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Calculus: Solve Distance & Velocity Word Problems


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About this Lesson

  • Type: Video Tutorial
  • Length: 22:06
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 238 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus: Final Exam Test Prep and Review (45 lessons, $64.35)
Calculus: Applications of Differentiation (15 lessons, $31.68)
Calculus: Position and Velocity (2 lessons, $4.95)

Taught by Professor Edward Burger, this lesson comes from a comprehensive Calculus course. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers limits, derivatives, implicit differentiation, integration or antidifferentiation, L'Hopital's Rule, functions and their inverses, improper integrals, integral calculus, differential calculus, sequences, series, differential equations, parametric equations, polar coordinates, vector calculus and a variety of other AP Calculus, College Calculus and Calculus II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, "Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas" and of the textbook "The Heart of Mathematics: An Invitation to Effective Thinking". He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The "Journal of Number Theory" and "American Mathematical Monthly". His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

About this Author

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Practical Application of the Derivative
Position and Velocity
Solving Word Problems Involving Distance and Velocity Page [1 of 4]
Okay, let's take a look at the derivative as a means of finding instantaneous velocity. Now, we looked at some of these problems earlier on when we were looking for the definition of a derivative. And I now want to take a look at another example just to get us back into the spirit of that. And now, you know one question that, that comes up again and again is the question of - well, okay, some physical thing happens, like my bicycle ride early on in our discussions. And I need to have a function that describes that activity, and in, in that example I gave it to you. And the natural question is where do those functions come from? How does, actually, someone find those functions in practice? And, so, let me just tell you really quickly how one could actually do that, and, in fact, in reality, how these are found.
The truth is that one has to perform an experiment, and whether it's fluid flowing through a pipe, or whether it's something going out into space, or whatever the activity is, one performs an experiment. And usually one collects a lot of data points, whether they take a lot of photographs, or videos, or time things and see -- use stopwatches compared to where locations are, and they collect a lot of data points. And, basically, they plot those data points on a graph. You have a graph, in fact, you have one right there, there's an example. And, now, I put down a whole bunch of data points, blip, blip, blip, blip, blip, blip, blip. See them all there? Okay, now, we have some sort of model, some sort of assumption that we're making, maybe that the thing is growing like a parabola, maybe the thing is growing like some other function. And then we use some sort of statistical software, some sort of program, and, actually, a little bit of calculus in there as well, believe it or not, but that's another story, to actually fit those data points with the right curve, a curve that fits them pretty well.
For example, if we pretend that this is actually quadratic, then watch what happens when I try to fit it, as best as I can, with a quadratic. Look, see that quadratic fits really well, and, in fact, the equation for that quadratic turns out to be, there it is. So, actually, I can get the equation and then come back, and then estimate the instantaneous velocities at any point in-between those particular points, or even at those particular points. So, in fact, in, in the real world one can find those, those formulas if one wanted to. But, in our special world, we'll just at least pretend that someone already did that work for us and they gave us the function.
Okay, well, let me give you an example and let's see if we can actually take a look at velocities. So, suppose that some disgruntled student that's taking Calculus decides that he's mad as all heck and he's not going to take it anymore. So, he goes out, takes his Calculus books, book, and just hurls it up into the air, throws it up into the air and it plummets back down onto earth. And suppose that the position function for that activity, the book rising up into the sky, and then plummeting back onto the earth, with gravity doing its part, is given by the following. P of t equals minus sixteen t squared, plus ninety-six t.
Well, now I want us to answer a few questions. The first question, so this is the position, so if you give me a time, and let's suppose that time, by the way, here is in seconds, and the position is given in feet. So, the first question I want us to resolve is how high, how high does the book attain as a maximum height. The book is going to go up, up, up, up, up, up, up. How high will it finally get before it comes down? That's the first question I want us to answer. And the second question I want us to think about is what time does it hit the ground? When does it hit the ground? So, how long do you have to wait for the book to return? And, in fact, here's a little follow up question we could ask. With what velocity does that book pound into the ground with? How fast does the book hit at that final velocity?
