 We're going to see if we can take this displacement time graph and work out what the velocity time graph, which corresponds to this graph, looks like. To do that, we're going to use the idea of tangents. A tangent is a straight line, so we're going to use a ruler to draw it, which touches the graph at only one point. And the slope of that tangent line that we draw will be the velocity at the point where the tangent's touching the graph. And I think that's one of those things where you hear it said it doesn't make a whole lot of sense, but when you see it in action, it's actually pretty sensible. So what I'm going to do to start off with is pick a point off my graph where I want to know the velocity. And I think I would like to know how fast this object is moving. When it's at, I don't know, I'm going to go to 0.5 seconds. You can pick a different time if you like on your graph. So at 0.5 seconds is right here on my graph. I'm going to use my ruler, and I'm going to draw a tangent line in. To draw a good tangent, you should have about the same distance between the edge of the ruler and the curve on either side. It should be kind of like splitting the difference, I suppose. So here I've got about the same distance between the ruler and the curve and the same sort of here. It's not an exact sort of science when we're doing it in Physics 20. There's a whole set of mathematical rules for drawing tangents that we don't even begin to get into. And mine could have been maybe a little bit more over, but it's going to work. It's going to be just fine. All right, so draw a tangent on your graph. The next thing we're going to do is we're going to work out the slope of that tangent. And to do that, we need two points off of the tangent line. So I'm going to go and I'm going to label two points here, which are more or less easy to read. And I'm going to put a box around them, a square, to identify that these aren't plotted points, but these are points off my tangent line. I'd like you to put the squares as well. Now you don't have to pick the same points as me, because our graphs might look a little different because we might have used slightly different increments. You might not get the exact same numbers, and that's totally fine. But I'm going to choose my first point here at 0.25 seconds. All right, that's what the x value or the time value is. And that's at 2.24, so 2.4 meters. There's my first tangent point. And I'm going to pick my second tangent point up here, put a box around it again, a little square, and I'm going to record that as 0.75 seconds, and it's 5.2 meters. All right, so to recap, so far I have selected a point on the graph, which I want to know the velocity. I decided I wanted to know the velocity at 0.5 seconds, and I put a tangent line down, touching the graph at 0.5 seconds. I made sure that the gap between the line and the curve was about even on either side, and now I've picked two points off the tangent. Now I'm going to use these points to work out the slope of the line. The easiest way to do this is to label the points with x1 and y1. Doesn't matter which one you decide to call 0.1 and which one you decide to call 0.2. So I'll, just for fun, call this bottom one x1, y1, and the top points I'll call x2, y2, and now off to the side I'm going to go and find my slope. So I'll sort of say slope of tangent at 0.50 seconds. That's what I'm finding. The slope formula is on your datasheet, so you don't have to worry about remembering that when it comes time to doing your quiz. It's y2 minus y1 over x2 minus x1. And if you did like I did and you labeled your points, it's pretty easy to go and just pop them into the formula. You don't have to worry too, too much about mixing them up. And it's very important that you put in the units. You'll notice that I'm not just putting in the numbers here. I'm putting in the meters, and I'm putting in the seconds, and I need you to do the same thing. Then I'm going to type that through my calculator, and I am getting a slope of 5.6 meters per second. Now that slope of the tangent line corresponds to the velocity at 0.50 seconds. So I can say the velocity at 0.50 seconds is equal to 5.6 meters per second. Now I know how fast the object was going at that one moment at 0.50 seconds. So I can put that piece of information down on my velocity versus time graph. So I'll quickly go and set up my v versus t as my velocity. There's my time. To make my life easier, I'll kind of use the same time increments. And I'm going to need a velocity of 5.6 meters per second. So I think what I'll do here is I'll go and count off some ticks. So I think every 5, I'll make one box sort of like 0.2. So that'll be 1 meter per second, 1, 2, 3, 4, 5, 2 meters per second, 3, 4, 5. So now I'm going to put my first point down on my velocity time graph. So one of the really common questions when you do this is how do we know where to put a point on this velocity time graph? What time do we use? What velocity? Well, we're going to use for our time 0.50 seconds because that's where we chose to draw our tangent on the curve. And for our y value, our velocity, we're going to use 5.6. So I'm going to go over to 0.50 seconds on my velocity time graph. And I'm going to go up to 5.6. 