Calculus: Antiderivatives and Motion

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Applications of Integration
Motion
Antiderivatives and Motion Page [1 of 1]
Remember that the derivative represents instantaneous velocity, instantaneous rate of change. And now realize, though, that we can now do is think about the issues we've thought about already backwards. Why we want to do that? Just for fun? Well, maybe, but also because this is incredibly practical, and in fact, now we are coming to some really serious and real world applications. To inspire that, let me first bring you back up to speed with all the stuff we've done early on about positions and velocity and acceleration.
So, a really quick review of what we've done a long time ago. First off, if I gave you a position function, so p(q) means that we have a function that the little machine, that if you give me time, it tells me my location or position. I want to now find the velocity, the instantaneous rate of change that point, what I will do is I will take the derivative. And we talked a lot about the fact that, in fact, the velocity function, which I can call v(p), is nothing more than the derivative, the instantaneous rate of change, of position. Great, we looked at all sorts of examples, I gave you position and asked how fast I was going, the very first question I asked you, was about my bike ride and was I breaking the law when I passed the twenty mile an hour speed sign.
Then a little later on we actually looked at the notion of acceleration, which is nothing more than how the velocity is changing. Are we accelerating with my velocity increasing, or am I decelerating with my velocity actually decreasing? And so, when you talk about acceleration, call that a(p), that's the acceleration function. You give me a time, I'll plug in here, and tell you what the acceleration is at that time, and that's nothing more than, well, the derivative of velocity, how velocity is changing. If you want to know if something changes, you take the derivative. The derivative would be the velocity prime, or the derivative of velocity. And of course, if you think about the fact that velocity is just the derivative of position, I could actually phrase acceleration in terms of position. It wouldn't be the first derivative; it would be the first derivative of the derivative, so it would be the second derivative. So I add one derivative to get into the velocity and if you take the derivative of that to get me back to acceleration. So, we have position with position, we take the derivative, we got velocity. With velocity, we took the derivative, we got acceleration. Okay, now why is all this relevant and how can we use that and how does it tie in with integration?
Well, the answer is that before I gave you position function and I asked you to deduce velocity or acceleration. Now, we're in power to do that process backwards. In particular, what we can now do is if I'm giving you the acceleration, if you integrate, that would give you velocity. Because velocity is a function whose derivative is acceleration. And if I integrate the velocity, what would I get? I would get position. Because notice that position is a function whose derivative is velocity. So, I can now move backwards as where before I was given P and got through a and now if I'm given a, if I anti-differentiate, I get to v, and if I anti-differentiate again, I get to P. So, that's the power. Let me give you an actual illustration of how this can actually be utilized. So, let me bring back, in fact, my bicycle and let me tell you what happened to me. What happened to me was: I was riding one day, not the same time that I talked about early on, but I was riding one day and it was a little wet, and I was probably going a little too fast to be honest with you. I was going actually 30 feet per second. Do you have any idea how fast that is? That is really, really fast. I was just cooking. I was going way too fast and the pavement was wet. What happened was I actually all of a sudden realized that I was going too fast. So, I was coming in like this, and what I did was, I actually jammed on the breaks. Let me show you exactly what happened to me. So, watch really close and you will see exactly what happened. So, I was coming in like this, you see, coming in way too fast, so I jammed the breaks, eeeeeek, and I finally stopped. Whew, it scared me so much I mean even right now, just talking about it my heart is going boom, boom, boom, boom, because it was really, really a long skid mark and it was really scary, and I was skidding out of control. However, I do know one thing about my skidding, I was skidding at a constant deceleration, which means, of course, I was slowing down and I was slowing down at a constant rate of 20 feet per second. So, I was slowing down, I was decelerating at a rate of 20 feet per second. And now, here's the question I want to ask you. Given the fact that when I started to skid when I was coming in, I was going 30 feet per second. I want to know how long is that skid mark? How long was I skidding before I finally stopped?
Well, that's an interesting question, actually a question that you might want to know the answer to for certain things. That is to say, I want to know how long this is. What do I know? I know that I was coming in and going 30 feet per second, I jammed the breaks, and then started decelerating at a constant rate of 20 feet per second; I finally came to rest. I now want to know how long that skid mark was. How do I do the problem? I don't know anything about position; all I know is acceleration. So let's see if we can use this idea somehow to get us up to figuring out the answer to this question.