Well, we're going to work all this out, but I thought it might be interesting for you to actually see this happen, so you can actually see what this looks like. So, I have here a sample Calculus book here that we found. By the way, you notice that Calculus books are always so darn thick, look how, look how thick that Calculus book is. Calculus, Math to Pump You Up. I think the only pumping that this book can do is if you actually just keep lifting it, you can sort of build muscles, but it's a thick, thick book. And, so, now, suppose that a disgruntled student takes this and wants to hurl it up in the air. And to give you a sense of perspective here, I brought in the Eiffel Tower. So you can see, this gives you, this shows you - look how big that Calculus book is compared to the Eiffel Tower. You can see these Calculus books are big. And, they basically cost about the same amount as it costs to build the Eiffel Tower. Have you noticed this cost you a good chunk of change? Well, anyway, so there's the Eiffel Tower to give you a sense of perspective. And now the Calculus student, who is right below us, hurls this up into the air and let's see what happens. Here we go. I'm going to do this for you live, by the way. So, I'm going to hurl it up and notice this will go up and come down. Here we go. Ready? Okay. And there's the Calculus student. Oh, no, oh, it was Mr. Bill. It was Mr. Bill, the Calculus student who apparently did not fare too well by that frustration in Calculus.
Okay, so let's try to attack this problem and think about how we're going to answer this question. The first question we want to ask is how high did the book get at its maximum height? Now, how are we going to figure that out? I want you to think about what the activity was. The activity was the book started here, went up, and then came down. What property does the -- what, what happens, what happened when the book attains its maximum height? What can you tell me about the book at its maximum height? Can you think of something? It goes up, maximum height, comes down. So, what actually is happening at the maximum height? In fact, let me give you an opportunity to actually try to guess an answer here. So, think to yourself, how would you classify, how would you describe, how would you know when the book is at its maximum height? What has to happen? Think about that and make a guess.
Well, let's think about that. The book, of course, as it's going through the air when it starts off has a velocity that's pushing it upwards. Now, how do we know when it has the maximum height? Well, it has the maximum height when, in fact, it's about to go down. It goes up no further and it hasn't come down yet. So, it's not going up and it's not going down. So, what's its velocity at its maximum height? Its velocity would actually be zero. It's no longer going up, it's no longer coming, it's not quite coming down. It's that moment where it just stops for a split instant before it started to fall. So, at that moment the velocity is zero. So, the maximum height, the maximum height happened when the velocity is zero, is at a point, whoops, is at a point when the velocity equals zero.
So, how do we find the velocity function? Well, we know that to find the velocity we need to take the derivative of positions. So, I'm going to write the velocity function right now, and I'm going to use V of t to indicate the velocity function. But, of course, we know the velocity is just the derivative of the position function, and what is that? Well, that actually equals minus thirty-two t, plus ninety-six. I just took the derivative. By the way, one thing you'll notice in our discussions together, I'm going to start to put less and less, and less emphasis on actually computing the derivative. We've worked on that and I hope that you feel a little bit more comfortable with it. I'm feeling comfortable with it, so I'm going to take the derivative as needed, but not harp on it anymore. Of course, I just harped on it now, but I won't do that anymore.
Okay. Well, that's the velocity function and the highest point the book attains is at the moment when the velocity is zero. It no longer is going up; it hasn't quite started coming down. It just goes up, stops for one split second and comes down. When does that happen? Well, set this equal to zero. So I set this equal to zero and solve. And if I solve that, what do I see? I see that t equals, if I bring this over I see minus ninety-six divided by, if I divide by the minus thirty-two, I see minus ninety-six divided by minus thirty-two, which equals what? Well, let's see, so the negative sines drop out, and then how would you cancel this? Well, probably some big thing goes into this. Let's see, certainly two goes into each of these. So that would be forty-five and, so it would be forty-eight on top, and on the bottom we have sixteen, and look, oh, and, so, in fact, four goes into this now. I'm very bad at arithmetic, by the way, but you'll watch me go through it right now. If I cancel the four out, I'll see a twelve here, and if I cancel a four out of here, I see a four, and, oh, that just equals three. Okay, you probably saw that immediately and I didn't, but it's live, folks and I can't do arithmetic and I don't mind.
