Lesson video

Hello there, I'm Mr. Forbes, and welcome to this lesson from the Measuring and Calculating Motion Unit.

In it, we're gonna be looking at calculations from motion graphs, and that involves using the graphs to find things like final displacement, distance travelled, speed and acceleration.

By the end of this lesson, you're gonna have looked at a wide range of motion graphs, and be able to take readings from those graphs to calculate things like distance displacement, speed, velocity, and acceleration.

You'll also be able to convert from one graph to the other through a series of calculations.

Here are the key words that will help you understand the lesson.

First of them is displacement time graph, and that's a type of graph that shows the displacement of an object over a period of time, where displacement is how far it is from its starting position.

It's also a velocity time graph, which shows the velocity of the object over the period of time.

Then there's velocity, and velocity is the rate of change of displacement, how fast the displacement is changing.

And finally, acceleration, and the acceleration of an object is the rate of change of velocity, how much the velocity is changing each second.

And here's some explanations of those keywords that you can return to at any point in the lesson if you need some help.

In the first part of the lesson, we're going to be comparing the two types of motion graph: velocity time graphs and displacement time graphs, and we'll see what type of information we can get from both.

In the second part of the lesson, we're going to be looking at one graph, and trying to convert information we can obtain from it into the other type of graph.

So we'll take a velocity time graph and use it to make a displacement time graph or a displacement time graph, and use that to make a velocity time graph.

And in the third part of the lesson, we can find information from both types of graphs, and use that in a variety of calculations.

So when you're ready, we'll begin by looking at the two types of motion graph: displacement time graphs, and velocity time graphs.

Both look very similar and give us similar types of information, but they're not identical.

They're very different in important ways.

So I've got a displacement time graph and a velocity time graph here, and I'm gonna show you the movement of the same object on both graphs at the same time.

So in the first stage of movement here, I've got a displacement increasing in regular amounts, and that indicates a constant velocity.

So if you look at the velocity time graph, I've got a constant level of velocity there.

Then I've got this secondary displacement graph where the displacement doesn't change, and that means the velocity is zero.

And as you can see on the velocity time graph, I've made the line go down to zero, then I'm gonna increase the velocity again.

But this time it's a small velocity and as you can see on the displacement time graph, the line goes up again, but this time the gradient of the slope is lower.

So both graphs show information about movement, but in different ways.

As you saw in those example graphs, the lines, even though they're representing the same movement, don't match each other on the graphs.

If I've got a displacement time graph that's got a line like this, this indicates that there's a constant velocity.

This object is moving forward at a constant rate, and that on the velocity time graph would be represented by a line like this.

The velocity is remaining constant at a steady amount, and it's producing a gradual increase in displacement, giving us that straight line.

Now, objects aren't always moving, and as displacement can stay at a constant value.

So if I've got a displacement here where the object's not changing displacement over the period of time, well, that must indicate that the object's not moving.

It's got zero velocity.

So on a velocity time graph, I'd have a velocity remaining constant at zero.

On those two earlier graphs, we had constant velocities, but velocity's not always constant.

Objects can accelerate, so we can have an object like this.

We've got an increase in velocity, showing that the object is accelerating in a regular way and that's gonna cause a change in displacement that's going to be greater for every second that passes.

So on a displacement time graph, I wouldn't get a straight line at all.

I'd get a curve like this.

And as you can see in the first part of the movement, I've got a very shallow gradient, and later on the object moving much faster and it got a much steeper gradient.

So increasing velocity produces curves on displacement time graphs.

Objects can also decelerate, so that would mean that the velocity is decreasing over time.

So on a velocity time graph, that could be represented with something like this.

A decrease in velocity means the change in displacement every second is gonna get smaller and smaller.

So I'd end up with a displacement time graph, something like this.

As you can see at the start, the displacement's increasing quickly, and towards the end of the graph later on in time, the displacement's flattening out.

The gradient is much shallower, so a decreasing velocity will give me a curve in that direction.

Okay, let's see if you understand the relationship between the shapes on displacement time graphs and velocity time graphs.

I've got three displacement time graphs here, and I'd like you to match them up with the corresponding velocity time graphs.

So A, B, and C are the displacement time graphs, and D, E, and F are the velocity time graphs.

