Calculating from displacement-time graphs (v = s ÷ t)

Hello there, my name's Mr. Forbes, and welcome to this lesson from the measuring and calculating motion unit. In the lesson, we're going to be analyzing different displacement-time graphs and using them to calculate speed and velocity. By the end of this lesson, you're going to be able to look at displacement-time graphs and take values from those graphs in order to calculate the speed or velocity of a range of different objects. Here are the three keywords and phrases that you'll need to understand to get the most from this lesson. The first is "displacement-time graph," and a displacement-time graph shows you the displacement of an object over a period of time. We're gonna be looking at a wide range of those in the lesson. The second is "gradient," and that's the steepness of a line on those graphs. And the third is "instantaneous velocity," which is the velocity at a particular moment in time. And here's a set of definitions of those keywords and phrases that you can return to at any point during the lesson. This lesson's in three parts, and in the first part, we're going to be reading information from displacement-time graphs and calculating the total distance traveled by different objects. In the second part of the lesson, we're gonna be reading similar information, but this time, we're gonna be calculating velocity based on that information. And in the third part of the lesson, we're going to be calculating average and instantaneous speed by looking at the gradients of lines on the graph. So when you're ready, let's start with calculating distance traveled. Displacement-time graphs can show positive and negative displacement. So you can see I've got a graph here, showing the movement of an object, and it's got both positive displacement in the first section there, negative displacement in the middle section, and then positive displacement again in the end section. Those positive and negative numbers show opposite directions, so opposite displacements, and they can be things like north and south or east and west, or even left and right. As long as those two words mean opposite things, then I'm okay. So here's a nice easy question to start with. This graph shows positive and negative displacement. A positive displacement on the graph represents up; I'd just like you to decide: what does the negative displacement represent? So is it sideways, down, or further up? Pause the video, make your selection, and restart. And as I said, that was a nice easy one. If the positive displacement is up, then obviously, the negative displacement is going to be down. So well done, if you got that. A displacement graph doesn't just show displacement. It can also be used to work out total distance traveled. So we'll go through an example of that here. So I'm gonna start with a displacement-time graph like this. It's empty at first, and a person at position zero with no displacement, and I've marked just the forwards direction on this number line. So if I follow a set of instructions about movement, if I go forwards two meters in 10 seconds, you can see I get a line like that on the graph. And that's actually showing constant speed during that motion there. And in the second section, well, stationary for 10 seconds. And you can see the little flat part of the line there means the displacement isn't changing over those 10 seconds. So stationary for 10 seconds looks like that on a graph. And then move again, this time moving forwards five meters in 20 seconds. So my displacement's gone up now. It's got a total displacement of seven meters. And in the final bit of movement: forwards two meters in 20 seconds. So you can see there, the final displacement is nine meters on the graph after 60 seconds. The distance traveled is also the sum of those distances. So I've got two meters, five meters, and three meters. So the total distance traveled is nine meters as well. So the displacement and the distance traveled is the same, because I've not changed direction at all during that motion. A quick check now: I'd like you to work out the total distance traveled by a car according to the graph here. Is it A, 200 meters; B, 400 meters; or C, 600 meters? Pause the video, make your selection, and restart, please. Okay, welcome back. Well, in this journey, it's 600 meters. The car traveled 400 meters, then stopped for a while, then another 200 meters. So the total distance traveled is 600 meters there. Well done if you got that. Now in the examples we've seen so far, the final distance traveled is equal to the total displacement, but that's only because there's no changes in direction. Let's have a look what happens when there is a change of direction in the motion. So again, I'm gonna be starting at zero, and I've got movements towards the right here as positive, and I can show different movements on the graph. So first of all, let's move right eight meters. So there's eight meters movement in 20 seconds, and you can see a constant speed on the graph there, a nice straight line. Gonna move again. Sorry, I stay stationary for 10 seconds, so there's no change in displacement there. And then the third part of the movement is, I'm gonna move backwards. I'm gonna move left five meters in 15 seconds. And you can see, in that instance, the displacement's gone down. The distance I've traveled, though, has increased, so they're no longer the same value. The third bit: stationary for five seconds, and then another movement towards the right there, three meters in 20 seconds. You can see the final displacement