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Acceleration vs. time graphs

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- [Instructor] All right, I wanna talk to you about acceleration versus time graphs because as far as motion graphs go, these are probably the hardest. One reason is because acceleration just naturally is an abstract concept for a lot of people to deal with and now it's a graph and people don't like graphs either particularly often times. Another reason is, if you wanted to know the motion of the object, let's say it was this doggie. This is my doggie Daisy. Let's say Daisy was accelerating. If you wanted to know the velocity that daisy had, you can't figure it out directly from this graph unless you have some extra information. You have to know information about the velocity Daisy had at some moment in order to figure out from this graph the velocity Daisy had at some other moment. So, what can this graph tell you about the motion of Daisy? Well, let's say this graph described Daisy's acceleration. So Daisy can be accelerating. Maybe we're playing catch. We'll give her a ball. We'll throw the ball. Hopefully she actually lets go and she brings it back. This graph is gonna represent her acceleration. So this graph, we just read it, it says that Daisy had two meters per second squared of acceleration for the first four seconds and then her acceleration dropped to zero at six seconds and then her acceleration came negative until it was negative three at nine seconds. But, from this we can't tell if she's speeding up or slowing down. what can we figure out? Well, we can figure out some stuff because acceleration is related to velocity and we can figure out how it's related to velocity by remembering that it is defined to be the change in velocity over the change in time. So this is how we make our link to velocity. So if we solve this for delta v, we get that the delta v, the change in velocity over some time interval, will be the acceleration during that time interval times interval itself, how long did that take. This is the key to relating this graph to velocity. In other words, let's consider this first four seconds. Let's go between zero and four seconds. Daisy had an acceleration of two meters per second squared. So that means, well, two was the acceleration meters per second squared, times the accel, times the time, excuse me, the time was four seconds. So there was four seconds worth of acceleration. You get positive eight. What are the units? This second cancels with that second. You get positive eight meters per second. So the change in velocity for the first four seconds was positive eight. This isn't the velocity. It's the change in velocity. How would you ever find that for this diagonal region. This is as problem. Look at this. If I wanted to find, let's say the velocity at six seconds, well the acceleration at this point is two but then the acceleration at this point is one. The acceleration at this point is zero. That acceleration we keep changing. How would I ever figure this out? What acceleration would I plug in during this portion? But we're in luck. This formula allows us to say something really important. A geometric aspect of these graphs that are gonna make our life easier and the way it makes our life easier is that, look at what this is. This is saying acceleration times delta t, but look it. The acceleration we plug in was this, two. So for the first four seconds, the acceleration was two. The time, delta t, was four. We took this two multiplied by that four and got a number, positive eight, but this is a height times the width. If you take height times width, that just represents the area of a rectangle. So all we found was the area of this rectangle. The area is giving us our delta v because area, right, of a rectangle is height times width. We know that the height is gonna represent the acceleration here and the width is gonna represent delta t. Just by the definition of acceleration we arranged, we know that a times delta t has to just be the change in velocity. So area and change in velocity are representing the exact same thing on this graph. Area is the change in velocity. That's gonna be really useful because when you come over to here the area is still gonna be the change in velocity. That's useful because I know how to easily find the area of a triangle. The area of a triangle is just 1/2 base times height. I don't easily know how to deal with an acceleration that's varying within this formula but I do know how to find the area. For instance the area here, though I have 1/2, the base is two seconds, the height is gonna be positive two meters per second squared. What are we gonna get? One of the halves, cancel. Well, the half cancels one of the twos and I'm gonna get that this is gonna be equal to two meters per second. That's gonna be the area that represents the change in velocity. So Daisy's velocity changed by two meters per second during this time. Now you might object. You might say, "Wait a minute. "I'll buy this over here because height times width "is just a times delta t, "but triangle, that has an extra factor of a half in it, "and there's no half up here. "How does this, I mean, how can we still make this claim?" We can make this claim because we'll do the same thing we always do. We can imagine, all right, imagine a rectangle here. We're gonna estimate the area with a bunch of rectangles. Then this rectangle, and this rectangle in your line like that looks horrible. That doesn't look like the area of a triangle at all. It's got all these extra pieces right here, right? You don't want all of that. And okay, I agree. That didn't work so well. Let's make them even smaller, right? Smaller width. So we'll do a rectangle like that. We'll do this one. You see we're getting better. This is definitely closer. This is not as bad as the other one but it's still not exact. And I agree, that is not exact so we'll make it even smaller rectangle and an even smaller rectangle here all of these at the same width but they're even smaller than the ones before. Now we're getting really close. This area is really gonna get close to the area of the triangle. The point is if you make them infinite testable small, they'll exactly represent the area of a triangle. Each one of them can be found with this formula. The delta v for each one will be the area, or sorry, the acceleration of the height of that rectangle times the small infinite testable width and you'll get the total delta v which is so gonna be the total area. Long story short, area on a, acceleration versus time graphs represents the change in velocity. This is one you got to remember. this is the most important aspect of an acceleration graph, oftentimes the most useful aspect of