So what I'd like you to do is to just pair them up.

So pause the video, pair up the graphs, and then restart, please.

Welcome back.

Well, let's have a look at those pairings.

The first pair is A and F.

And in A, you can see that the displacement is increasing all the time, but it's gradually slowing down, so that maps to F, where the velocity is decreasing.

The second pair is B and D, and in that I've got a constant velocity, so a constant increase in displacement, and that leaves the final pair of C and E.

And in C and E, you can see that the velocity must be increasing, 'cause the displacement is increasing more and more rapidly as they go.

Well done if you've got those pairings.

So far, we've just looked at simple sections of a graph where we've got just one phase of motion, but graphs can show multiple phases of motion.

So we can have a displacement time graph like this, where I've got a steadily increasing displacement and then displacement stops increasing over time.

So we've got two phases there, and that would match a velocity time graph, something like this.

So in the first part of the motion, we've got a constant velocity, and that represents the slope on the displacement time graph, and a flat section of a velocity time graph.

And then we've got the velocity falling to zero, because the displacement no longer increases.

The graph can show gradual changes in motion as well.

So I'm going to try and plot a displacement time graph that shows a constant velocity, and then gradually slowing to a stop instead of suddenly stopping, and a velocity time graph that matches it.

So here's the first section, where we've got some constant velocity, which results in that slope on the displacement time graph.

And then I've got a gradual slowing here.

So that's represented by a curve on the displacement time graph, and a straight line on the velocity time graph as the velocity gradually falls to zero.

And then when it reaches zero, I'll put that final section of the velocity time graph in, and that gives me a displacement, and that doesn't change.

So I've got that final section, on the displacement time graph.

So I've got a complete picture of an object that's moving at steady speed, gradually slowing down to a stop.

Okay, it's time for you to try and interpret a graph that got several phases of motion.

So I'd like you to decide which of the lines, A, B, C, and D on the displacement time graph, which is on the right there, matches the motion shown on the velocity time graph, which is on the left.

So pause the video, have a think about that, and then restart when you've made your selection.

Welcome back, and the correct answer to that was answer B.

Well done if you selected that.

We've got a velocity that's constant, so that gives us a straight gradient upwards at first.

Then we've got a velocity that decreases.

So on the displacement time graph, we're gonna get a curved section.

And then in the final bit, we've got another constant velocity.

So we're looking for another straight section of the graph, but this time a shallower gradient, 'cause the velocity is less.

So well done if you selected B.

And now let's check if you can convert the graphs the other way.

So which line on the velocity time graph matches the motion shown on the displacement time graph? You can see there's a displacement time graph there on the left, and I've got four different lines on the right.

I'd just like you to decide which of those four lines on the right velocity time graph matches that displacement time graph.

So pause the video, make your selection, and restart please.

Okay, welcome back.

And the correct answer to that was D.

Well done if you selected that.

We can see from the displacement time graph in the first part of the motion, I've got a steadily increasing displacement, and that indicates a constant velocity.

Then I've got a curved section of the graph, and that indicates that the velocity is decreasing.

And then I've got a section of the graph, where the displacement remains the same, so a zero velocity, and the only line that matches that was line D.

Well done if you selected that.

So far, we've only looked at graphs that have got positive velocity.

But there are plenty of situations where we can have negative velocity, an object moving backwards.

So we're gonna look at what that would produce on a displacement time graph here.

So I've got a velocity time graph and a matching displacement time graph, and we'll look at each section of the motion and try and explain it.

So in the first section of the motion, I've got a positive velocity of two, and that's gonna cause an increase in displacement.

It's gonna increase by the same amount every second.

So I'm gonna get a straight line section of the graph sloping upwards like this.

And the next part of the graph, in this part I'm going to be decelerating towards zero velocity, so that's going to change the gradient on my displacement time graph.

It's gonna get gradually flatter and flatter.

So as you can see, it curves until we've got a sort of peak of the graph there.

In the next section of the motion, I've got accelerating backwards.

I've got a velocity that's increasing, or becoming more negative, I should say, and that's going to cause another curved section of the graph.

But this time, it's curving in the other direction.

And in the final section, I've got a constant negative velocity.

So that's gonna give me a straight slope downwards.

So we've got four sections of motion there, where the object is going forwards, then slowing down and then starting to accelerate backwards, and then finally, moving backwards.

