Calculating from displacement-time graphs: including tangents (v = x ÷ 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 speed or velocity of a range of different objects. This is a set of the keywords and phrases you'll need to understand to get the most from the lesson. First is displacement-time graph, and that's a graph showing the displacement of an object over a period of time, and we'll looking at a wide range of those during the lesson. Second is a gradient, and the gradient is the steepness of a line. We'll measure some of those. Third is instantaneous velocity, and the instantaneous velocity is how fast something's going at a particular moment in time and what direction. And finally, tangent, and a tangent is a line drawn to a curve that allows us to measure the gradient. And here's a set of explanations about those keywords 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 looking at displacement-time graphs and trying to read information so that we can calculate the total distance traveled by objects. In the second part, we'll be reading similar information but then using it to calculate the velocity of an object when it's moving at constant velocities. And in the third part, we're going to be looking at instantaneous and average speed and finding that from the gradient of the graph, which also includes using tangents to find that gradient. So when you're ready, let's begin with finding 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 first and a person at position 0 with no displacement, and I've marked just the forwards direction on this number line. So if I follow a set of instructions for that movement, if I go forwards 2 meters in 10 seconds, you can see I get a line like that on the graph. That's actually showing constant speed during that motion there. And in 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 the graph. And then move again, this time moving forward's 5 meters in 20 seconds. So my displacement's gone up now, it's got a total displacement of 7 meters. And in the final bit of movement, forwards 2 meters in 20 seconds. So you can see there, the final displacement is 9 meters on the graph after 60 seconds. The distance traveled is also the sum of those distances. So I've got 2 meters, 5 meters, 3 meters, so the total distance traveled is 9 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 and stopped for a while, then another 200 meters, so the total distance traveled is 600 meters there. Well done if you've 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 0 and I've got movement towards the right here as positive. I can show different movements on the graph. So first of all, let's move right 8 meters. So there's 8 meters movement in 20 seconds, and you can see a constant speed on the graph there, a nice straight line. When I 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 5 meters in 15 seconds. And you can see in that instance, the displacement's gone down. The distance of travel though has increased so they're no longer the same value. The third bit, stationary for 5 seconds, and then another movement towards the right there, 3 meters in 20 seconds. You can see the final displacement is 6 meters according to the graph, but that's not the same as the distance I traveled. I traveled 8 meters at first, then another 5 meters, and then another 3 meters. That gives me a total distance I've traveled of 16 meters. But as I've said, the final displacement is just 6 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 changes of direction. So I've got, is it A: 2 meters, B: 6 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 2 meters, they moved 6 meters forward at first according to the graph and then another 4 meters backwards. You can see the displacement has decreased by 4 meters there, but the total distance that Sam moved is 6 meters plus the 4 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 we'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 0 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 0, so definitely not the same as the distance traveled. So now a slightly more complicated example for you. I'd like you to find again the total distance traveled, this time by Sam walking. Is it 3 meters, 30 meters, 23 meters or 25 meters? Pause the video, work that out, and then restart, please. Okay, welcome back. Well, the total distance traveled there was 23 meters. If we look at the graph, we can see each section of movement. There's a forwards movement of 5 meters there, a backwards movement of 5 meters, then another backwards movement of 5 meters, and another forwards movement of 5 meters, and finally a further 3 meters forward at the end. That gives us a grand total of 23 meters. So well then if you got that. Okay, it's time for the first task of the lesson now, and what I'd like you to do is to draw a displacement-time graph based upon the movement instructions shown in the box here. Once you've drawn the graph, I'd like you to state the final displacement after the journey. So you should be able to read that off the graph quite easily. And the third thing I'd like you to do is to calculate the total distance traveled during that journey. So pause the video, try those three questions and then restart, please. Hello again. And your graph should look something like this. As you can see, there's quite a lot of different phases during the movement, so you've got to be quite careful to draw it. I've got one, two, three, four, five, six different phases of movement there and in two different directions. The final displacement when you read it off the graph is 300 meters. So if you just look at the end point of the graph back to the displacement axis, you can see we are 300 meters west of the starting point. And the total distance moved is the sum of all those individual movements that were carried out, 