Calculating from displacement-time graphs: including tangents (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 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 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 be looking at a wide range of those during the lesson. Second is the 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. 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 or 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. The 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 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 two meters in 10 seconds, you can see a line like that on the graph. 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 the graph. And I 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 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 gone two meters, five meters, 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 going to 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 eight meters, so there's eight meters movement in 20 seconds, and you can see constant speed on the graph there, a nice straight line. I'm gonna move again, so 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 going to 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 of travel, 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, the 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 travel 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 out the total distance traveled by Sam according to the graph, bearing in mind changes of direction, so I've got 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 is 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 displacement, but we can still work out the total distance traveled by analyzing the movement. So, I've got movement here. If I look at the first section of the graph, 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 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 three meters, 13 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 second of movement. There's a forwards movement of five meters there, a backwards movement of five meters, then another backwards movement of five meters, and another forwards movement of five meters, and finally, a further three meters forward at the end. That gives us a grand total of 23 meters, so well done 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 endpoint of the graph, back to the displacement axis, you can see we're 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. 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 combine that as a set of symbols as v equals s divided by 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 looking at how you can find the velocity for a complete journey or the average velocity for a complete journey. And 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 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. 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. 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, and 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 a section between 120 seconds and 150 seconds. I'd like you to find the average velocity for just that section of route 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 got to 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. 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. Welcome back. Well, let's look at the complete journey first. So, we've got a final displacement of minus three meters at a time of 60 seconds. Substituting those in gives us a velocity of minus 0. 05 meters per second. Then we're just looking at this section of the graph, the first 30 seconds of movement, and you can see the displacement at the end of that is two meters, and we've got 30 seconds, so we've got a velocity of 0. 07 meters per second. And in the final part between 40 and 50 seconds, you can see there's a change in displacement of minus five meters in a time of 10 seconds. That gives us a velocity of minus 0. 5 meters per second. Well done if you've got all of 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 0 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 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. 2 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 four meters, so we've got 20 seconds of four 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. 2 meters per second. Well done if you've 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 is 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 the 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. 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 or 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 nine 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 three seconds, I 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 three 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 gradients of them. Okay, I'd like to see if you understood what a tangent is. I'd like to decide which of those lines shows a tangent to the curve at 11 seconds please. So pause the video, make your decision, 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. So, once I've drawn a tangent to the curve, what I need to do next is to find the gradient of that tangent line, so I've got a tangent line drawn in blue here. And to find its gradient, what I do is select two different points that are along that line, and I choose them so that I can easily read off the values from the axis, the time and displacement values. So, I'm gonna select this point here and I'll look carefully at that point, I can find out the time is 14 seconds and the displacement is 9. 2 meters. Then I pick a second point. I'm gonna pick this point here down along the axis. And the time is 4. 3 seconds and the displacement is 0, so I've got those two values. The next thing I do is I can calculate the speed from the change in displacement and the change in time between those two points. So, the speed is the change in displacement divided by the change in time. I put those values in, the change in displacement is 9. 2 meters minus 0 meters, and the change in time is 14 seconds minus 4. 3 seconds, so it's the difference in displacement and the difference in time I put in there. 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. Well, to do that, we'll draw a tangent to the black line there at 20 seconds, so it's just touching the line at 20 seconds. I can write out the values for two points along that line, and that gives me an approximate speed of 0. 15 meters per second there. There'll be some variation because it's not that easy to draw tangents. 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. Okay, welcome back. And to find the average velocity for the complete journey, your calculation should look something like this. To find the instantaneous speed of the drone at time 250 seconds, you should have drawn a tangent similar to this one, read off the values for the change in height and the change in time, and that will give you a calculation giving you an instantaneous speed for the drone of 1. 1 meter per second. Well done if you've got those two. 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. In the graph I've got there, I've traveled 23 meters, but I'm only three 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.