Okay, so, the first thing I want to do is the following: let us write down exactly what we want. So, what I want is the length of this skidding trip. So the length--find the length of the skidding trip--so that length. Okay, that's what we want to find. What do I know? Well, what I know is that first of all, as I was skidding, I was decelerating at a rate of 20 feet per second. So, that tells me about my acceleration. So in fact, I know my acceleration is what? Well, it was 20 feet per second, so you might be tempted to write down 20 feet per second, and in fact, acceleration, you need a feet per second squared because it is a second derivative, so its feet per second squared. So, you may think that in fact is my acceleration, but that would be incorrect, because this is the process where I was breaking and I was actually slowing down. So, I actually need a negative sign here to tell you that I was decelerating. If I were accelerating, if I was speeding up, then in fact, it would be positive, but it's not I was slowing down, so I need a negative sign there, need a negative sign when you decelerate, you need a negative sign. Okay, well, that's my acceleration, so I'm told that. What else do I know? Well, I know something about velocity; I know that velocity at the very moment that I entered the skid. What was the velocity at the very moment I entered the skid? Well, I told you that was 30 feet per second. So, how can I write that down? Well, at the very beginning of this skid time, my velocity was 30 feet per second. So, I could write it this way: velocity at time zero, when I started the skid, is actually equal to 30 feet per second. Now, look what I wrote here, I put down v(0) because that's my initial velocity. The moment I started to skid, I know I was going 30 feet per second. So, I put in P = 0 and I know that that V, the velocity, at the initial time was 30 feet per second. So, I know that and I know the acceleration, and if you were going to start the measurement, here by the way, if we were going to start measuring from that point on, where am I when I first start? Well, my position when I first start is at zero. Where I start to skid. So, I could also write down that the position function is valued at zero is zero feet because at that instant, that's where I'm starting my measurement from.
So, let's recap all the things we know. I know that the acceleration function is -20, the minus because I'm decelerating and 20 is what I was going at a constant rate, constant deceleration at. The initial velocity, my velocity at the moment I started to skid, was 30 feet per second, and then I also know that my initial position, my position when I started to skid was zero. I'm starting to measure from there, and I want to know how long I traveled from that point up. Well, if I want to find out my distance traveled, I want to find out the length, I have to figure out the position function. So, we need to figure out the position.
So, how far did I skid? I need to know that. Well, how do we do that? We need to find position. All I know is acceleration, so I have to go from the acceleration and work my way backward. Well, that's now where we can use the notion of an anti-derivative, or integral, because if I take acceleration, which I know to be the derivative of velocity, if I were to integrate that, I would get velocity out. Once I have the velocity function, if I find the anti-derivative of that, that would give me position. So, in fact, this is a problem set up perfectly for anti-derivatives. So, we are going to begin by integrating or finding the anti-derivative of acceleration, and the recipe that I'm going to use here is that if I integrate acceleration, I get velocity plus a constant, and this just comes out of this fact right here. If acceleration is the derivative of velocity, then if I integrate acceleration, I get the velocity. So, let's start right here and we'll come back to this page. We know that acceleration of -20, so I want to integrate -20dt, because everything here is in terms of time. Well, what's the integral of -20, well, the integral of -20--is it zero by the way? Is an integral a constant zero? No. The derivative of a constant of zero, but the integral of a constant is actually -20t, because the derivative of that would just give you the constant minus 20. But, I have a constant that I have to add here, plus c. That's the integral or the anti-derivative of acceleration, so that's actually velocity. So we just found the velocity function. Great, we're almost done.
Except now the c is very mysterious. How do I actually figure out the value of c? What does this equal? Well, how can I figure out that value of the constant? Is there anything I know about velocity at all? Well, if we go back to what we already know, I remind you that we do know the initial velocity, the velocity was t=0, is 30 feet. So, in fact, if I let t be zero, I know that the velocity has to be 30. So, in fact, if I come back to here, since I know that v at zero is 30, then this must be satisfied when I plug that into here. So, what does this mean? I let t be zero and this whole thing has to be 30. Well, if I let t be zero, and I put zero in for t and I see that -20(0) + c has to equal 30. Well, that's just zero and look what I discover? I discover that c has to equal 30. So that constant, which we didn't know what it was, we now see for sure is 30. It's the initial velocity; it's the velocity that I had the moment my skid began. And I got that by just plugging in t for zero and knowing my velocity there has to be 30, and if I solve that I get c=30. Well, great, well now I know my velocity for certain, my velocity function is -20t + 30. What I really want, though, is my position. So, I've got to find p(t). Well, what I remember about p(t), what I know is if I were to integrate velocity that would give me p(t) plus the constant, and why? Well, because this is the function that if you take the derivative of it, you get the velocity. So, now I'm going to integrate, or anti-differentiate, the velocity. What do I get? I see the integral of -20t + 30dt, and I'll just take it a little bit at a time.
