1

concept

## Conceptual Problems with Position-Time Graphs

9m

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Hey, guys, when you're solving motion problems involving motion graphs, a lot of times you have to interpret these graphs to solve some conceptual questions about the position, velocity and acceleration. This can confuse some students because the hardest part is figuring out what exactly you're looking for on the graph. So what I'm gonna do this video is give you a list of steps to follow so that you get the right answer every single time. The best way to do this is actually just by looking at this example together. So we're gonna do that here. We've got a motion graph for a moving object. We've got a bunch of letter points A through G. We've a bunch of questions to solve. Might seem like a lot, but once we get the hang of it, that's actually gonna go pretty quickly. So the first thing the first problem here says at which letter, point or points is the object at the origin. So here's we're gonna do for every single one of these problems. The first thing you have to do is identify which motion variable you're talking about. There's only three possibilities. You're talking about the position the velocity or the speed, which was related and the acceleration. So let's take a look at the problem. Where's the object at the origin? Well, the origin, remember, is just a coordinate. It's a reference point, which you're starting from, so it's a location. So between position, velocity and acceleration, that's actually going to be a position. So that's this first step. The second step says, identify the graph feature. And again, there's only three possibilities. So we're gonna be looking at the values, the slopes or the curvature is to figure out which one of these were actually just gonna use this table down here that summarizes everything we know about position graphs on a position graph. Remember, the position is gonna be in the y axis positive values air here. Negative values air here, and you're zero when you're on the line. And so if you're looking for the position, you're just looking for the values of each of these points. So that's what we're gonna look at here. So we're gonna look at the values. So what I like to do is for each one of these points here you're basically looking for Where are you on the Y axis. So what I like to do is just draw little lines straight down here. And the length of each of these lines is your position here at point you're on the access, so there is no length. And then f n g look like this. So that's the second step. The third step says now, the qualifier, which qualify we're gonna use the qualifier is basically just what about the slope? Are you are sorry? What about the value are you looking for? We've got this big list here, but every single one of these problems is gonna boil down to one of these options. We have to figure out the right one. So we're looking for where the object is at the origin. And remember, the origin is just basically when you're on the access. So that's when your position is equal to zero. So in this list over here between positive negative zero up down sign changes, maximums and minimums, the qualified that we're looking for here is where the value is at zero. So that's the third step. And now the last thing we just have to do is interpret this from the graph. Where are the values equal to zero. And this only happens to two points here where you're at the beginning and then here at point E. But this isn't one of our letter choices. So we're gonna delete that one and instead is just gonna be Option E. That's the only place where the object is at the origin. That's the fourth step. Interpreter from graph. Let's keep going. So now we're supposed to find out where the object is the farthest away from the origin. So let's just go through the list of steps again. First identify which variable we're talking about. Remember, Origin means we're gonna be looking for the position, which means if we're looking for the position, the second step says, we're gonna look at the values. So that's the first two steps right there. And now we have to look at the qualifier. Well, the thing that's different about this problem isn't now we're looking now. We're not looking for where it's at. The origin were looking for where it is farthest away from the origin. And so in this list here, what is farthest away mean? Well, farthest away means the most. It means the most away from the origins. You know, inner list here. It's not gonna be positive. Negative or zero. Up or down, it's gonna be a sign change. It's gonna be a maximum value. So we're actually looking for where the value is the maximum. So now the last step is interpreting it from the graph. So remember that the positions basically just the length of each of these lines. So which lines the longest? Well, it's just this one right up here. It's just D D is the farthest away from the origin. So that's your answer. Cool. All right, let's keep going. So this third part here now is Where is the object moving forwards? So again, let's go through our list of steps. Identify the variable. We have some new keywords here. Now we're talking about moving, and we have forwards, which is a direction. So we're talking about motion and we have a direction. So that means that we're not talking about velocity. We're actually going to be talking about the I'm sorry. We're not talking about the position. We're actually talking about the velocity