Position-Time Graphs and Instantaneous Velocity

by Patrick Ford
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Hey, guys. So we've already seen situations in which the position time graph is curved like this, which means the velocity is changing. Now we already know how to calculate the average velocity, but in some problems, you need to calculate something called the instantaneous velocity. So in this video, I'm gonna show you the two different types of velocities and how to calculate them. And the big difference guys is that the average velocity is always calculated between two points, whereas the instantaneous velocities always calculated at one particular point or at one particular instant. That's why we call it instantaneous. For example, let's say we wanted to calculate the average velocity between zero and three seconds over here on this graph. Then basically, we just need a line that connects these two points from 0 to 3. So the line connecting them looks like this, and the slope of this line here is gonna be our average velocity. So now what if I wanted to calculate the velocity at a particular instant? Let's say T equals three. So I need to calculate the slope of a single point. But how do I do that? Well, it turns out I need a new kind of line called a tangent line and the tangent line looks a little bit like this. So this is the tangent line. It's a line that touches the graph Onley once. And one way I like to think about the tangent line is if I were to just trace along this graph over here with my pen without lifting it and then when I get to this point instead of following the curvature of the graph, basically keep on going in a straight line like this, that would be the tangent line. So then what happens is the instantaneous velocity is gonna be the slope because the velocity is always gonna be a slope. But it's gonna be the slope of this tangent line that I've made. So let's take a look at the difference here. When I got the average velocity, it's the slope between two points, so I can always figure out the rise of the run. But what's tricky about the instantaneous velocity and this tangent line here is that we actually had to draw it ourselves. So a lot of problems aren't gonna give you with this tangent line. Looks like so you're always just gonna use an approximated line. So basically, we're just using, like, a best guess, or like a rough estimate as to what this instantaneous velocity will be. Anyway, that's all Really all there is to a guys. So let's take a look at an example. So you've got this position time graph over here, and let's take a look at part. A party is asking us to calculate the velocity between 10 and 25 seconds. So the first thing is, what kind of velocity are we talking about? Well, this is a velocity that is between two points in time, so that means it's going to be an average velocity, which means we need the slope Delta X over Delta T off the line that connects these two points. Well, AT T equals 10 and 25. Those are my two points right here. So this point is that T equals 10. And then this point over here is AT T equals 25. So I draw the line that connects them. That's gonna be this line over here. And then this is my average velocity. It's the slope of this line over here, so I needed to make this triangle My rise over my run. My Delta X is gonna be my final position minus my initial. I end up at 60 over here, and I started off at 30. So that means that my Delta X is 60 minus 30 which is just 30. And then over here, my delta T is 25 minus 10 which is seconds. So that means that my Delta X over Delta T is 30/15 and that's 2 m per second. Let's move on to part B. Now, Part B is asking us to calculate the velocity AT T equals 10. So now that's actually one point that they're giving us, Which means this is an instantaneous velocity here, So I'm gonna write the instantaneous AT T equals 10. So first I need to figure out what the slope of the line looks like, which is gonna be the tangent. That line AT T equals 10. So we're still gonna look at this point over here. That's t equals 10. Now we have to draw the tangent line. So again, we're gonna trace along this graph here. And when I get to this point instead of following the curvature, I'm gonna keep going in a straight line like this, so that's gonna be the straight line. And this kind of looks a little funky because it's gonna overlap a little bit with the graph. But this is kind of like a best guess. So now I need to calculate the slope of this line over here. This here is my instantaneous velocity. So I need to calculate the Delta X over Delta T. So it's still gonna be the calculating the slope. But we're gonna use is we're gonna use these two points over here to calculate it. So even though we're calculating the instantaneous line at one point, we're still gonna use two points to calculate a rise over run. So basically, I'm gonna use this point and this point over here, so I have to calculate the Delta X over Delta T. So my final position is gonna be 15. My initial position is gonna be 45. So that means that my Delta X over here is 15 minus 45 that's negative. 30. So this is my negative 30 and then the time the time over here is between five and 15 seconds. So this delta T is equal to 10. So that means that my instantaneous velocity here is negative 3 m per second and it's negative again because it's going downwards like this. So that's sort of, you know, again, this just ah, best guess on approximation as to what this instantaneous velocity will be. So let's move on to part C calculate the velocity AT T equals five. So what you're gonna do the same thing now except T equals five is actually right over here. So we have to calculate the instantaneous velocity at this point. So I'm gonna do this in Green s. So basically, I'm gonna draw the the line like this, and when I get to this point, I'm gonna keep going, as if I were basically to go off in a straight line instead of following the curvature. So what I end up with because it's sort of hear the top of this little hill is I just end up with a straight line so V, when T equals five is gonna be Delta X over Delta T. But if you'll notice that a straight line we know from previous videos the Delta X for straight line is just zero, and it doesn't matter what this delta T is. So let's say already. Use these two points over here. Delta T is 10 seconds. The velocity is 0 m per second. So what that means, guys, is that the velocity is equal to zero at the peaks and the valleys of your position. Time graph. Anytime you have a peek like this, the top of a hill or any time you have the bottom of the Valley like this, there's gonna be a moment in which the instantaneous velocity is flat. And so therefore at zero. Alright, guys, that's what we need to know. For this video, let me know if you have any questions.