Conceptual Problems with Position-Time Graphs - Online Tutor, Practice Problems & Exam Prep

Understanding motion graphs involves interpreting position, velocity, and acceleration. Key steps include identifying the motion variable, analyzing graph features like values and slopes, and applying qualifiers such as maximums or zero values. For instance, an object is at rest when the slope is flat, indicating zero velocity. Positive acceleration corresponds to upward curvature, while negative acceleration relates to downward curvature. Mastering these concepts enhances comprehension of motion dynamics, essential for solving related physics problems effectively.

Conceptual Problems with Position-Time Graphs

Hey, guys. When you're solving motion problems involving motion graphs, a lot of the times you'll 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 going to do in 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 going to do that here. We've got a motion graph for a moving object. We've got a bunch of lettered points a through g. We have a bunch of questions to solve. It might seem like a lot, but once we get the hang of it, this is actually going to 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 what we're going to 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 are only 3 possibilities. You're talking about the position, the velocity or the speed, which is 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 are only 3 possibilities. So we're going to be looking at the values, the slopes, or the curvatures. To figure out which one of these, we're actually just going to use this table down here that summarizes everything we know about position graphs. On a position graph, remember, the position is going to be on the y-axis. Positive values are here, negative values are here, and you're at 0 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 going to look at here. So we're going to look at the values. So what I like to do is, for each one of these points here, you're basically looking for where you are on the y-axis. So what I like to do is just draw little lines straight down here and the lengths of each these lines is your position. Here at point E, you're on the axis so there is no length. And then f and g look like this. So that's the second step. The third step says now the qualifier. Which qualifier are we going to use? The qualifier is basically just what about the slope are you, or sorry. What about the value are you looking for? We've got this big list here, but every single one of these problems is going to boil down to one of these options. We just 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 axis. So that's when your position is equal to 0. So in this list over here, between positive, negative, 0, up, down, sign changes, maximums, and minimums, the qualifier that we're looking for here is where the value is at 0. So that's the 3rd step. And now the last thing we just have to do is interpret this from the graph. Where are the values equal to 0? And this only happens at 2 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 going to delete that one. And instead, it's just going to be option E. That's the only place where the object is at the origin. That's the 4th step, interpret it from the 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. Well, remember origin means we're going to be looking for the position. Which means, if we're looking for the position, the second step says we're going to 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 is now we're not looking for where it's at the origin, we're looking for where it is farthest away from the origin. And so, in this list here, what does farthest away mean? Well, farthest away means the most. It means the most away from the origin. So now in our list here, it's not going to be positive, negative, or 0, up or down. It's not going to be a sign change. It's going to be the 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 is basically just the lengths of each of these lines. So which line is the longest? Well, it's just this one right up here. It says d. D is the farthest away from the origin. So that's our answer. Cool. Alright. 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 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 going to look at our table down here. So remember that when you are looking for the velocity in a position-time graph, you're actually going to be looking at the slopes of the graph. And there are a couple of rules to remember. Whenever you have positive slopes, like upward slopes like this, you're going to that's going to be a positive velocity. So these are going to be positive velocities. Flat slopes are when you have zero velocity. And then downward slopes are when you have negative velocities. The other thing to remember is that if you have more vertical lines, then you're going faster. The magnitude of this velocity is higher. So if you were to call this V1 and V2, then V IsEthernet higher than V2 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 going to look at the slopes of the graph. So now the third step is the qualifier. So positive or negative, 0, up or down, sign changes, maximums, or minimums. Well, what does again moving forwards mean? Well, moving forwards means you have a positive velocity. And again, from our rules, we know that that's going to happen when your slope is upward. So the qualifier 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 going to draw the slopes really quickly. For A it's going to look like this. At B. At C, and for these curvy parts, you have to draw the tangent lines or the instantaneous. So it's going to look like that. And then F is going to 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 are only 3 points. It's going to be A, C, and G. So those are our three options. So A, C, and G. So that's where we interpret it from the graph. Alright.