Let's see if you understood my description involving positive and negative velocity.

I've got a graph here of a drone, and it's got positive and negative velocities.

I'd like to think carefully, and decide at what time would the displacement of that drone be its maximum value.

So pause the video, make a selection from those on the left there, and then restart, please.

Welcome back.

You should have chosen T equals 40 seconds because at that point, the velocity is just about to become negative, so the drone's going to start moving backwards at 40 seconds, and that means that the displacement's gonna start to decrease at that point.

So well done if you spotted that.

It's time for a more challenging question about velocity, time and displacement time graphs.

So this one involves negative velocities.

What I'd like you to do is to look carefully at the two lines on the velocity time graph, labelled A and B, and select the two lines on the displacement time graph that match that motion.

So pause the video, make your selections and restart, please.

Okay, welcome back.

And you should have noted that A matches line E.

So if you look carefully at line A, we've got a gradual increase in velocity.

So what we'd be expecting is a curve upwards on the displacement time graph, and that is line E.

For line B, that matches line C on the other graph.

You can see there's going to be an increase in displacement while the velocity is positive, and then the displacement's going to start decreasing when the velocity is negative.

Well then, if you spotted those two, and now it's time for the first task of the lesson.

What I'd like you to do is use everything you've learned so far to sketch a pair of graphs for me.

So I'd like you to sketch a velocity time graph that matches the motion shown on the displacement time graph on the left there.

And I'd like you to sketch a displacement time graph that matches the motion shown on the velocity time graph on the right there.

Now there's no numbers on the graphs, I just want the general shapes.

So pause the video, sketch your graphs, and then restart please.

And welcome back.

The velocity time graph you should have sketched for part one should look something like this.

We've got a constant forwards velocity at first, so constant positive velocity, and that gives us that upward straight slope.

Then we've got a velocity of zero, because you can see the displacement doesn't change, so the object's not moving, it's got a velocity of zero, and then we gradually curve upwards, and that means a gradual increase in velocity, and that gives us that upwards diagonal line on the velocity time graph.

Well done if you sketched something that shape.

And here's the solution to the second part of that.

Your graph should look something like this.

At first we've got a high constant velocity, so we get a straight line going upwards like that on the displacement time graph.

Then we have a decreasing velocity.

So that's a curved section on the displacement time graph.

And finally, we've got a constant lower velocity.

So well done if your graph looks something like this.

So let's move on to the second part of the lesson now, and in it we're going to be converting from one motion graph to another.

So we'll be reading information from one type of graph, a displacement time graph, and then using that to construct a velocity time graph, or the other way around.

So we're gonna be making those conversions from one graph type to another, and we need to follow fairly standard procedures to do that.

That involves reading data from the first graph, and then we use that data in the speed or the velocity equations.

And finally, from the results of those equations, we can plot the results on the second graph.

So let's start by looking at a few example graphs to see if we can do that.

So far, we've just sketched the shapes of the graphs.

Now we'll try and plot an accurate displacement time graph using velocity time graph information.

And we're going to use the change in displacement for each part.

We're gonna calculate that using data from the velocity time graph, and then we can plot it on our displacement time graph.

So for example, if I've got a velocity time graph like this, what I should do is calculate a change in displacement in section of motion.

So for the first two seconds, and then from two to five seconds, and then from five to eight seconds.

So let's do that.

So in this first section of the graph, I can calculate the change in displacement using the speed equation, and I get the change in displacement, and the distance travelled is three metres for that section.

It's moving at 1.

5 metres per second for two seconds, and that gives us a change in displacement of three metres.

So that allows me to plot the value on the graph over here.

After two seconds, the object's moved three metres, so we can draw the red line there.

In the second section of motion here, well, you can see the velocity is zero, so there's not gonna be any change in displacement between two and five seconds.

So I can transfer that to the displacement time graph, and that will give me a line that looks a bit like this.

There's no change in displacement there.

And for the final section of the graph, again I calculate the change in displacement using the speed equation there.

The change in displacement is two metres per second times three seconds.

That gives me an increase in displacement of six metres.

And it's important to note that's an increase in displacement.

That's not the final displacement.

So I need my graph to go up by six more metres.