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. The average velocity of an object can be found using information about its change in displacement and time and using the equation here, which you may have seen before or variations of it. Average velocity is change in displacement divided by time. If we write that out in symbols it's usually written as v equals x divided by t. And change in displacement is represented by x, which is measured in meters. Velocity is v, measured in meters per second, and time, t, is measured in seconds. Now if you look carefully at the equation, you should notice that displacement and time, the two values that we need to calculate velocity can both be found on displacement-time graphs, and that's exactly what we're going to do to find velocity. So let's find the average velocity for a complete journey first so we've got a graph here showing the complete journey of an object and we need to read two values from it. We need to read displacement and time. So, starting with displacement, we can just look at the end of the graph, the final point looking across that gives us a displacement of 600 meters. So x is 600 meters. Similarly, we need to find the time it took for that journey and looking down from the final point in the graph, that's 500 seconds, so t is 500 seconds. And then we can calculate the average velocity quite simply, write out the equation using symbols to save a bit of space: v equals x divided by t, substituting those two values and then complete the calculation 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. Hopefully you found the answer was 0. 05 meters per second. And if you look at the graph and find the end point, you'll find the final displacement was 2 meters looking across there. And looking down, you'll find the final time was 40 seconds. If we substitute those into the equation v equals x divided by t, that gives us a final answer of 0. 05 meters per second. Well done if you got that. We can also find the average velocity for part of a journey by looking at the change in displacement and the change in time. So let's try that here. If you look at just this section of the journey, between 300 seconds and 500 seconds. and we find the change in displacement first, so looking at that section, you can see the displacement's gone up from 400 meters to 600 meters and we can calculate that as a 200 meter change. The next thing we do is find the change in time for the section of the graph, and you can see, again, we've got a change in time of 200 seconds, up from 300 seconds to 500 seconds. And then we substitute those two values, those two readings from the graph, into the equation v equals x divided by t, put them in, and we get an average velocity for just that section of the graph, of 1 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 a 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, welcome back, let's go through the answer to that, and it was 6. 7 meters per second, and the way we found that answer was looking at the graph and we can see there's a change in time of 30 seconds. It's gone up from 120 to 150 seconds and there's a change in displacement of 200 meters, up from 100 meters to 300 meters. We can substitute those two changes into the equation, and we get a velocity of 6. 7 meters per second. Well done if you got that. Sometimes we've got situations where the displacement is decreasing, and that going to gives us negative velocities moving towards the origin point. So let's have a look at an example of that. We've got a section here of 100 seconds between 200 and 300, and you can see that the displacement has gone down during that section. It was 600 meters and it decreases down to 0 meters back to the start. So we can find the change in displacement, by looking carefully and seeing it's a negative value, it's minus 600 meters and we've gotta represent that by using the minus sign. We then can find the change in time. The change in time here is 100 seconds, And then we can just put those two values into the equation remembering to include that minus sign for the displacement or change in displacement. So putting those values in, that gives us a value of -600 meters per second. So we have got negative velocity here the displacement is decreasing during that time. 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 what you should have selected was 0. 2 meters per second. If you look carefully at the graph, you can see there's a change in time of 20 seconds and a change in displacement of -4 meters. So substituting those into the equation gives us a velocity of -0. 2 meters per second. Well done if you got that. It's time for the second task of the lesson now, and what I'd like you to do is to calculate some average velocities based on this graph here, showing the motion of a robotic arm. So I'd like you to calculate the average velocity for the complete 60 seconds of movement, please, then the average velocity between 0 and 30 seconds, and finally the average velocity between 40 and 50 seconds. So pause the video, work those out and then restart, please. Okay, welcome back, and let's look at the complete journey first. You can see there's a final displacement of -3 meters and a time of 60 seconds. That gives us a velocity of -0. 05 meters per second using those two values. Then we are just for the first 30 seconds of movement, you can see the displacement is 2 meters at the end of that and we've got an increase overall then of 2 meters and 30 seconds and that gives us a velocity of 0. 07 meters per second. And for just this section between 40 and 50 seconds, we have a change in displacement of -5 meters in a time of 10 seconds. And that gives us a velocity of 0. 5, sorry, -0. 5 meters per second. Well done if you got those three. 