is six meters according to the graph, but that's not the same as the distance I traveled. I traveled eight meters at first, then another five meters, and then another three meters. That gives me a total distance of traveled of 16 meters. But as I've said, the final displacement is just six meters to the right. So when there's a change in direction, the final displacement and the distance traveled are not the same. Okay, I've got a graph of motion for Sam here. I'd like you to work at the total distance traveled by Sam according to the graph, bearing in mind the changes of direction. So I've got, is it, A, two meters; B, six meters; C, 10 meters; or D, 12 meters? Pause the video, make your selection, and then restart, please. Okay, welcome back. And the answer to that was 10 meters. Although Sam ended up with a final displacement of just two meters, they moved six meters forward at first, according to the graph, and then another four meters backwards. You can see the displacement has decreased by four meters there, but the total distance that Sam moved is six meters, plus the four meters, and that's 10 meters. Well done if you got that. As you've seen in some earlier examples, displacements can be negative as well as positive. So I've got a graph here showing a negative displacement as well as positive displacements, but we can still work out the total distance traveled by analyzing the movement. So I've got a movement here. If I look at the first section of the graph, I've moved, well, I've decided to call the positive displacement north here, so north 600 meters in 100 seconds. And then stayed still for a bit, stationary for 100 seconds. Then I moved south. So I'm getting back to zero displacement here. Moved south 600 meters in 100 seconds. Then further south, and that's what gives me negative displacement. I'm south of my starting point now. So, south 400 meters in 100 seconds. Then I stopped again for 100 seconds, and then moved north 400 meters in 100 seconds. So the total distance I've traveled there is the sum of all those separate distances in each phase of movement. So the distance traveled is 2,000 meters there. But my final displacement is zero, so definitely not the same as the distance traveled. So let's see if you can find another example of total distance traveled. I've got a bicycle riding here, and I'd like you to find the total distance traveled by it over those 180 seconds. Is it 100 meters, 200 meters, 300 meters, or 500 meters? Pause the video, make your selection, and restart, please. Okay, welcome back. Hopefully, you chose 500 meters. As you can see, there are three separate phases of movement here where the distance, sorry, the displacement's changing: I move forwards 200 meters, backwards 100 meters, then forwards, again, 200 meters. That gives me a total distance traveled of 500 meters. The sum of those values. Well done if you got that. Okay, now it's time for the first of the tasks. I'd like you to have a look at the information shown in the box there and draw me a displacement-time graph based upon those movement instructions. I'd like you to also state the final displacement after the journey, and you'll see that from the graph. And third of all, calculate the total distance traveled during that journey, please. So pause the video, draw that graph and answer the other two questions, and then restart, please. Welcome back, and here's the graph you should have drawn. As you can see, there's quite a few different phases of movement there. I've got movement forward, stopped, backwards, forwards, stopped, and then backwards a little bit again. So well done if you've drawn a graph that looks that shape. The final displacement, if you drew the graph accurately, should be 700 meters, and you can see that on my graph as well. And finally, the total distance traveled is gonna add together all of the movements of each phase. Well done if you got that. Now it's time for the second part of the lesson, and in it, we're going to be looking at how to calculate velocity by reading information from displacement-time graphs. Let's have a look at how we calculate velocity. The average velocity can be found using this equation, which you may have seen before. Average velocity is change in displacement divided by time. So we can write that as a set of symbols as "v = s ÷ t." And the change in displacement is s, measured in meters. Velocity is measured in meters per second. That's symbol v. And time, t, is measured in seconds. From that, you can see that both displacement and time can be read from a displacement-time graph, and that's what's going to allow us to calculate average velocity. Let's start by finding out how you can find the average velocity for a complete journey. As you saw in the equation, we need to have a displacement and a time. So all we really need to do is to find the displacement from the graph. So, looking at this graph for the complete journey, after 500 seconds, the displacement is 600 meters. So we get s equals 600 meters. I can also look at the end of the graph and find the total time for the journey, and that was 500 seconds. So t is 500 seconds. I can then calculate the average velocity for the complete journey by writing out the equation: v equals s divided by t, substituting those two values I've taken from the graph, and then calculating the final answer, which gives me 1. 2 meters per second. Okay, let's check if you can find the average velocity for a journey. I've got a journey here that took 40 seconds. I'd like you to find the average velocity for it, please. So pause the video, try and work out the average velocity, and then restart, please. Okay, welcome back. Hopefully, you selected 0. 