here, so that's the first step. Now, the second step says Identify the graph feature. And to do that, we're gonna look at our table down here. So remember that when you are looking for the velocity in a position time graph, you're actually gonna be looking at the slopes of the graph. There's a couple of rules to remember. Whenever you have positive slopes like Sorry, upward slopes like this you're going to that's gonna be a positive velocity. So these are gonna be positive velocities, flat slope. So when you have zero velocity and then downward slopes or when you have negative velocities the other thing to remember is that if you have more vertical lines than you're, you're going faster. The magnitude of this velocities higher. So if you were to call this V one and V two than V one is greater than V two because it is a steeper line, so steeper line means faster. So going back to our problem now, where is the object moving forwards? Well, the second step says we're gonna look at the slopes of the graph. So now the third step is the qualifier. So positive or negative? Zero up down sign changes, maximums or minimums. Well, what is again. Moving forwards means well moving forwards means you have positive velocity and again from our rules. We know that that's gonna happen when you slow slope is upward. So the qualify that we're looking for is where the slope is upwards. So that's the third step. Now we just have to go and interpret that from the graph. So we're just gonna draw the slopes really quickly for a it's gonna look like this at be at sea. And for these Kirby parts, you have to draw the tangent lines or the instantaneous. So it's gonna look like that. And then F is gonna look like this because it's at the bottom of the valley and then G looks like that. So where are these slopes? Positive. Well, there's only three points. It's gonna be a C and G. So those are three options. So a, C and G. So that's where you do interpreted from the graph. All right, so now where's the object moving backwards? So again, let's just go through a list of steps really quickly. Moving backwards means velocity. Velocity means you're looking at the slope now. The qualifier. Well, the qualifier for when we're moving forwards was we're looking at upwards. Slopes backwards is gonna be the opposite. So now we're just gonna look at downward slopes. So now in the graph, where do we have a downward slope? There's actually only one point is right here at point E. All the other ones are either upwards or flat. So that's just gonna be pretty straightforward for this last one. Where is the object or sorry for this next one? Where's the object? At rest? Well, at rest. Uh, it means that it's actually not moving at all. So even though it doesn't say moving, we're actually still talking about motion. So we're still talking about the velocity, which means we're still going to look at this slope. And so now the qualifier. Well, remember that when an object is at rest, the velocity is equal to zero. And from our rules for slopes, we know that you have zero velocity when the slope is flat like this. So we're just looking for where the slope is equal to zero or where the slope is flat. So now from the graph, where does that happen? Well, if you look through here, we have a flat slope B, D and F. So that's either the average or the instantaneous velocity doesn't necessarily matter which one. So B, d and F those were answers. So now for these last two, where is the objects? Acceleration. Positive. So now there's in a couple of key words there. So we have acceleration now. So let's go through a list of steps. Which motion variable We're talking about position, velocity, acceleration. It's pretty straightforward. We're gonna be talking about the acceleration. So which graph feature are we going to look for? Well, let's go through our diagram. Our table, if we're looking for the acceleration now in a position time graph were actually looking for, is we're looking at the curvature. So the curvature, whether it's a smiley face or a frowny face, the rules are pretty straightforward. Remember that when you have a smiley face like this, then that's going to be positive acceleration. And when you have a frowny face, that's going to be negative acceleration. So we're looking at acceleration, which means that from our second step, we're looking at the curvature. So now our third step is the qualifier. What qualify? What about the curvature are we looking for? Or remember that positive acceleration, positive acceleration. You means that you're going to have an upward slope. So that's the qualifier we're using here. So where do we have an upward slope? Well, it's actually to be right over here where we have a little smiley face. So the two points over there are F and G. And now, lastly, where's the objects? Acceleration. Negative. Well, now we're looking for acceleration again. Go through the list of steps, which means we're looking at the curvature. And if positive acceleration meant we're looking at upwards slopes or smiley faces, the negative acceleration means we're looking at where the curvature is downwards or frowny face. So that happens at two points here. We're looking at C and D, where you have a little hill like this. This is going to be that a debt downwards curvature. So the two points there are C and D. Those air answers. Alright, guys, hopefully got a lot of practice with this. We have another example that's coming after and let me know if you have any questions