So now where's the object moving backward? So again, let's just go through the list of steps really quickly. Moving backward means velocity. Velocity means you're looking at the slope. Now the qualifier. Well, the qualifier for when we were moving forwards was we were looking at upward slopes. Backwards is just going to be the opposite. So now we're just going to look at downward slopes. So now on the graph, where do we have a downward slope? There's actually only one point. It's right here at point E. All the other ones are either upward or flat.

A Moving Car

Hey, guys. Hopefully, you gave this one a shot. We're going to get some more practice with position diagrams. Let's take a look at this example together. We have a position diagram for a moving car, with points a through e. Let's start off with the first one. Where is the car moving the fastest? So again, we'll just stick to the steps. Which variable are we talking about? Position, velocity, or acceleration? Well, we're talking about motion here. Where are we moving the fastest? So it's not going to be the position. And remember, the acceleration is a change in motion. So we're actually not going to talk about acceleration either. It's going to be the velocity. That's the first step. Now, the graph feature, we just get by using the table down here. When we are looking at a position-time diagram, we are looking for the velocity. That means we're going to look at the slopes of the graph. There are a couple of rules to remember: upward slopes are going to be positive, flat is 0, and downward is negative. And the steeper you are, the faster you're going. That's the rule we're going to use. We are looking for the slopes here. The third step is the qualifier. In which of these terms over here in this list are we going to use? Well, the fastest is the keyword here, and the fastest means like the highest velocity. In this list, that's actually going to be the maximum value over here. We figured out that we're looking for the maximum slope. Notice how the question also doesn't mention forwards or anything like that. So we're just looking for the most vertical slope. Now, we just have to interpret it from a graph. We have to draw out the slopes for each of these points by looking at the tangent lines. Those are going to be the slopes for each of these points. Now I just have to figure out which one of these points is the steepest, and if we take a look, that's actually going to be point c. So that's our answer.

Moving on to the second one. Where is the car moving the slowest? Again, guys, if you go through all the steps, where moving the slowest means we're going to 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 going to look for the minimum. And if the steepest slope was the maximum, the flattest slope is going to be the minimum. So let's look at the graph. Where do we have flat slopes? Well, it's actually pretty easy. At points b and d, we have flat slopes. So those are going to be points b and d. Those actually also are where you have no motion at all. It's v equals 0. So if you're not moving, that's the slowest you go.

Now, move on to part 3. Where is the car turning around? What does turning around mean? Well, if you turn around, you're going to be moving in one direction, you stop, and then you move in the other direction. So we're still talking about motion here, which variable are we going to use? We're going to use the velocity. So we're talking about velocity, which means we're going to talk about the slope. Those are the first two steps. Now, we just have to look at the qualifier. Remember what turning around means. It means you're moving in one direction, and then you turn around and 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 going to be positive velocity. That means you're moving forward. Downward slopes are going to be negative velocity, which means you're moving backwards. So when you're looking for a direction change, you're actually looking for a sign change. At point a, because it's upward, you have a positive velocity. Point b, you have v equals 0, and then point c, because it's downward, you have negative velocity. Which means somewhere inside of this little point over here, you were moving forward, you stopped, and then you turned around and started moving backward. That actually happens right here at point b. It happens when you stop momentarily and then right before you start moving in the opposite direction. So that means that happens at point b. There is no other point with a direction change, so point d is not a turning point here.

Last two parts, where is the car speeding up and slowing down? Let's just go through our variables. Which are we talking about? Position, velocity, or acceleration? Well, you might think, oh, we're talking about speed, so we're going to be looking at speed. But actually, what's happening is speeding up means the speed is increasing. It's talking about a change in that speed. So we're actually going to be talking about the acceleration here. Now we just have to figure out which graph feature we're talking about. Again, going down to our table, we're going to see that when we talk about the acceleration in an x-t graph, we're looking at the curvature. There are a couple of rules to remember. When we have a smiley face upwards like this, that's going to be positive acceleration. A frowny face is going to be negative acceleration. We have to remember that if you break up these smiley faces and frowny faces into two halves, then on the left side, you're always going to be slowing down. The right side, your slopes are becoming more vertical here, so you're actually going to be speeding up. So let's go back to the question. We're going to be looking at the curvature over here for speeding up for the acceleration. So which qualifie...