So, t equals three, this is three seconds. So the answer is at three, after three seconds, or at t equals three seconds, book is at max height. What is the maximum height, by the way? How high does the book actually get? Well, we could actually figure that out by doing what? Going back to the position and plugging in time equals three and seeing what the position is how high off the ground it is. Let's do that just for fun. So p of three equals minus sixteen, notice I have to go back, not to the derivative, but to the position function because I want a position, three squared is nine, plus ninety six times three. And what does that equal? Well, nine times six is fifty four, and nine, and then I have to add a five, which would give me, I guess, one forty four with a minus sign in front, and then plus here I get eighteen, twenty seven and one is twenty eight. So, I get two hundred and eighty eight minus one hundred and forty four, which actually equals one hundred and forty four feet. Wow! So, Mr. Bill actually hurled that thing up in the sky a hundred and forty four feet. That was the highest point it got and it reached there after exactly three seconds. So that, now, analyzes and takes care of the first question I asked.
The second question I asked was when exactly did the book hit the ground, or more precisely, when exactly did the book hit Mr. Bill? Well, let's think about that. When does the book hit the ground? What condition, what property, or what fact must hold for the book to hit the ground? Well, it starts at the ground and then it went up, and then it came down. So, its position, once it hits the ground, must be equal to what? Well, it must equal to the position that we started at, and we're assuming we're starting at ground level. So, in fact, we have to ask the question, when is the position zero? When is the position, when is the location zero? When is it the same as where we started? So, I don't set the derivative equal to zero there. Instead, I take, actually, the original position function and set that equal to zero.
Let's try that right now. Let's take the position function and set it equal to zero. So, let's set the position function equal to zero to find out when the book hit -- struck Mr. Bill. And, so the position function, I remind you, is minus sixteen t squared, plus ninety-six t, and I set that equal to zero and solve. Well, it's a quadratic I want to factor and you see that I can factor. I can factor out a t from this. So, we factor out a t from this, so I take out a t and I see minus sixteen t, plus ninety-six equals zero and so I see there are two solutions here. Either this term is zero, or that term is zero. If this term is zero, that means t equals zero. And the other possibility is that this term is zero, which I'm going to actually solve in my head. If this term equals zero that means that minus sixteen t would equal negative ninety-six. So, if I divide both sides by negative sixteen, I would see that t equals minus ninety-six over minus sixteen, which, now, I'm going to have to reduce again. These signs go away, and what do I see here? I see that -- well, four goes into ninety-six how many times? Six and four goes into here, four times, and so, here I see three over two, or one and a half seconds.
So I have two answers. Does that make sense? Absolutely, because when we started the book was actually at ground level and that was the time equal zero, and then it went up and came down and it must have come down one and a half seconds later. Great. So, that takes care of that question.
And, now, the last question that I wanted to ask was what was the impact? How fast did the book hit the ground? How would you find that? Well, now I want a velocity. I want to know what the velocity is at one and a half seconds. By the way, is this really right, because I don't actually believe, I don't believe that's right, by the way? Do you believe that's right? In fact, that can't be right, because this is completely, you know, by the way, that was the sound man that whispered that, because it can't be right. In fact, let me tell you why, of course, this can't be right. Of course, this is completely idiotic, let me put it down here. This is Mike, the soundman. You have to acknowledge -- yes, you know, and, you know, when you're, when you're, when you're writing a scholarly paper or you're writing a paper, you have to acknowledge all of your sources. And so I want to acknowledge my source. I don't want you to think that somehow I said that, because that would actually be plagiarism. So, in fact, Mike, the sound guy said that.
So, let's see how we could actually see, even if we didn't know anything about arithmetic, as I don't. Let's just see that, in fact, this, this fact here is actually a falsehood, and let's see if we can verify that. The reason why we can verify that is the following. What Mike is claiming is that one and a half seconds later, it strikes the ground. Well, what have we already found correctly? We've discovered correctly that it reaches its highest point after three seconds. So, how could it reach its highest point after three seconds, yet strikes the ground after only one and a half seconds? One and a half seconds, we know exactly where it is. It's on its way up to that maximum. You see that? So, in fact, this actually brings up an important point. And I don't think that we should be too mad at Mike -- although once the cameras are off, I'll have a couple of select words for him, but I won't share those with you right now.
But, there's an important lesson here. And the lesson is when you get an answer you should think to yourself, does the answer make sense? The moment you see one and a half seconds, that's the time it finally strikes the ground, that's strange because it takes three seconds for it to go up. So, in fact, this mistake is a great mistake, all mistakes are great. So, we're going to give him a star for this and let's just try to do the arithmetic correctly, by the way, to finish this off.