2

example

## A Moving Car

6m

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Hey, guys, hopefully you give this one a shot. We're gonna get some more practice with position diagrams, but take a look at this example together, we have a position diagram for a moving car. We've got points a through E. Let's just start with the first one. Where is the car moving the fastest. So again, we're just going to stick to the steps. Which variable are we talking about? Position, Velocity or acceleration? What? We're talking about motion here. Where we moving the fastest. So it's not gonna be the position. And remember, acceleration is a change in motion. So we're actually not gonna talk about acceleration either. It's gonna be the velocity. So that's the first step. Now the now the graph feature. We just get by using the table down here when we're looking at a position Time diagram. We are looking for the velocity. So that means we're gonna look at the slopes of the graph. There's a couple of rules to remember. Remember that upward slope? They're gonna be positive flat zero. And the negative is downward or downward is negative and steeper you are, the faster you're going. So that's the That's the rule that we're gonna use. So we're looking for the slopes here, So that's the second step. The third one is the qualifier. So the qualifier, in which of these terms over here in this list are we gonna use? Well, the fastest is the key word here. The fastest means like the most where you're moving the most fast. So in this list, that's actually gonna be the maximum value over here. So that's the third step. We figured out that we're looking for the maximum slope. Notice how the question also doesn't say forwards or backwards anything like that. So we're actually looking for here. Remember, maximum slopes are gonna be the steepest slopes, and it doesn't matter whether it's upward or downward. We're just looking for the most vertical slope. So now we just have to interpret from the graph. And to do this, we actually have to draw out the slopes for each of these points. So let's just go ahead and quickly do that by looking at the the tangent lines. That is gonna be the slopes. Treat each of these points. Now I just have to figure out which one of these points is the steepest. If you take a look, that's actually just gonna be point C. So that's our answer. Moving on to the second one. Where is the moving the slowest again? Guys, if you go through all the steps, we're moving the slowest means we're gonna talk about velocity. We know that means we're looking at the slopes. So now for the qualifier. Well, if the fastest meant we were looking at the maximum, then the slowest means we're just gonna look for the minimum. And so if the steepest slope was the maximum, the flat ist slope is gonna be the minimum. So let's get the graph. Where do we have flat slopes? Well, it's actually pretty easy at point B and appoint d. We have flat slopes, so those are gonna be points B and D. There's actually also are where you have no motion at all. It's V equals zero. So if you're not moving, that's the slopes, Ugo. So now move on to part three. Where is the car turning around? So what does this mean? What is turning around mean? Well, if you turn around, you're gonna be moving in one direction. You stop and then you move in the other direction. So we're still talking about motion. Here's which. Which position? Sorry, which variable we're gonna use. We're gonna use the velocity. So we're talking about velocity, Which means we're gonna talk about this slope. Those are the first two steps, and now we just have to look at the qualifier. So which one of these in this list is gonna be the right qualifier? Well, again, remember what turning around means means you're moving in one direction and then you turn around and you move in the opposite direction. You're talking about a change in direction. And so remember, when you're talking about the slopes, when you have upward slopes, that's gonna be positive velocity. That means you're moving forward. Downward slopes is gonna be negative velocity, which means you're moving backwards. So when you're looking for a direction change, where you're actually looking for is a sign change. That's the qualified we're gonna use when you use a sign change. So now we just have to interpret from the graph at point A. Because it's upward. You have a positive velocity point B, you have the equal zero, and then point see because it's downwards. You have negative velocity. Which means somewhere inside of this little point over here you are going forwards. You stopped and then you turned around and started moving backwards. That actually happens Very like right here at point B. It happens when you stop momentarily and then right before you start moving in the backwards or opposite direction. So that means that happens at B. Now, is there any other point? Well, at D were also the equal to zero. But what happens is it's a constant zero for a long time. There is no change or to flip in the sign so d is not a turning point here. Okay, so that's it. Now for the last two, where is the car speeding up and slowing down? So let's just go thru are variables Which are we talking about? Position, velocity or speed or acceleration? Well, you might think Oh, we're talking about speed, so we're gonna be looking at speed. But actually, what's happening is speeding up means the speed is increasing its talking about a change in that speed. So we're actually gonna be talking about the acceleration here. So that's the first step Now we just have to figure which graph feature we're talking about again going down to our table. We're gonna see that when we talk about the acceleration in the next a graph. We're looking at the curvature and there's a couple of rules to remember when we have a smiley face upwards like this, that's gonna be positive. Acceleration of frowny face is gonna be negative acceleration. And then also we have to remember is we have to remember that if you break up these smiley faces in frowny faces into two halfs, then on the left side you're always going to be slowing down. So in these pieces right here, you're going to be slowing down. Because as you move towards the center, your slopes are becoming more flat, which means you're sort of going to zero. Whereas on the right side, your slopes are becoming more vertical on here. So you're actually going to be speeding up? That's what you need to remember. So left is slowing down and right is speeding up. So let's go back to the question we're gonna you know, we're gonna be looking at the curvature over here for speeding up for the acceleration. So which qualifier makes the most sense? Well, this is the only case where the qualifier doesn't really make a whole lot of sense. Uh, there's no really good choice that we can pick from this from this table That will help us here. Instead, we're gonna use this rule that we talked about about speeding up and slowing down. So here we're gonna go to the graph, figure out all the places where you have curvature. So, for instance, right here and also at E. We also have, like, another sort of half smiley face, and then we just break up each curve into two halves. So just break this up into two halves and this one's already kind of a half, and then we're looking at speeding up. That's just gonna be any points to the rights of those curves. So, for instance, at sea where that's gonna be to the right of the curve and also an e. This is also to the right of the curve, So C and E are our choices for speeding up. You can also tell because the slopes of these lines are becoming mawr vertical as you move along here. All right. So finally, the last one, where we slowing down guys, that's just gonna be the opposite. We know we're looking at acceleration, which means we're looking at the curvature. And so now, instead of looking for the right sides of those curvature, is we're just gonna be looking at the left sides. So the left is when we're is when we're slowing down, The only point that fits here is actually gonna be point a again. Remember, this is sort like a half. Uh, this is like a half smiley. It's not really a full one, so d doesn't count. So that means that point is our answer choice, and that's it. Let me know if you guys have any questions.