05 meters per second. And you should see that the displacement there was two meters at the end of the journey, and the time taken was 40 seconds. So if we substitute those into the equation, we get 0. 05 meters per second. Well done, if you found that. You can also find the average velocity for part of a journey instead of the complete journey just by looking at the changes in the values for the displacement and the change in the value for the time. So we're gonna do that here with an example. We're gonna look at this part of the journey from 300 seconds to 500 seconds. So we start by finding the change in the displacement during that time. And as you can see, the displacement was 400 meters and then went up to 600 meters. That gives us a change in displacement of 200 meters, and I've marked it on the graph. You can find the change in time, and as I mentioned, we've got from 500 seconds and 300 seconds. That gives us a change in time of 200 seconds there. And then we substitute those values into the equation just like we do for the other calculations. So write out the equation, put the values in, that gives us an average velocity for that part of the journey of one meter per second. Okay, let's see if you can find an average velocity for a section of movement. I've got a graph here, and I've highlighted the section between 120 seconds and 150 seconds. I'd like you to find the average velocity for just that section of movement, please. So pause the video, work that out, and then restart. Okay, let's have a look at the solution to that, and it's 6. 7 meters per second. And we can find that by taking readings from the graph. There's a change in time of 30 seconds there and a change in displacement of 200 meters. And so we can substitute those two values into the equation, giving a value of 6. 7 meters per second. Well done if you got that. Sometimes displacement is decreasing, and that gives us negative velocities, and we have to be very careful with doing calculations involving negative velocities because we've got negative numbers. So let's have a look at an example of that. We've got a section of a graph here, and you can see clearly that the velocity is negative, because the displacement is decreasing between 200 and 300 seconds there. So we can find the change in displacement, and this time we've gotta be very careful because it's a negative value. We've gone down by 600 meters, so we'd represent that as minus 600 meters. And then we can look at the change in time, and the change in time is 100 seconds, time's always positive there. So then we can calculate the velocity by substituting the values into the equation as we've done before. But being very careful, because we've got to use that value of minus 600 meters. So we've got minus 600 meters divided by 100 seconds, that gives us minus six meters per second. And the minus is telling us we've got negative velocity with moving back to the origin point. So I'd like you to find the average velocity for the highlighted section of the graph here between 20 seconds and 40 seconds. So pause the video, find the velocity for that section, the average velocity, and then restart, please. Okay, welcome back. And you should have chosen minus 0. 2 meters per second. You can see there's a change in time of 20 seconds and a change in displacement of minus four meters there. We substitute those into the equation, and that gives us minus 0. 2 meters per second. Well done if you got that. And now it's time for the second task of the lesson, and I've got a graph showing the motion of a robotic arm here, and you can see it's a displacement-time graph. What I'd like you to do is to calculate the average velocity for the complete 60 seconds of action there. Then I'd like you to calculate the average velocity between 20 and 30 seconds. And finally, the average velocity between 40 and 50 seconds. So pause the video, work out those average velocities, and restart, please. Welcome back, and let's have a look at the complete journey first. So for the complete journey, the total displacement is six meters and 60 seconds. So we can substitute those into the equation. We'll get a velocity of 0. 1 meters per second. For the next section of the graph, between 20 seconds and 30 seconds, just this section here, we can see there's a change in time of 10 seconds and a change in displacement of three meters. So we get a velocity of 0. 3 meters per second. And for this section of the graph here, we get a change in time of 10 seconds, and a change in displacement of minus two meters. So that gives us a velocity of minus 0. 2 meters per second. Well done, if you've got those. And now it's time for the final part of the lesson, and in it, we're going to be looking at how we can use the gradient of a displacement-time graph to find the instantaneous speed. The instantaneous velocity of something is how fast it's going at a particular moment in time and in what direction. And we can find that from the gradient at a specific time on a displacement-time graph. So I've got a graph here, and there's several different sections to it. And in each of the sections, the object's moving at constant velocity. So it's giving me a straight line there. And the instantaneous velocity is going to be the same as the average velocity for that section. So if I can find the average velocity, that will give me the instantaneous velocity as well. So, for example, we can look at this first section here, and we