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2. 1D Motion / Kinematics

1

concept

7m

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Hey, guys, A very common problem that you'll see involving motion graphs is how to sketch one graph from another. So in this video, we're gonna get a ton of practice and you're going to see how you graph position, velocity and acceleration graphs from one another. Let's check it out. So, guys, when you're graphing or you're sketching out these motion graphs, all we need to do is just remember a couple of simple rules involving the three motion variables. So we got position, velocity and acceleration. And so the slope of the position graph is gonna equal the value of the velocity. And we've seen that before. And when we talked about velocity time graphs, it was a very similar rule. The slope of the velocity graph gave us the value of the acceleration. So this is really all we need. We won't need any equations because we're really just gonna be sketching. It doesn't have to be super precise, so we just need to remember these rules. The best way to learn this is just by doing a bunch of examples, So let's check it out. We've got these diagrams below, and we're gonna be given the we're gonna use the given graph to sketch the missing graphs. So let's just get right to it for part A. We've got this position Time graph over here. We need to sketch out with the velocity and acceleration. Look like So, let's just remember the rules for our motion variables. We know that the slope of this position graph is gonna be the value of our velocity graph. So, what's going on in part? A. Well, we've got a slope that is constant everywhere along this line has the same exact slope. So that means that the value is also just gonna have the same value. So that means we're just gonna draw a straight flat line like this. So whenever we have a constant slope like we do over here, that's gonna mean that we have a constant value for the velocity. Now, again, this is kind of just a sketch. We don't have to be super precise about what the value is, but it does need to be a straight line like I've drawn over here. So now what's the acceleration? What, we're gonna use the exact same principle. The slope of our velocity time graph is gonna be the value of our acceleration graph. So what's going on with the slope of this line? Well, if we draw these lines here, the slope is just perfectly flat, and it's also constant. So here we have a zero slope. And what does that mean for the acceleration? Well, that just means we're gonna have a zero acceleration. So that means when I draw my acceleration, it's just going to be a flat line at zero. It's gonna be a constant zero because the slope of this line is constant and flat. Let's move on to part B. So here in part B, we actually have the velocity graph. And so we need to calculate, or we need to figure out or sketch the position and the acceleration graphs. Now, to be honest, you could go in whatever direction that you want. But I personally like to go from here down. I like to go down from velocity to acceleration because I feel like it's easiest reason through, So let's check it out. So just remember the relationships. The slope of our velocity graph is gonna give us the value of our acceleration graph. And so, if you look at the slope of this line here. This is gonna be a constant slope that has basically the same slope all throughout this line. It's exactly like what we did over here with this position graph. And so a constant slope here is just gonna mean a constant value for the acceleration, not zero. It's just gonna be a constant value. And so let's say we could just draw this who are here, so this literally looks exactly like it did over here. So now how do we calculate, or how do we sketch out the position graph? Well, remember that the value of this velocity graph is going to give me the slope of the position graph. So what's going on with the values of this graph? Well, if we take a look here, the values are actually constantly increasing. The slope is constant, but the values with numbers are actually going up. So that means that the slope is also gonna have to go up. So a changing value here and actually in this case it's increasing means that we're gonna have a changing slope and our slope is also going to be increasing. So what does that look like? Well, if you take a look here, these values as they get Well, first of all, we actually where where do we even start from? Eso remember, this is just a sketch. But the problem actually tells us that for parts B and C, we can assume that we're starting at rest and from the origin at X equals zero. So the first thing here is that we're going to start here at the origin. And so as the values are increasing, we know that the slope is going to increase. So, for instance, the slope is gonna look like this and it's gonna look like that, and then it's gonna keep on going up and up. And if you connect these points over here, what we're gonna end up with is we're gonna end up with a curved position graph now, remember that when we first talked about curved position graphs, we said that the velocity was constantly changing, and that was because there is some acceleration. So this should make This should make sense, because here we have a constant acceleration, and therefore we end up with a curved position graph. Let's move on to the last one here. So we actually have a Sorry. It might be blocking this for a little bit. So we have a constant value for our acceleration. And now we want to figure out what our velocity graph looks like. So let's just remember our relationships here. The value of our acceleration graph is going to be the slope of the velocity graph. So here, what's the value? Our value is a Constance. We have a constant value for the acceleration, which means we're gonna have a constant slope for the velocity. But here the acceleration is negative. And what we know about this slopes of when they're negative is that they go downward. So the velocity graph is actually going to start where we start. Well, any anything negative would look just look like this. So where do we actually picked the point that we start from? Well, again, the problem says that for parts B and C, we're gonna assume that we start from rest and we start at the origin. So here we're gonna start at V equals zero. And now all we have to do is just draw a straight line that angles downwards like this. So this here is a constant slope that points downwards. So we're good there. So now finally, what is the value of our position? Well, or sorry. How do we sketch our position graph? Well, we're just gonna use the same principle. The value of R V T is gonna be the slope of our X t. Now here, the value is actually constantly changing as we go along. This graph here, the value is changing. So we can say that this value is decreasing, right or changing, and so therefore we're gonna have a decreasing slope. So again, we're just gonna start from the origin, which is gonna be over here. And what does a decreasing slope look like? Well, decreasing means that we're gonna start off by, you know, going relatively flat, and then it's gonna go and it's gonna get Mawr and Mawr Mawr vertical. But it's gonna point downwards. That's what the slope decreasing is gonna look like. So this is just gonna look also curved, but it's actually going to be curving downwards now. This should also make some sense because we said that when we had a positive acceleration than that was going to be a smiley face curve. When we had negative accelerations, that was gonna be a frowny face. So here everything makes sense and we also have a curved position graph. That's it for this one. Guys, let's get a couple more practice problems. Let me know if you have any questions.