I've got a final displacement of nine metres on the graph there.

So I've successfully converted a velocity time graph to a displacement time graph.

Okay, let's see if you can match up a velocity time graph to the displacement time graph that should be produced.

So which line on the displacement time graph, which is the graph on the right, shows the same motion as the velocity time graph on the left there.

So pause the video, make your selection and restart, please.

Okay, welcome back.

The answer you should have selected was line C.

So if you look carefully, the velocity is at one metre per second there.

So after two seconds, the object should have moved two metres, then the velocity increases to two metres per second.

So the gradient would increase on my displacement time graph and I end up with a total distance after four seconds of six metres.

And that indicates that the red line is the correct one.

Well then, if you selected that.

We've looked at just positive velocity so far.

So let's have a look at a case where there's some negative velocities as well.

So I've got a graph here of motion, and I've got positive velocity for the first three seconds.

Then the velocity goes negative, and then it goes back to positive again.

So we'll look at each phase of that motion.

So in this first section of the graph, I've got a velocity of plus three metres per second, and that will mean after three seconds I've travelled nine metres.

So that would give me the line there shown in red on the displacement time graph.

In the second phase of motion here, I've got a negative velocity, so the displacement's going to decrease by two metres every second.

So I get a line like this showing a velocity of minus two metres per second.

So, my displacement will decrease by two metres every second though.

And in the third section of motion, again I've got positive velocity again.

So my displacement's going to increase by one metre per second here, and that'll give me the final line like this, plus one metres per second.

So you can see how to convert from a velocity time graph to a displacement time graph, and the velocity is positive and negative.

So a fairly challenging question here now.

I'd like you to decide which line on the displacement time graph below shows the same motion as the velocity time graph.

So pause the video, make your selection, and then restart please.

Welcome back.

You should have selected line B, the dotted black line there.

And as you can see in the velocity time graph during the first two seconds, the velocity is three metres per second.

So I should see the displacement increase by three metres every second.

So after two seconds, the displacement would be six metres.

And that shows me that the black dotted line was the correct one.

Well done if you selected that.

We've been using velocity time graphs to produce displacement time graphs, and now it's time to do the opposite to reverse that process.

So I've got a displacement time graph here.

What I'm going to do is try and make a velocity time graph from it, and we use a very similar procedure.

We're going to calculate the velocity for each part of the motion, and then plot it on a velocity time graph.

So in the first section of the motion here, the first two seconds, you can see the displacement's increased to six metres.

So I can calculate the velocity during that section using the speed equation.

And I get a velocity of three metres per second.

It went up by six metres in two seconds.

That gives us three metres per second.

So all I need to do is to go to my velocity graph and plot a line when it's a constant velocity of three metres per second.

So it looks like that.

In the second section of the graph here, you can see the displacement doesn't increase, and that gives us a nice simple velocity.

It's zero metres per second, it's not moving at all.

So in that section, I can draw in this part of the graph, it goes down to zero metres per second there.

In the third section of the graph, again I can calculate the velocity by reading the change in displacement, and the change in time.

The displacement's gone up by four metres, and the time is four seconds, from four seconds to eight seconds there.

And that gives me a velocity of one metre per second.

And I can plot that in this section of the graph here.

And that gives me the final part of the graph.

And I've successfully translated a displacement time graph into a velocity time graph.

Let's see if you can match up a displacement time graph and a velocity time graph.

So which line on the displacement time graph shows the same motion as the velocity time graph, which is on the right there? So pause the video, make your selection and restart please.

Welcome back.

You should have selected line A there.

So in the first part of the motion, the first two seconds, I've got a velocity of one metre per second.

So you would expect the displacement to be two metres after two seconds, which the blue line has.

Then you've got a velocity of zero for another two seconds.

So you should have a flat part on the blue dashed line, and you do.

And then we've got another increase in velocity to three metres per second.

And finally the final section of the line between six and eight seconds, we've got no increase in displacement.

So that was the blue dashed line.

Well done if you've got that.

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

And what I'd like you to do is use the techniques I've shown you to draw some accurate graphs.

So I'd like you to draw an accurate displacement time graph using the information shown on the velocity time graph there in red.

And then I'd like you to draw an accurate velocity time graph using the information from the displacement time graph, which is shown in the greeny blue there on the right.

