to 0 metres per second.

And identify when the robot has the highest acceleration, well, we look for the steepest gradient, and that's between 20 seconds and 30 seconds.

Well done if you got those three.

And now we've asked to find the acceleration of the robot between 20 and 30 seconds, so we look at the values from the graph, identifying the initial and final velocity, finding the change in velocity and the change in time there.

We substitute those into the equation.

That should have given you an acceleration of 0.

3 metres per second squared.

Well done if you got that one.

And now we're onto the final part of the lesson, and in it we're going to use a velocity-time graph to calculate the distance moved by an object over several phases of motion.

We're going to start by finding the distance travelled by an object that's got a constant velocity, and that's the simple graph like this.

So we've got constant velocity, and that gives us a straight line on the graph.

So the velocity's unchanging there at three metres per second over the full six seconds of motion.

Now, if you remember the distance travelled equation, the distance travelled is the velocity times the time when the velocity is constant.

So that's fairly simple to work out the distance travelled for this graph, because it's just three metres per second, that constant velocity, multiplied by the time of six seconds, and that gives us a distance travel of 18 metres.

That is the same as the area beneath the line on the graph here.

If you look, I've got the area, it's a three by six block, and that gives us an area of 18.

So the distance travelled is the same as the area beneath the line.

Now if we've got an object that's moving with a constant acceleration, we can use the average velocity to calculate the distance travelled, and that's halfway between the initial velocity and the final velocity.

So I've got a graph like this, and as you can see the velocity is increasing as time goes on.

We've got a uniform acceleration, a constant acceleration there.

And we can find the average velocity by looking at the initial velocity and the final velocity.

So the average velocity is five minus zero metres per second, divided by two to find average, and that's 2.

5 metres per second.

And I can use that in my equation for calculating distance travelled.

The distance travel is the average velocity times the time.

Substituting those values in, it was 2.

5 metres per second, a time of six seconds, that gives us a distance travel to 15 metres.

But again, if we look at the area beneath the line, that's a triangular shape.

And if we calculate that area, the area is 1/2 times 5 times 6, and that's 15.

So again, we've got the area beneath the line is equivalent to the distance travelled.

And now I'd like you to calculate the distance travelled by this object between 0 seconds and 8 seconds.

So look carefully at the graph, pause the video, make your selection and restart.

Welcome back.

You should have selected 20 metres for the answer there, and again, we can calculate that using the area of the triangle part of the graph.

So the area is 1/2 times 5 times 8 and that's 20, so that gives us a distance travel of 20 metres.

We could have done the same calculation using the average velocity there, which would've been 2.

5 times eight seconds, and that would've given 20 metres as well.

So well done if you got that.

So we've seen that the distance travelled when an object's going at constant velocity is the area beneath the line.

We've also seen that the distance travelled when an object has got a uniform acceleration is also the area beneath the line, and we can use the idea to find the total distance travelled for an object that's moving at different velocities and different constant accelerations.

So I've got a graph here that shows an object moving at two different velocities with a deceleration between four and eight seconds there.

And to find the total distance travelled by that object, we can break that area up into simple sections.

So I've got a rectangle here section, I've got a triangle section here, and I've got another rectangular section here.

And the total area travelled would be the sum of all of those three areas.

So all I need to do to find the distance travelled is to work out the area of each of those shapes.

So for the first shape, I calculate the area, and it's a rectangle, so it's its width times its height, that's an area of 20.

For the triangular shape, we've got a base of four and a height of four, and the area of a triangle is half base times height, so that gives an area of eight.

And for this rectangular section, again, it's a rectangle, so base times height gives the area.

So I've got three separate areas.

The total area is those three added together.

So this object has travelled 34, and I've gotta look carefully at the velocity units there, that's in metres per second, so the distance is measured in metres.

So it's 34 metres.

Right, now I'd like you to calculate the total distance travelled between 0 seconds and 10 seconds for this object.

So use the technique I've just shown you to work it out.

Pause the video, make your selection, and then restart, please.

Welcome back.

Hopefully you selected 35 metres, and we can show that by looking at the shapes.

I've got this triangular shape.

It's got an area of 15, because half times the base, which is six, times the height, which is five gives 15.

And then we've got this rectangular area, and again, the area there is 24 times 5.

So adding those two together will give a total of 35 metres.

Well done if you've got that.

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

So I've got a graph showing the movement of a boat.

I'd like you to find the acceleration at time equals 50 seconds, and I'd like you to find a total distance travelled during the 60 second journey, please.

So pause the video, try and answer those two and restart.

Welcome back.

Let's start by finding the acceleration at time 50 seconds, and I've highlighted that section here.

And we used the technique we learned earlier in the lesson.

We find the gradient of that section of the graph.

So the initial velocity is six metres per second, the final velocity was 0 metres per second, that gives us a change in velocity of minus six metres per second.

And we've got a change in time of 20 seconds there.

So we use those values in our equation for acceleration, and that gives us a final acceleration of minus 0.

3 metres per second.

Well done if you've got that.

And for the second part, I've divided the area underneath the line up into three sections, two triangles and a rectangle, and I work out the area of each of them like this.

And then the total distance travelled can be found by adding those three areas, looking carefully at the velocity-axis to find out what unit I'm gonna be using for the distances.

And so I get a total distance of 225 metres.

Well done if you got that.

And now we've reached the end of the lesson, and here's a quick summary of everything we should have learned.

Our velocity-time graph shows changes in velocity over a time period.

I've drawn a graph there with a couple of lines on.

The total distance travelled by an object is the area beneath that line on the graph.

We can find the acceleration from the gradient of the line on the graph, and deceleration is shown by a negative gradient.

A curved line will show changing acceleration, whereas those straight lines show constant acceleration.

Well done for reaching the end of the lesson.

I'll see you in the next one.