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 a 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 6 meters per second there, so the velocity is constant at 6 meters per second. So at any time between 0 and 100 seconds, the velocity is going to be 6 meters per second. In this section here, again I can find the velocity is a constant 4 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 6 meters per second and 4 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. Hopefully you chose 0. 3 meters per second. To find the instantaneous speed, we look at this section of the graph where the 30 seconds is, so 30 seconds is right in the section, in that section of that line, and we can find that there's a change in time of 20 seconds, and a change in displacement of 6 meters, and we can use those to find the instantaneous speed of 0. 3 meters per second. And it was okay to use the 6 meters instead of -6 meters because I just wanted the speed there so the direction wasn't important. 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 is 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 0 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 is 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. A tangent is used to find the instantaneous velocity of speed when the gradient of the line is changing on a displacement-time graph. So I've got a displacement-time graph here and it's a curve. And to find the instantaneous speed, I need to draw a tangent to the line that just touches it at 90 degrees. What I mean by that is a line like this. This purple line just touches that curve at 9 seconds and it's at 90 degrees to it, and it's a straight line I've drawn with a ruler. If I tried to draw a tangent at time equals 3 seconds, I'd end up with a line like this, this red line. And again, that just touches the curve and is at 90 degrees to it at exactly 3 seconds. The gradient of that tangent line is what's going to give me the velocity at that time. So I need to draw these lines and then I need to find the gradient of them. Okay, I'd like to see if you understood what a tangent is, I'd like you to decide which of those lines shows a tangent to the curve at 11 seconds, please. So pause the video, make your decision, and then restart please. Okay, welcome back. You should have selected A, the red dashed line, that just touches the curve at 11 seconds and is at 90 degrees to it. Well done if you selected that. Now once I've drawn a tangent, I can find the speed at a particular point. So I've drawn the tangent line drawn in blue here, and what I need do is to select two different points along that line and ones I can read easily off the graph. So I'm gonna select this point here, and this is the first point, and it's at 14 seconds. And the displacement at that point is 9. 2 meters. So I've written those two values down. Then I select a second point that's also along that blue tangent line. And I'm gonna select this one because it's easy to read off the graph, and there I've got a time of 4. 3 seconds and a displacement of 0 meters. So I've got my two values. What I can do then is can calculate the speed from the change in displacement and the change in time. So the speed is the change in displacement divided by the change in time. And my change in displacement is the difference between those two displacements, and my change in time is the difference between these two times. So I've written those out here. So what I'd like you to do now is to try and estimate the instantaneous speed of this object at time equals 20 seconds. So pause the video, try that, and then restart, please. Okay, welcome back. And the first thing we'll need to do is to draw a tangent line. And once I've drawn the tangent line, I can pick two points and record the values of those two points along the line and you can pick any two points that easy to read off. And from that, I can then do the calculation, giving me a speed of about 0. 15 meters per second. Your answer might not be exactly the same because it can be quite difficult to choose points and draw that tangent line perfectly. But well done if you got close to that. And now it's time for the final task of the lesson, and I've got a graph here showing the vertical movement of a drone. So you can see it's moving up there, but it's not going at a constant speed or a constant velocity. What I'd like you to do is to calculate the average velocity for the complete journey of that drone for the 500 seconds time it was moving. Then I'd like you to draw a tangent and find the instantaneous speed at the time of 250 seconds, please. So pause the video, try and find the answers to those two and then restart. Welcome back. And to find the average velocity for the completed journey, your calculation should look something like this. And to find the instantaneous speed of the drone you drew a tangent like this, read off the change in height, 550 meters and the change in time, 500 seconds, and that should give you a calculation that looks a little like this, giving a average, sorry, an instantaneous speed of 1. 1 meter per second. Well done if you got that. Right, we've reached the end of the lesson now, so here's a quick summary of everything we've covered. Displacement-time graphs show positive and negative displacement, as you can see on the example graph there. The total distance traveled is equal to all of the changes in displacement added together. And as you can see, the distance traveled and the final displacement are not the same. And the graph I've got there, I've traveled 23 meters, but I'm only 3 meters away from the starting point. Instantaneous velocity is equal to the gradient of a displacement-time graph or to the gradient of a tangent if we're dealing with a curved graph. Instantaneous speed is equal to the magnitude, the size of the gradient. So, well done for reaching the end of the lesson, and I'll see you in the next one.