2

example

3m

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Hey, guys, good to see you. Let's get some more practice with graphing all of these position, velocity and acceleration graphs. So we're gonna start from the origin were shown our velocity time graph. We're going to sketch the position and acceleration graphs. Let's just get to it. So we have the velocity time graph that's shown for us. And so if we're trying to get from the velocity to the acceleration and the position, I always like to start off with the acceleration first because I think it's the easiest. All we have to do is we just have to look at the slope of this line and what it's doing. So the slope of this line is constant and positive, and what that means is that we're gonna have a constant positive value for acceleration. So for this little piece right here, up until it gets to this point, the acceleration is just gonna have some positive value up until over here. But then what happens is that we have some changes, some abrupt change in sort of what the velocity graph is doing. So I've drawn this little like, dotted line here to sort of indicate that This is like a point where something happens in something changes here. So I'm gonna actually just gonna draw that all the way down the motion Over here. This dotted line. Okay, so now what happens is that instead of the velocity continuing to increase like it normally would. Now what happens is that we have it flipped, and now it starts to decrease. The velocity is now going down towards zero again. So that means that in this section right here, we have a negative constant slope, which means that we're gonna have a constant negative value for the acceleration. So that would look like, actually basically the same exact flat line. But it's now negative instead of positive. So there's some abrupt change here in the acceleration, and this is what it would look like. So that's the velocity and the acceleration. Now, what is the position? Time graph look like? The first question is where we're going to start in the first place and told we're told in text that we're gonna start from the origin, which means we're gonna start basically right from 00 So now, in order to get from the velocity to the position. We need to look at what the values are doing here. Now, the values for the velocity graph are actually going to be increasing right. The values are continuing to increase, which means that in the position graph, the slope is going to be increasing, so increasing values means increasing slope over here. So what that means is, if we have a increasing velocity, then that was gonna introduce, um, curvature. And again, we've already seen this piece right here. So now what happens in this section? Well, if increasing increasing values may increasing slope and what happens is decreasing values here, so decreasing values are going to result in a decreasing slope. So what would that look like? Would it look like this, or would it look like this or something like that? Well, the velocity is still going to be really high here, and then it's gonna eventually go down towards zero. So what happens is the velocity steepness is still going to continue on being the same. But eventually it's going to flatten out and become zero over here. So whenever this line here becomes flat, then that's going to corresponds to the velocity equaling zero on the graph. So that means that basically, it's gonna take this weird like s looking shape over here. And this is actually what the position time diagram would look like. Alright, guys, hopefully this sort of wraps this up. Let me know if you have any questions. That's it for this one.

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