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

Two Bicycles

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 have two moving objects or two 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 graph, then from our table of conceptual points and 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 at what time or times on this time axis over here are the bicycles at the same exact value? So let's take a look here. Well, bicycle B is going to be at this value over here, and then bicycle A is going to 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=1 is one of our times. Let's just see if it happens again later on in the diagram. Well, bicycle A will just go on like this and the values are over here. Whereas bicycle B, the values are going to be over here. Notice how these are not the same. But eventually, what happens is B is going to catch up to A or the lines are going to 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, therefore, the bicycles are at the same position. So it's t=1 and t=4seconds. Those are our two times. There's never anywhere else on the diagram in which these two lines will cross each other. So let's move on to B.

Now at what time or times do 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, we're looking for the slope. So what this question is really asking is where do the bicycles, or here. They don't have the same slopes here. So let's check it out. 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. It's constant velocity. Whereas for B, what happens is it's a curvy position graph, so we know that there's going to be some acceleration and the velocities are going to be constantly changing. So where do these two lines have the same velocity? Well, I can basically trace out what the instantaneous velocity looks like here. It's kind of hard to visualize it, or see it really clearly. So this is like the instantaneous velocity. And then right around here, it starts to trend upwards like this. And I'm basically looking for where it matches this, you know, approximate sort of steepness here. And right around at 3, 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 and 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 ahead and actually write this, in blue, just so you can see it 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=3. Then what happens is the slope is going to continuously increase for B, and it'll never be the same as A again. So that means that there's only one time that this happens. It's at t=3seconds right over here. So notice how these two slopes are the same. Alright, guys. That's it for this one. Let me know if you guys have any question.

Here’s what students ask on this topic:

How do you determine the object's position at the origin on a position-time graph?

To determine the object's position at the origin on a position-time graph, follow these steps: First, identify the motion variable, which is position in this case. Next, look at the graph feature, which is the value of the position. The origin corresponds to a position value of zero. Therefore, you need to find the points on the graph where the position (y-axis) is zero. These points are where the graph intersects the x-axis. By identifying these intersections, you can determine the object's position at the origin.

How can you find where the object is moving forwards on a position-time graph?

To find where the object is moving forwards on a position-time graph, follow these steps: First, identify the motion variable, which is velocity. Next, look at the graph feature, which is the slope of the position-time graph. Positive velocity corresponds to an upward slope. Therefore, you need to find the segments of the graph where the slope is positive. These segments indicate that the object is moving forwards. By analyzing the slopes, you can determine the intervals where the object has a positive velocity.

What indicates that an object is at rest on a position-time graph?

An object is at rest on a position-time graph when its velocity is zero. To determine this, follow these steps: First, identify the motion variable, which is velocity. Next, look at the graph feature, which is the slope of the position-time graph. A flat slope (horizontal line) indicates zero velocity. Therefore, you need to find the segments of the graph where the slope is flat. These segments represent the intervals where the object is at rest. By identifying these flat slopes, you can determine when the object is not moving.

How do you identify positive acceleration on a position-time graph?

To identify positive acceleration on a position-time graph, follow these steps: First, identify the motion variable, which is acceleration. Next, look at the graph feature, which is the curvature of the position-time graph. Positive acceleration corresponds to an upward curvature (smiley face). Therefore, you need to find the segments of the graph where the curvature is upward. These segments indicate that the object is experiencing positive acceleration. By analyzing the curvature, you can determine the intervals where the object has positive acceleration.

How can you determine where the object is moving backwards on a position-time graph?

To determine where the object is moving backwards on a position-time graph, follow these steps: First, identify the motion variable, which is velocity. Next, look at the graph feature, which is the slope of the position-time graph. Negative velocity corresponds to a downward slope. Therefore, you need to find the segments of the graph where the slope is negative. These segments indicate that the object is moving backwards. By analyzing the slopes, you can determine the intervals where the object has a negative velocity.