First, I'll do this one, -20t, what's the anti-derivative of that? Well, I add one to the exponent, so I have a two exponent now, and I divide through by two, and so I see a -10t^2 plus a derivative of a constant 30 is just 30t. Don't forget the constant of integration. Don't believe me? Let's check our answer really fast. The derivative of this is -20t, the derivative of this is 30, the derivative of that is zero. Check. Look how great. You can always check your answer, no mysteries here. No mysteries. Well, that is the position function. So, the position function, okay, we're almost done here, is equal to -10t^2 + 30t + c. Now, of course again, I'm running into the same problem we had before, I don't know what c is. Is there anything I know about position? Well, the position that I was located at when I first started to skid is my starting position in terms of my skid. And so, like we said before, I'll just remind you, my position at times zero, I'm calling that zero feet because I want to measure this skid from the beginning to the end. So, if I let t be zero, my position should equal zero. Let's plug in t for zero and see what I get, if I know that p(0) equals zero, I plug in zero here, I get zero. If I plug in zero, I get zero. So, our c here is just 0 + c and that has to equal zero. So, this constant turns out to be the number zero. Well, that's pretty easy, so now I know what position is, position's pretty easy to figure out. Position is equal to -10t^2 + 30t + 0, well, that's position. Well, great, well, now armed with position, I can try to answer the question, because the question, you remember, was how long was the skid? Well, now I'm seeing I have position in terms of time. So, I have to actually figure out what time I stopped skidding. Once I know that time, then I can plug into the position and figure out how far the skid was.
So how can I figure out how long I was skidding? Well, what happens after the end of my skid? How do I know when I stopped skidding? The answer? I stop moving. So, I have to first find the time when I end my skid, and that's the exact time that I stopped moving. And what does it mean to stop moving? It means that my velocity must be zero. So, I have to figure out when was my velocity zero? Well, that's actually not too hard because we've already figured out the velocity function, the velocity function, we did that over here. I'll remind you of that. The velocity function was right here, -20t + 30. So, I want to find out when is that zero? So, my velocity function is -20t + 30, and I want to know, when did I stop? When do I stop? So what do I do? I find out, I solve for when the velocity is equal to zero, and that's -20t + 30 = 0, and what do I see? Well, I see the t would equal -30/-20 which equals 3/2, and we want it in seconds, so this is actually 1 seconds. So, that means that I stop, I stopped after 1 seconds, and I found that out by examining the velocity and asking when is the velocity zero? That's when I stopped, when the velocity of zero. That means that the length of time I traveled here was 1 seconds. But the question I asked was, the length of the skid itself.
Well, now that I know the time it took me to stop, I can go back to the position function and ask where is my position after 3/2 seconds. So, now I just find the position at 3/2 seconds, and I plug into the position, -10 x 3/2^2, which is 9/4 + 30 x 3/2 and what does this equal? Well, we'll do some canceling here, and I see that this becomes a 2 and this becomes a 5, and so I see here 5 x 9, which is 45/2, so this equals -45/2, and here I get a little cancellation, and here I see this is a 1 and this is a 15, and so I see 15 x 3 which is 45, so now I add on 45, and so what's -45/2? Well, that's going to equal 45/2. Which is roughly 22 feet. So, in fact the answer to my question, how far did I skid? I now see I skid 22 feet. That was a really long skid. How did I find it? I just knew acceleration. I integrated to find velocity, and then to get the constant, I actually plugged in what I knew about velocity to solidify the constant. Then once I had the velocity, I integrated the velocity to get position, I solidified the constant again. I found out where the velocity was zero when I stopped. Once I had that, I plugged that value of t in for p and found our position. So, you can really see the power and the utility of the anti-derivative. Just knowing the acceleration of velocity gives me location. We'll see more practical example of this up next. I'll see you there, bye for now.