So, let's do this now, correctly. And I'll do this one, and I'll have to do it by hand very slowly. So, let's see, two goes into here, it's forty eight, two goes into there, it's eight. And now when I cut it, I see twenty-four. That actually sounds a little bit better to me. And that's going to be twelve over two, which equals six. And that sounds a lot better. If it goes up you see three seconds, and then it comes down another three seconds later. So, the answer is t equals six seconds. So at six seconds, and it hits the ground. So I'll do the whole thing for you. It goes up, three seconds -- it's here, comes down, six seconds there.
The last question I wanted to ask you is what was that final impact? How did it impact the ground? A natural answer that you may guess is well, zero, because it stopped. But that's actually not correct, because remember that when it hits the ground -- in fact, if you take your hand, or if you take an object and you throw it up, when it comes down you feel it. You feel it. You feel pressure, and what you're feeling is the fact that your hand or an object wants to keep going. There's an impact. Now, at -- the moment after it hits the ground it is no longer moving. So, when it hits the ground, at that instant, it wants to keep going down. My hand doesn't know there's something there. It wants to go right through it. So it pushes down with a particular force, and that force is the velocity.
So, how would I find that? I would go back to the velocity function, which we've already found. The velocity function we saw, we computed that earlier and saw that's minus thirty-two t plus ninety-six. And what do I want to do? I want to find the velocity at that final impact. Well, what time did that happen? We just discovered that happened when t equaled six. So I want to find the velocity when t equals six. So, I plug in six here and I see minus thirty two times six plus ninety-six. Let's not ask Mike to do this. Let's do it in our heads. Six times two is going to be a two, this is going to be eighteen and one is nineteen. So, that'd make it minus one hundred ninety two, and then plus ninety-six. Now, this is going to be taxing, it's going to be negative, and this will be a six, and that will be an eighteen and nine is minus ninety-six. Ninety-six of these units here are feet per second. So the answer is minus ninety two feet per second.
Now, let's think about that for a second. The velocity is minus ninety two feet per second. What does that mean? What does the negative sign mean? The negative sign actually gives you an indication of how the object was moving. Negative means in a negative direction. You think about how we have axis, that's negative and that's positive. So, the fact that we see negative ninety-six tells us that that must be going in this direction. And that, of course, is the truth. We know what happens. It goes up and then it comes down. And so the point here is that the sign actually tells us the direction, which the object is moving in. A negative means down. If we would have got the answer to be ninety-six, we would know now that would be wrong, because, of course, the book is not going up at that point, it's going down. So, a positive rate in a vertical motion means going up. A negative rate means going down. You get a real sense of the, the sign of that by looking at that.
Now, what about acceleration? Well, acceleration is just the change in velocity, so, the change in velocity. And how do you measure change? Change is just the derivative. So to find the acceleration, what do we do? We take the derivative of velocity. We want to see how velocity is changing and so we take the derivative of velocity. And the derivative of velocity, in this case, here's the velocity function, the derivative of that is minus thirty-two, and the units here, by the way, would be feet per second squared. So, acceleration is nothing more than the derivative of velocity. And notice we could actually say this in a different way. How does acceleration, how would we build acceleration from the position function? Well, in some sense, if you have the position function, you first need velocity and then you need to take the derivative of the velocity. So, you need to take the derivative of position one to get the velocity, and then take the derivative of that to get the acceleration. So, in fact, another way of thinking about this -- it's actually, well, what compared to the position, it's not the first derivative, it actually requires two derivatives. One derivative to get me to the V and another derivative to get me to the V prime. So, in fact, I could write this to mean a second derivative, a second derivative, which just means the derivative of the derivative. So, in fact, acceleration is the second derivative of position, or the first derivative of velocity. And acceleration measures the rate of change of velocity. So, when you decelerate, you're slowing down, the velocity is slowing down. When you accelerate, you're increasing; you're speeding up your velocity.
By the way, here's a little trivia point for you. What do you think we call rates of change of acceleration? Which, I guess, would be a third derivative of position, seeing how the acceleration changes? Do you know what we call that? We call that a jerk, it's a jerky motion. A jerk actually means the acceleration is changing. So, when velocity changes, when you accelerate, or decelerate, that's a change in velocity, it's called acceleration. When you actually change your acceleration then you're jerking, anyway, but I digress.
Why don't you take a look at some of the, the examples here? Try some of them on your own and see if you can get a sense of how the velocity and acceleration all fit together with position. We'll see you soon.

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