5.2, 4, 6, right there. So here's where that information went. I took my slope, I took the time that I found the slope, and I turned that into a point on the graph. Now it's only one point, we still need at least another point so we can make a graph out of it. There's your first step. What a great question. If I didn't draw a tangent line, but instead I picked two of the points that I plotted on the graph, will I still get a velocity? You will, but it won't be the correct velocity. You'll have to draw these tangent lines to know how fast the object's moving at any given one point in time. When you pick two points off of the graph, those are pieces of information about two different points in time. So it won't tell you how fast it's going at any one given time. And you wouldn't have any spot to put on your velocity time graph. You'd have to choose between the two times that you put down, it wouldn't work out. So that's why we have to make the tangent. That was so much fun, let's do it again. We've done one tangent, we now have to do a second tangent. Now just to make things a little easier to keep things sort of symmetrical, I'm going to make my next tangent at 0.75 seconds. So I'm going to go over here to 0.75, there we go. This is where I'm going to draw my tangent. Oh thank you, 1.5. Try to get my tangent looking nice, there we go. Not bad. I'm going to pick two points off again and label them. So I have drawn another tangent. The second tangent that I drew was at 1.5 seconds. So I'm going to label again what I'm doing. So slope of tangent at 1.50 seconds. And the two points that I pulled off of my tangent line are at 1 second and 4.8 meters and 1.75 seconds and 2.3 meters. So I'm going to do another slope calculation. Don't forget to put your units in as you're doing this. So I'm getting negative 3.3 meters per second at a time of 1.5 seconds. That's the second point that I'm going to graph on my velocity time graph. And annoyingly, I set up my velocity time graph without thinking too hard. I didn't leave myself much space in the negative on the grid. But that's OK. I'll just extend it down some. Negative 1, negative 2, negative 3, negative 4. That'll work. So now I'll go to 1.5 seconds, which is right here. And I'll go down to about 3, negative 3.3, put a point down. So we now have two points and we can connect those two points with a line. And that's what our velocity versus time graph looks like. It is a nice negative slope, showing that the object is going to be slowing down, slowing down, slowing down, reaching a velocity of 0, and then speeding up, speeding up, speeding up. So this is the procedure you're going to need to use in order to go from making your displacement time graph into a velocity time graph. The last step to get your acceleration time graph is a little bit quicker. And it's very similar. You'll remember from last class we were discussing that the slope of a velocity time graph gives you the acceleration. So all I need to do is find the slope of this graph just one time. I don't need to do a tangent. I don't need to do multiple slopes. I'm just going to find the slope one time. And that'll tell me where to draw my horizontal acceleration time graph. So I'll put down a couple of points. Again, I'm not using those plotted points. I'm using points in boxes that I choose off of the line of best fit. So that's 0.75 seconds, comma 3.2 meters per second. I'll use that as x1, y1. And I'll pick another point that actually shows up in my grid. I don't know. I'll go maybe point. I'll go to 1. It's about 1.0 seconds. I've got this thing going at about 1.2 meters per second. You can pick different points than I am. That's just fine. Pick any two points off of your line that are not your plotted points and work out the slope. So I'm getting negative 8.8 meters per second squared. Pardon me, meters per second squared is my slope. And the slope of your velocity time graph is the acceleration. Now that's not too bad. Is it perfect? No. I would have liked that to be negative 9.81 meters per second squared, because that is what the acceleration should be. But it's pretty close. Pretty close for me graphing by hand. Now to do our acceleration time graph, thankfully, it is very fast. Thankfully, it is very easy. So let's quickly go and do that acceleration time graph. You're going to like this one. We know the shape and acceleration versus time graph should be. We know it should be a nice horizontal line, which is straight. So all we need to do for this graph, and all you'll need to do when it comes to doing your quiz tomorrow with this as well, is simply set up your graph like this, put a horizontal line showing where the graph would be, and label that with whatever your acceleration is. So mine was negative 8.8. I'm not even going to put increments on, because it doesn't really matter what increments I use. It's only going through one point anyways. I'm not going to put any time increments on, because the line is always, at any time, at that same acceleration. So very fast. All I had to do for my acceleration time graph was take my slope of the velocity time graph. Take that slope, and just since the slope of the velocity time graph is the acceleration, graph that up. So these are the steps we'll use to take a displacement time graph and turn it into a velocity time graph and an acceleration time graph.