it, the way you analyze it. So why do we care about change in velocity? Because it will allow us to find the velocity. We just need to know the velocity at one point then we can find the velocity at any other point. For instance, let's say I gave you the velocity Daisy had. For some reason I'm gonna stopwatch. I start my stopwatch at right at that moment. At t equals zero, Daisy had a velocity of, let's say positive one meter per second. So Daisy was traveling that fast at t equals zero. That was her velocity at t equals zero seconds. Now I can get the velocity wherever I want. If I want the velocity at four, let's figure this out. To get the velocity at four, I can say that the delta v during this time period right here, this four seconds. I know what that delta v was. That delta v was positive eight. We found that area, height times width. So positive eight is what the delta v is gotta equal. What's delta v? That's v at four seconds minus v at zero seconds. That's gotta be positive eight. I know what v at zero second was. That was one. So we can get that v at four minus one meter per second is equal to positive eight meters per second. So I get the velocity at four was positive nine meters per second. And you're like, phew, that was hard. I don't wanna do that every time. Yeah, I wouldn't wanna do that every time either so there's a quick way to do it. We can just do this. What's the velocity we had to start with? That was one. What was our change in velocity? That was positive eight. So what's our final velocity? Well, one plus eight gives us our final velocity. It's positive nine. Well it's just gonna take this change in velocity of this area which represents the change in velocity which is gonna add our initial velocity to it when we solve for this final velocity. for instance, if I didn't make sense, for instance, if we want to find the velocity at six, well, we can just say we started at t equals four seconds with a velocity of positive nine. We start here with positive nine. Our change was positive two so we're gonna end with positive 11 meters per second. You might object. You might say, "Wait a minute, hold on now. "If we want delta v, "right, and that's positive two, "shouldn't delta v be the whole thing "from like zero to six seconds? "Shouldn't I say v at six seconds minus v at zero "is positive two meters per second?" I can't do that. The reason I can't do that is because look at what I did on the left hand side, my time interval goes from zero to six but on the right hand side, I only included the area from four to six. That's the area, there's a yellow triangle right here. If I wanted to put six and zero on this left hand side, I could do that but from my total area, I wouldn't use that. I have to use the total area. In other words, the total are from zero all the way to six because that's what I define on this side. These sides have to agree with each other. So from zero to six, my total area would be, this area here was eight, right? We found that rectangle was eight. This area here was two. So my total area would be 10. I can do that if I want. I could say v at six minus v at zero was, well v at zero we said was one because I just gave you that, equals 10 meters per second. I get that the v at six would be 11 meters per second just like we got it before. So you can still do it mathematically like this but make sure your time intervals agree on those sides. Now let's do the last part here. So we can find this area. This area and the area always represents the area from the curve to the horizontal axis. So in this case it's below the horizontal axis. That means it can negative area. The reason is it's a triangle again. So 1/2 base times height. So 1/2, the base is one, two, three seconds. The height is negative three, negative now, negative three meters per second squared. I get that the total area is gonna be negative 4.5 meters per second. All right, now Daisy's gonna have a change in velocity of negative 4.5. If we want to get the velocity at nine, there's a few ways we can do it. Right, just conceptually, we can say that Daisy started at six with a velocity of 11. Her change during this period was negative 4.5. If you just add the two, you add the change to the value she started with. Well you're gonna get positive 6.5 if I add 11 and negative 4.5 meters per second or, if that sounded like mathematical witchcraft, you can say that, all right, delta v equals, what, negative 4.5 meters per second. Delta v would be, all right, you gotta be careful, this negative 4.5 represents this triangle so it's gotta be the delta v between six and nine. So v at nine minus v at six has to be negative 4.5 meters per second. V at nine minus the v at six we know, v at six was 11. So I've got minus 11 meters per second equals negative 4.5. Wow, we ran out of room. V at nine would be negative 4.5 plus 11. That's what we did up here. We got that it was just 6.5 meters per second and that agrees with what we said earlier. So finding the area can get you the change in velocity and then knowing the velocity at one unknown at a time can get you the velocity at any other moment in time. Just be careful. Make sure you're associating the right time interval on both the length and the right side. They have to agree. One more thing before you go. The slope on these graphs often represents something meaningful. That's the same in this graph. So the slope of this graph, let's try to interpret what this means. The slope on an acceleration versus time graph. Well the slope is always represented as the rise over the run and the rise is y two minus y one over x two minus x one except instead of y and x, we have a and t. So we're gonna have a two minus a one over t two minus t one. This is gonna be delta a, the change in a over the change in time. What is that? It's the rate of change of the acceleration. That is even one more layer removed from what we're used dealing with, right? Velocity, velocity is the change in position with respect to time. Acceleration is the change in velocity with respect to time. Now we're saying that the something is the change in acceleration with respect to time. What is it? It's the jerk. So this is often called the jerk. That's the name of it. It's not used all that often. It's quite honestly not the most useful motion variable you'll ever meet and you won't get asked for that often most likely on test and whatnot but it has its application sometimes that exist and it has a name that's called the jerk. So recapping, the area, the important fact here is that the area under acceleration versus time graphs gives you the change in velocity. Once you know the velocity at one point, you could find the velocity at any other point. The slope of an acceleration versus time graph gives you the jerk.