3

Problem

The position of an object vibrating on a moving spring is represented by the diagram below. Which of the following options is true of the velocity and acceleration at Point P?

A

The velocity is positive and the acceleration is positive

B

The velocity is positive and the acceleration is negative

C

The velocity is negative and the acceleration is positive

D

The velocity is negative and the acceleration is negative

4

example

## Two Bicycles

3m

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Hey, guys, let's check out this example. Problem. Work it out together. I've got a position time graph that is shown over here. Except now. It's not for just one object. I actually two moving objects or to moving bicycles labeled A and B. Now, the first question is asking us at what time or times do the bicycles have the same position? So what does that actually mean on this diagram here on this graph? Well, if I'm looking at a position Time Grafton from our table are of conceptual points in position graphs, then we know we're just going to be looking at the values. So what this question is really asking us for Is that what time or times on this time access over here are the bicycles at the same exact value. So let's take a look here. Well, bicycle B is gonna be at this value over here, and then bicycle is gonna have this value over here later on. What happens is when the two lines will cross. This is one of the points where they have the same value, so t equals one is one of our times. Let's just see if it happens again and later on in the diagram we'll be or sorry, bicycle A. We'll just continue on like this and the values over here whereas bicycle be the values, they're gonna be over here. Notice how these are not the same. But eventually what happens is B is gonna catch up to a or the lines are gonna meet again right over here. Here is where the values are also going to be the same. So these two points here correspond to the same values. And so therefore there at the bicycles are at the same position. So it's t equals one and T equals four seconds. Those there are two times. There's never anywhere else on the diagram in which these two lines will cross each other. So let's we want to be now. What time or times did the bicycles have Roughly the same velocity? Well, on a position time diagram. Remember? Now we're looking for the velocity, which means we're not looking for the values were looking for the slope. So what this question is really saying is, where do the bicycles or what point to the bicycles on this time diagram. Do they have the same slope, not the same values. It's not going to be here or here. They don't have the same slopes here, so let's check it out. Well, the slope for a is actually just a straight line. So in other words, it's always going to be this line over here. It never changes its constant velocity, whereas for be what happens is it's a curvy position graph. So we know that there's gonna be some acceleration and the velocities air constantly gonna be changing. So where do these two lines have the same velocity? Well, I could basically trace out with the instantaneous velocity. Looks like here it's kind of hard to visualize it or c it really clearly. So this is like the instantaneous velocity. And then right around here starts to trend upwards like this. And I'm basically looking for where it matches this approximate sort of steepness here and right around at three. If I were to draw the tangent line, the tangent line looks like this. So let me erase all the other ones, so it just looks a little bit clearer right here. So at this point right here, the velocity of B is this line on the velocity of a is this line. So that means that that is actually the point at which they have the same velocity. So let me go and actually write this in blue just so you can see it's super super clearly so the same slope is going to be right here at this point over here. So that means that the time is T equals three. Then what happens is the slope is gonna continuously increased for be and it will never be the same as a again. So that means that there's only one time that this happens. It's a T equals three seconds right over here. So notice how these two slopes of the same Alright, guys, um, that's it for this one. Let me know if you guys have any questions.