can see that there's 100 seconds of time passed and the object's moved 600 meters, and I can get a velocity of six meters per second there. So the velocity is constant at six meters per second. So at any time between naught and 100 seconds, the velocity is going to be six meters per second. In this section here, again I can find the velocity is a constant minus four meters per second. So we've got the velocity there. The instantaneous speed is going to be the same as the instantaneous velocity, but without the direction. So I've got an instantaneous speeds there of six meters per second and four meters per second. So I'd like you to find the instantaneous speed at 30 seconds on this graph, please. So find the time 30 seconds, and work out what the instantaneous speed would be there. Pause the video and then restart when you're done. Okay, welcome back. And you should have found the instantaneous speed there is 0. 3 meters per second. If we look at this section of the graph, where the 30 seconds is, in the middle of that section there, then we can find that the time in that section is 20 seconds. The change in displacement is six meters. So we've got 20 seconds and six meters there, so we can find the instantaneous speed. Remember, that doesn't have direction, which is why I'm using six meters, and I get an instantaneous speed of 0. 3 meters per second. Well done if you got that. In the graphs we've looked at so far, we've just looked at constant velocities or constant speeds, and that gives us straight line sections on a graph. But on many graphs, there are curved lines showing that the velocity's actually changing. So in this graph, I've got a series of curves, and in this first section, you can see the gradient is increasing in those first 200 seconds; the object is speeding up, its speed is increasing. And in this section of the graph and the object is actually slowing down; the gradient is decreasing. So the speed and the velocity is decreasing there. Objects can speed up as they're moving towards you as well. And in this graph, I've got an object changing its speed as it's moving towards you. So in this section of the graph, the gradient of the graph is becoming more steep, and so the object's speed is increasing; the velocity is becoming more negative. We've got negative velocity here because the object is moving towards you. In this section of the graph, the object is slowing down; the gradient's becoming shallower, and the speed is decreasing there. And as you can see towards the end, it's becoming almost flat. The velocity is approaching zero at the end there. Okay, let's see if you understand what a curve on a displacement-time graph represents. So I've got a graph here of a roller skater, and what I'd like you to do is to identify in which of those four parts the speed of the roller skater is increasing. Not the velocity, the speed of the roller skater. So pause the video, select whichever options you think are correct, and then restart, please. And welcome back. You should have selected A. In that section, you can see the graph is becoming more steep, so they're speeding up. The velocity's actually increasing there. And also, C, because at that point, the velocity is in the opposite direction. But again, we've got the graph becoming steeper in a downwards direction, so we've got an increase in speed there as well. So well done if you selected those two. And now it's time for the final task of the lesson. And what I'd like you to do is to have a look at this graph here, which shows the vertical movement of a drone. And I'd like you to calculate the average speed for the complete journey, the full 500 seconds of movement there. Then I'd like you to calculate the instantaneous speeds at these points: t equals 50 seconds, or time equals 50 seconds; t equals 250 seconds; and t equals 450 seconds. So pause the video, work out the answers to those, and then restart, please. Okay, welcome back. And here's the solution to the first part of that: the average speed for the complete journey. So I'm looking for average speed here, so I need to work out the total distance traveled. As we learned earlier in the lesson, we add up all those small distances, and that gives us a total distance traveled of 1,300 meters, and we divide that by the time, which was 500 seconds. And that gives us an average speed of 1. 4 meters per second. Well done if you got that. And for question two, we have the solutions here. And as you can see, for each section, I found the time and the change in displacement and have calculated speed for that section. So the average speed for each of those sections because that's going to be equal to the instantaneous speed as well. So well done, if you've got these three. Now we've reached the end of the lesson, and here's a quick summary of all the information we've covered. So displacement-time graphs can show positive and negative displacement. The total distance traveled in any journey is equal to all of the changes in displacement added together. Total distance is usually different from the final displacement, because the direction of travel can change. As you can see in this graph, the direction of travel is changing several times. I've got a distance traveled of 23 meters, but a final displacement of only three meters. Instantaneous velocity is equal to the gradient of a displacement-time graph at a point, and instantaneous speed is equal to the size of that gradient. So we've calculated several gradients during the lesson. Well done for reaching the end of the lesson. I'll see you in the next one.