So pause the video, draw your two graphs accurately, and then, restart please.

Welcome back.

And for the first graph you should have drawn something like this.

We've got zero velocity for the first second, so there's no change in displacement.

Then we're moving up one metres per second for two seconds.

So we should get up to two metres after that three second mark.

Then we've got a velocity of three metres per second.

So that steeper line there.

And finally back down to a velocity of 0.

5 metres per second for the end two section.

Well done if you've got that.

And for the second part of the task, you were asked to draw a velocity time graph, and you should have come up with a velocity time graph that looks like this.

Well done if you've got that.

And now it's time to move on to the final part of the lesson where we're going to be finding information from different motion graphs.

So we're gonna read values from the graphs, and use them in different calculations to find things like distance travelled, velocity and acceleration.

We have a displacement time graph, where the motion is only in one direction where the direction of travel doesn't change.

It's quite simple to find total distance travelled.

So if I've got a graph like this, and as you can see I've got changes in velocity there, but I've not got a change in direction, I've got no decrease in displacement.

I can just read off the total distance travelled.

It's the same as the final displacement like this.

So the distance travelled there would be 10 metres.

This graph, the distance travelled would be 60 kilometres.

So as long as the direction of travel doesn't change, the displacement and the total distance travelled are the same.

In a lot of motion, we've got direction of travel changing.

Well as long as the direction of travel is just along the straight line, so north to south or east to west or something like that, the distance travelled can still be found by just adding the separate phases of motion together.

So I've got a displacement time graph here.

We've got displacement kilometres and time in hours, and as you can see the displacement goes up and then down and then up again.

So I've got changes of direction here.

And so to find the total distance travelled, what I have to do is add up all of the changes in displacement.

So I've got an increase in displacement of 60 kilometres there, and then I travel backwards 40 kilometres, and then forwards again 30 kilometres.

And I can find the total distance travelled just by adding those three together.

So the distance travelled is those three values, and that gives a total distance travelled of 130 kilometres.

Let's see if you can find a total distance travelled for an object that's moving in two directions.

So I've got a graph showing the displacement and time for an elevator here.

As you can see, it goes upwards and downwards.

So what I'd like you to do is find the total distance travelled from its journey where it starts at 50 metres to the ground, which is at displacement of zero.

So pause the video, make your selection from the list on the left and then restart, please.

Welcome back.

You should have selected 130 metres.

In the first part of the motion, the elevator goes down 20 metres, and the second part of the motion it goes back up 40 metres.

So we've already got 60 metres there, and then down from its maximum height of 70 to the ground, so 70 metres down.

And if we add those three values together, we get a total distance of 130 metres.

Well done if you got that.

The distance travelled for any section of a velocity time graph can be calculated from the area beneath the line.

And you may have seen that in a previous lesson.

So if we wanted to calculate the distance travelled between 10 and 30 seconds on this graph, what we could do is break down that part of the motion into two simple areas that we can calculate, and then we can calculate those two areas.

For area A, it's 20 seconds by three metres per second.

That gives us 60.

And area B, that's a triangle, so we've gotta remember half base times height, or half five, which is the difference in velocity there, between three and eight times 20 seconds.

We get those two areas, and if we add those two areas together, we get the total distance travelled between 10 and 30 seconds, which is 80 metres.

Okay, I'd like you to use the same technique to calculate the total distance travelled between nought seconds and 30 seconds on this graph.

So you can ignore the rest of the graph, just concentrate on that first section between nought and 30 seconds.

So pause the video, make your selection from the list on the left there, and then restart.

And welcome back.

You should have selected 180 metres.

If we break down that section into two areas, a triangle and a rectangle, we can see area A the rectangle.

That's four high by 30 across, that's 120, and the area of the triangle a bit.

And that again is four high from four to eight and 30 across, but it's half base times height.

So that gives us an area of just 60.

Adding those two together gives 180, so it's 180 metres.

Well done if you got that.

We can find the value of a constant acceleration from a velocity time graph using the change in velocity and change in time.

So I've got a graph here, and between two and four seconds, there's a constant acceleration.

You can tell it's constant, because we've got a straight line section there.

And you may have seen the acceleration equation already.

Acceleration is change in velocity divided by change in time, and that gives us an expression in symbols as A is delta V, delta V for change in velocity there, divided by the change in time T.

