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Displacement of a Car

Patrick Ford
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Hey, guys, let's work out this example. Together we're showing the velocity time diagram. So that's V versus Tear the velocity time graph of a moving car. And we're told that the initial position is negative. 21 were supposed to find out what's the cars? Final position AT T equals five seconds of final position. So the variable that we're looking for is going to be X or X final. So that's gonna be what we're looking for here. So how do we find the final position? Well, we find position by finding the displacement. Remember, Delta X. Delta X can be written as X final minus X initial. But when we have a velocity time graph Delta X, the displacement is also the area that is under the curve. And so what's the variable amongst looking for here? I'm looking for the final position so I can rearrange this formula really quickly and say that the final position is going to be the initial position plus Delta X, in other words, plus the area that's under the curve. So if this is what I'm looking for, already have the initial position, which is 21 or negative 21 m. So now all I have to dio is I just have to figure out what the area that's under the curve, I have to figure out. The displacement is. So let's go ahead and do that over here. The displacement Delta X is just going to be the area that's under the curve. Four between the two times I'm looking for it says, What's the cars? Final position AT T equals five seconds. And we're also told that the initial positions is Teoh t equals zero. So what this really means is we're looking for the areas are the curve off this entire graph over here? So the area that's under under the curve is just gonna be all of this stuff that's highlighted in yellow. So let me see if I could do that real quick. Yeah, it's pretty good. All right. So how do we find now the area of this really complicated shape? Remember, any time we do this, we just have to break it up into a bunch of little smaller shapes that are easy that are a little bit more manageable. So I was like to break it up into the smallest number of shapes. I see a big triangle over here, and if I keep this line going, I actually have a smaller triangle and then a big rectangle. So it's only three shapes I need to worry about, and that's good. So if I want to figure out the total displacement over here now, I've got these three smaller shapes, so I'm just gonna label them Delta X one. Let's call this guy Delta X two and then we'll call the smaller one Delta X three. Okay, so let's just get to it. Uh, in the total displacement is just gonna be by adding up all of those all of those areas, they're all of those smaller displacements up. And so Delta X one is gonna be a triangles. We're gonna use one half base times height. Now we just have to figure out the base and height is, and so we're going from zero all the way to three. So the base here is three, and the height is gonna be going from to all the way up to 10. So that means that this height of this triangle is gonna be eight. So it means we have one half of three times eight and that's 12. Let's move on to part the second one. The second one is gonna be from zero all the way to five. It's a rectangle, which means the equation we're gonna use is base times, height, and we're gonna use s so we have that. The base is five and the height is gonna be too right here. So we have just displacement is 10. So we've already got these first two. Now we just have to look at the last one Delta X three. It's a small triangle. So we're still going to use the same equation. One half, one half. Wow. Okay, one half base times height. So we've got one half the base is going to be, too, because going from 3 to 5 and the height is also going to be from two, because we're going from two up to four. So we have one half of two times to one half and two will cancel Lee, just leaving one factor of two there. And so now we just put everything together, all of these areas, and then add everything up. The Delta X is gonna be 12 plus 10 plus two. And so that's just gonna be 24. So now the question is, are we done? Is this our answer? No, it's not. It's not our answer. Remember, this just represents the displacement. We actually to take this number, and we have to plug it back into this formula over here to figure out the final position. So you can remember that last. You can't forget that last step there. So we're gonna take this number over here and we'll pop it into this equation. And so, for the final answer, the final position is gonna be the initial position of negative 21 plus 24 which is the displacement. And so what we get is positive. 3 m. That is our final position. Alright, guys, let me know if you have any questions and I'll see in the next one.