Graphing Position, Velocity, and Acceleration Graphs: Study with Video Lessons, Practice Problems & Examples

Understanding motion graphs involves recognizing the relationships between position, velocity, and acceleration. The slope of a position graph indicates velocity, while the slope of a velocity graph indicates acceleration. For instance, a constant slope in a position graph results in a constant velocity, leading to zero acceleration. Conversely, a changing slope in a velocity graph indicates acceleration, resulting in a curved position graph. Mastering these concepts is essential for analyzing motion effectively, as they illustrate how these variables interact in various scenarios.

Graphing Position, Velocity, and Acceleration Graphs

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 going to 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 3 motion variables. So we've got position, velocity, and acceleration. And so the slope of the position graph is going to equal the value of the velocity and we've seen that before. And then 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 going to 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 going to be given the, we're going to 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 what 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 going to 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 going to have the same value. So that means we're just going to draw a straight flat line like this. So whenever we have a constant slope like we do over here, that's going to 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? Well, we're going to use the exact same principle. The slope of our velocity time graph is going to 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 that we're going to have a 0 acceleration. So that means when I draw my acceleration, it's just going to be a flat line at 0. It's going to be a constant 0 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 can 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 to reason through. So let's check it out. So just remember the relationships. The slope of our velocity graph is going to give us the value of our acceleration graph. And so if we look at the slope of this line here, this is going to 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 going to mean a constant value for the acceleration, not 0. It's just going to be a constant value. And so let's say we can just draw this over here. So this literally looks exactly like it did over here. So now how do we calculate or 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, the numbers are actually going up. So that means that the slope is also going to have to go up. So a changing value here, and actually in this case, it's increasing, means that we're going to 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, well, actually, where do we even start from? So 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 0. 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 going to look like this and then it's going to look like that, and then it's going to keep on going up and up. And if you connect these points over here, what we're going to end up with is we're going to end up with a curved position graph. Now remember that when we first talk about curved position graphs, we said that the velocity was constantly changing and that was because there is some acceleration. And so 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 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 constant. We have a constant value for the acceleration, which means we're going to have a constant slope for the velocity. But here, the acceleration is negative. And what we know about the slopes, when they're negative is that they go downwards. So the velocity graph is actually going to start Where do we start? Well, anything negative would just look like this. So where do we actually pick the point that we start from? Well, again, the problem says that for parts b and c, we're going to assume that we start from rest and we start at the origin. So here we're going to start at v equals 0 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 going to use the same principle. The value of our v t is going to 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 going to have a decreasing slope. So, again, we're just going to start from the origin, which is going to be over here. And what does a decreasing slope look like? Well, decreasing means that we're going to start off by, you know, going relatively flat, and then it's going to go it's going to get more and more and more, vertical, but it's going to point downwards. That's what the slope decreasing is going to look like. So this is just going to 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, then that was going to be a smiley-faced curve. When we had negative accelerations, that was going to 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 us a couple more practice problems. Let me know if you have any questions.

Graph Velocity and Acceleration

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 going to start from the origin. We're shown our velocity-time graph, and 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 going to 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 going to have some positive value up until over here. But then what happens is that we have some change or 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 and something changes here. So I'm actually just going to 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 0 again. So that means that in this section right here, we have a negative constant slope, which means that we're going to 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 to the acceleration. Now what does the position-time graph look like? The first question is where are we going to start in the first place? And told we're told in the text that we're going to start from the origin, which means we're to start basically right from 0. 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 an increasing velocity, then that was going to introduce some curvature. And again, we've already seen this piece right here. So now, what happens in this section? Well, if increasing values make increasing slope, then 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 going to eventually go down towards 0. 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 0 over here. So whenever this line here becomes flat, then that's going to correspond to the velocity equaling 0 on the graph. So that means that, basically, it's going to 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.

Here’s what students ask on this topic:

How do you sketch a velocity-time graph from a position-time graph?

To sketch a velocity-time graph from a position-time graph, follow these steps: First, identify the slope of the position-time graph at various points. The slope of the position-time graph represents the velocity. If the position graph has a constant slope, the velocity is constant, resulting in a horizontal line on the velocity-time graph. If the slope of the position graph changes, the velocity will change accordingly. For example, if the slope increases, the velocity increases, and if the slope decreases, the velocity decreases. By plotting these slopes over time, you can sketch the corresponding velocity-time graph.

What is the relationship between acceleration and the slope of a velocity-time graph?

The relationship between acceleration and the slope of a velocity-time graph is direct. The slope of the velocity-time graph at any point gives the value of the acceleration at that point. Mathematically, acceleration is the derivative of velocity with respect to time. If the velocity-time graph has a constant slope, the acceleration is constant. If the slope of the velocity-time graph changes, the acceleration is changing. For example, a positive slope indicates positive acceleration, while a negative slope indicates negative acceleration.

How do you determine acceleration from a position-time graph?

To determine acceleration from a position-time graph, you need to first find the velocity-time graph. The slope of the position-time graph gives the velocity. Once you have the velocity-time graph, you can determine the acceleration by finding the slope of the velocity-time graph. Mathematically, acceleration is the second derivative of position with respect to time. If the position-time graph is a curve, the velocity will change, indicating acceleration. By analyzing the changes in the slope of the position-time graph, you can infer the acceleration.

What does a curved position-time graph indicate about velocity and acceleration?

A curved position-time graph indicates that the velocity is changing over time, which means there is acceleration. If the curve is concave up (smiley face), it indicates positive acceleration, meaning the velocity is increasing. If the curve is concave down (frowny face), it indicates negative acceleration, meaning the velocity is decreasing. The steeper the curve, the greater the change in velocity, and thus, the greater the acceleration. This curvature reflects the non-linear relationship between position and time due to the presence of acceleration.

How do you sketch a position-time graph from an acceleration-time graph?

To sketch a position-time graph from an acceleration-time graph, follow these steps: First, determine the velocity-time graph by integrating the acceleration-time graph. The area under the acceleration-time graph gives the change in velocity. Next, use the velocity-time graph to find the position-time graph by integrating the velocity-time graph. The area under the velocity-time graph gives the change in position. If the acceleration is constant, the velocity changes linearly, and the position changes quadratically. By plotting these changes over time, you can sketch the corresponding position-time graph.