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I can read information from velocity-time graphs and calculate acceleration and distance travelled.

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Licence

This content is © Oak National Academy Limited (2023), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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Key learning points

A velocity-time graph shows how velocity changes with time.
Acceleration is equal to the gradient of a velocity-time graph.
A falling line shows decreasing velocity called deceleration, which is a negative acceleration.
The gradient of the line gives the instantaneous acceleration.
The distance travelled is equal to the area under a velocity-time graph.

Common misconception

Pupils often consider time in the acceleration equation to be moment in time (perhaps read from a graph) rather than a change in time.

Label the value of t in the acceleration as a change of time rather than just time.

Keywords

Displacement-time graph - A displacement-time graph shows the displacement of an object over a period of time.
Velocity-time graph - A velocity-time graph shows the velocity of an object over a period of time.
Acceleration - The acceleration of an object is the rate of change of velocity. This is how much the velocity changes per second.
Deceleration - Deceleration is used to describe a change in velocity where the object slows down.

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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 and objects 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-time 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 five to 20 seconds.

As I've said, velocity-time grafts and displacement-time grafts 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 axis 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 north to six metres per second and the times 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 bicycles 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 40 there, 35 kilometres per hour.

Well done if you've 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 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've 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 in 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 of highlighted on the graph there.

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

Pause 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 it speeds 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 with stops.

Well that's car C.

It's velocity is zero at end of race.

The other two cars, A and B, are still moving.

Identify which cars reached the highest velocity and when.

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

5 seconds.

State the velocity of each car at time equals four seconds where we have to look carefully up from the four seconds mark here and we should be able to see that car A is good 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've 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.

So to start this part of the lesson, we need to revise how to calculate acceleration.

So the acceleration of an object is the rate of change of velocity.

That is how much of velocity is changing every second.

So for example, an acceleration of four metres per second squared would mean that velocity is changing by four metres per second every second.

If we have an equation for that, it's acceleration is changing velocity divided by time.

And as symbols that's written as a = v - u divided by t.

And in that equation we've got acceleration, or A, and that's measured in metres per second squared.

And we've got initial velocity, which has got the symbol U, and final velocity with the symbol V.

Both of those are measured in metres per second and we have to be very careful when we write those down because they look quite similar.

And finally we've got time, t, measured in seconds.

And so we can use our velocity-time graph to find acceleration.

What we need to do is to read off the initial and final velocity from the graph and use it in that acceleration equation we've just seen.

So let's try and find the acceleration during the first five seconds of motion for this object.

Well first of all we find the initial velocity, U, and that's 0 metres per second here.

And then we find the final velocity after those five seconds and that's four metres per second.

Then all we need to do is to write out the expression for acceleration here, substitute those two values in carefully, also put in the change in time, there is five seconds at the bottom, and that gives us an acceleration of 0.

8 metres per second squared.

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.

So pause the video, do your calculation, and then restart.

Welcome back.

Well the first thing we need to do though is to identify the initial velocity and final velocity.

I've done that on a graph here; not metres per second and four metres per second.

And then all we need to do is to substitute those two values into the equation that gives us a calculation like this, remembering that the time is eight seconds, not the full 10 seconds shown on the graph, just the eight seconds we were asked about.

Well done if you got that.

It's important when you're calculating accelerations to use the change in velocity and not the absolute value, the difference between the initial and the final velocity.

So, let's say we're trying to find the acceleration of this object, which has got several changes in velocity.

I've got to find the acceleration between six seconds and 10 seconds.

So what I need to do is to identify the initial velocity at six seconds.

So U is five metres per second there.

And then I find the final velocity and that's two metres per second.

So, when I calculate the change in velocity, I've gotta use those two values.

So now we look for the acceleration equation, substitute those two values in and that gives us a value of -0.

75 metres per second.

So, this object is slowing down at 0.

75 metres per second squared.

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

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

So take the information from the graph and calculate the average acceleration please.

Pause the video, do that calculation and then restart.

Welcome back.

Well, the initial velocity was two metres per second, as you can see marked on the graph there, and the final velocity was five metres per second.

So I've got my two values there and then substitute those into my expression for acceleration, v - u over t, putting in the change in time as well, which was five seconds between two seconds and seven seconds.

And that gives me an acceleration of 0.

6 metres per second squared.

Well done if you've got that.

In our calculations of acceleration so far we've just used positive velocities, but we also know that velocities can be negative as well.

And so we might be dealing with situations where we've got both positive and negative velocities.

And we've gotta be extra careful when we find the change in velocity in that case.

So we've got a graph here and as you can see the motion changes from a positive velocity to a negative velocity.

So there's an acceleration section of that graph where the slope is.

So we're going to try and find the acceleration between five seconds and 19 seconds.

So, we identify the initial velocity, it was moving at four metres per second, and then we identify the final velocity, it ends up moving at minus three metres per second.

And we can calculate the change in time as well as a 14 second interval during that acceleration.

And then we can just substitute those, very carefully, into our equation.

So we write out our equation, but then when we put in the values, remember V was minus three metres per second and then we're subtracting the plus four metres per second.

So we end up with an expression like that, and once we've calculated that, we can see that acceleration is 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 second 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.

We find the initial velocity, four metres per second, and the final velocity, minus two metres per second, and the time difference, the time change, 0.

1 second.

We substitute those into our equation for acceleration and that gives us a final acceleration of -60 metres per second squared.

Well done if you've 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.

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

5 metres per second squared between six and 10 seconds.

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

75 metres per second squared.

Well done if you've 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 it 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 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 time, 15 seconds, we look carefully at the graph, find 15 seconds and look across to find the velocity is two metres per second.

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

Well, the robot was decelerating.

You can see the velocities going down there from five metres per second to not 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've got those three.

And we were asked to find the acceleration of the robot between 20 and 30 seconds.

So, we've identified the initial and final velocities like that and the change in time and then we substitute those into the equation and that should have given you an acceleration of 0.

3 metres per second squared.

Well done if you got that.

And now we're onto the final part of the lesson and in it we're gonna 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's constant.

So that's fairly simple to work out the distance travel 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 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.

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

So the average velocity is 5 - 0 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 travelled is the average velocity times the time.

Substituting those values in was 2.

5 metres per second, a time of six seconds, that gives us a distance travelled of 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 a 1/2 x 5 x 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 eight 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 x 5 x 8 and that's 20.

So that gives us a distance travelled of 20 metres.

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

5 x 8 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 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, but it's a rectangle, so it's width times its height, that's an area of 20.

For the triangle of 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 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 that is 24 x 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 to find the total distance travelled during the 62nd journey please.

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

Welcome back.

To find the acceleration at time equals 50 seconds I need to look at this highlighted section of the graph.

And notice it's a straight line so I can use the gradient of that line to find the acceleration.

And so we use the technique we learned earlier in the lesson.

We find the initial velocity, u, six metres per second, the final velocity is zero metres per second, and the change in time is 20 seconds.

We substitute those two values into the equation and that gives us an acceleration of -0.

3 metres per second squared.

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 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 a change in 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.

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