So we can use those values like this.

So let's look at between two and four seconds, and we can find acceleration during that section.

The change in velocity, well, it went up from two to five, and the change in time is from two to four seconds, so we can put those values into the equation.

Acceleration is five minus two, so gone up by three metres per second there.

And the time, four seconds minus two seconds.

We've got a time period of two seconds there.

We do that calculation, and that gives us an acceleration of 1.

5 metres per second squared.

Okay, I'd like you to find out an acceleration for yourself now.

I've got our velocity time graph here.

As you can see, there's acceleration going on between one and four seconds, so I'd like you to find the acceleration between one second and four second for the journey shown.

So pause the video, calculate the acceleration, and then restart, please.

Welcome back.

You should have selected 1.

33 metres per second squared, and if we read the values from the graph in this section, we can use the equation A equals delta V over T substituting the two values we've read off the graph for the velocities, and the change in time and gone there.

And that gives us 1.

33 metres per second squared.

Well done if you got that.

Acceleration isn't always a constant value, and we can see that by a change in gradient on a velocity time graph.

So if I've got an object that's got a changing acceleration, I'll end up with a curve on a velocity time graph like this.

And that one there has got the lowest acceleration because the curve isn't changing much, it's gradually increasing.

If I've got a slightly higher acceleration, then I'm going to have a curve that becomes more steep, more rapidly.

So that dotted black line there would be medium acceleration.

And as you'd expect, if I've got a higher acceleration again, I'm going to get a curve that becomes steeper much more rapidly, like that.

Let's see if you can identify the greatest deceleration here.

I've got an object that was moving at six metres per second, but which of those lines shows the object that is decelerating of greatest deceleration? Pause the video, make your selection from the lines below and then restart, please.

Welcome back.

You should have selected D, that becoming steepest more quickly.

The object is reaching a velocity of zero more rapidly, only takes four seconds, so that must have the greatest deceleration.

Well done if you selected that.

And now we're onto the final task of the lesson, and there's five questions in total over these next two slides.

The displacement time graph below shows the motion of a camera drone, and what I'd like you to do is to look carefully at that graph, and find the total distance travelled between nought and eight seconds.

So that's the full length of the graph there.

Then I'd like you to find the greatest speed of the camera drone, identify when that happens, and then work out what that speed is.

Then I'd like you to look at the velocity time graph here, which shows the motion of a boat.

I'd like you to find the total distance travelled between nought seconds and four seconds there.

So the first half of that graph, really.

Then I'd like you to find the deceleration between two seconds and four seconds, between two seconds and four seconds.

There you can see that downward slope.

And finally, I'd like you to try and describe the motion between five seconds and eight seconds.

So pause the video, work your way through those five questions and then restart, please.

Welcome back.

Well, the total distance travelled between nought and eight seconds was 33 metres.

It's a sum of those three phases of motion there.

The greater speed would've happened here with the gradient steepest, and we can calculate that from the change in displacement and the time.

And that gives us nine metres per second, or minus nine metres per second.

I just asked for the speed, so I didn't need the direction there.

Well done if you've got those two.

The total distance travelled between nought and four seconds is the sum of these areas here.

So I've worked out the areas of each of them, added them together.

That gives me 10 metres.

The deceleration between two seconds and four seconds.

I can read the values off the graph, and I find that deceleration, minus one metres per second squared.

Well done if you've got those.

And the final question here, describe the motion between five seconds and eight seconds while the toy boat's accelerating, because you can see the velocity's increasing, but you can see it's a curve there.

So it's not uniform acceleration, it's increasing acceleration.

So the rate of acceleration is increasing over time.

Well done if you got that.

Now we've reached the end of the lesson, so here's a summary.

Motion graphs can show the same information in different ways.

So I've got a displacement time graph there, and constant velocity is shown by a straight line on that graph.

Change in velocity is shown by a curve, and for a constant velocity you can find the distance travelled, S is V times T from readings from the graph.

On a velocity time graph, constant velocity is shown by a horizontal line, and constant acceleration is shown by a straight line.

And we've got an equation for acceleration that we can use if we can read off two values of velocity and time difference.

Well done for reaching the end of the lesson.

I'll see you in the next one.