1. Course Introduction and Newtonian Mechanics
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Shankar, Ramamurti
Professor Shankar introduces the course and answers student questions about the material and the requirements. He gives an overview of Newtonian mechanics and explains its two components: kinematics and dynamics. He then reviews basic concepts in calculus through two key equations: x = x0 + v0t + ½ at2 and v2 = v02+ 2 a (x-x0), tracing the fate of a particle in one dimension along the x-axis.
 This is a first part of the year-long course introducing you to all the major ideas in physics, starting from Galileo and Newton, right up to the big revolutions of the last century, which was on relativity and quantum mechanics. The target audience for this course is really very broad. I have always been surprised at how broad the representation is. So I don't know what your major is. I don't know what you're going to do later. So I picked the topics that all of us in physics find fascinating. Some may or may not be useful, but you just don't know. Some of you are probably going to be doctors, and you don't know why I'm going to do special activity or quantum mechanics. But you don't know when it'll come in handy. If you're a doctor and you've got a patient who's running away from you at the speed of light, you'll know what to do. Or if you're a pediatrician with a really small patient who will not sit still, it's because the loss of quantum mechanics don't allow an object to have a definite, cannot have a definite position in momentum. So these are all things you just don't know when they will come in handy. And I tease them because these are the things that turned me on and got me going into physics, and whether or not you go into physics, you should certainly learn about the biggest and most interesting revolutions right up to present-day physics. All right, so that's what the subject matter is going to be. And I'm going to tell you a little bit about how the course is organized. First thing is this year it's going to be taped. You can see some people in the back with cameras. It's part of an experimental pilot program funded by the Hewlett Foundation. And at some point they will decide what they will do with these lectures. Most probably they'll post them somewhere so people elsewhere can have the benefit of what you have sitting in the Yale classroom. So I've been told that from now on we just ignore the camera and do business as usual. Nothing's going to be changed. I try to negotiate a laugh track so that the jokes don't work. We can superpose some laughter. I was told, no, I just got to deal with it as it happens. So it's going to be, it's like one of the reality shows where things are going to be as they are. And hopefully after a while we'll learn to act and behave normally and not worry about this presence. Then, coming to the rest of the details of the course, by the way, there are more details on the website that I posted that was given to me by the university if you want to know more about what all this is about. The course organization is fairly simple. We're going to meet Monday and Wednesday in this room, 11.30 to 12.45. I will give you some problems to do on Wednesday and I'll post them on the website. You guys should get used to going to the classes website. I'm really dependent on that now. I finally learned how to use it. I will use that to post information. Maybe once in a while send email to the whole class. If you want to get those emails, you've got to sign up for the course because I push a button and it goes to anybody who's signed up there. The homework will be given on Wednesday and it's due before class the following Wednesday. And let me introduce you to our head TA, Mara Daniel, who's recently Mara Barabas. Right. She's, will you come a little, okay, so Mara's going to be the person who will see you after class and she will take the problem sets that you have submitted before class and she'll give you the graded ones from after class. Okay, that will be sorted out. It will be up there. So you should drop the homework before you come into class rather than furiously working on it during class. And the solutions will be posted the same afternoon. So there is not much point in giving a homework that's late. But once in a while, you know, you come up with a reason that I just cannot argue with. You got married, getting a transplant, whatever it is, that's fine. You got a transplant, I want to see the old body part. You got married, I want to see your spouse. If something happened to a grandparent, I'm counting, and up to four, I don't get suspicious. Go five, six, seven, eight. I will have to look into the family tree. But you know, look, any reasonable excuse will be entertained. And the relative importance given to these different things is 20% for your homework, 30% for the midterm, which will be sometime in October, and 50% for the final. That'll be the weighted average. But I have another plan called the amnesty plan in which I also compare just your final grade, what you did in the final exam, and whichever is higher up the 2 is what I will take to determine your overall course grade. This is something I used to announce near the end, but then some people felt that it's not fair not to know this from the beginning. So I'm telling you from the beginning, but don't dream and think that somehow the final is going to be so much different from your regular day-to-day performance, but to give you some reason to live after the midterm. You feel that is hope, I can change everything overnight. It does happen. I put that in for a reason because sometimes some of you have not taken a physics course, you don't know how to do well in physics, and slowly you catch on, and by the time it's final exam, you crack the code, you know how to do well. As far as I'm concerned, that's just fine. If at the end of the semester you take a three-hour exam in a close environment and you answer everything, I don't care what you did in your homework or your midterm, that's not relevant. So that's how the grading will be done. We have Mara's group of TAs. She's the head TA and she's the one you should write to whenever you have a problem. Then we also have two faculty members. One is our post-doctoral fellow, Mark Caprio. Mark, okay. So he will have a discussion section on Tuesdays between one and two in Sloan Lab. And Steve Fulanetto, I don't know if Steve is here or not. There's Steve, our new assistant professor. He will have a section on Tuesday night in Dunham Lab, Room 220. Tuesday night is a night when you people realize homework is due on Wednesday. So we know that, so he will be there to comfort you and give you whatever help you need. All right, my own office hours, I'm not determined yet. I'll have to find out when it is good for you. I used to, you know, I live and work out of Sloan Lab up on the hill, and it was easy to have office hours before or after class, but now you have to make a special trip. So just give me a little bit of time to find out maybe by soliciting email responses from you. What would be a good time for my office hours? But for any procedural things like, you know, this problem set was not great or properly and so on, there's no point emailing me because I'm going to send it to Mara anyway. So directly deal with the powers that be. Okay, finally, I want to give you some tips on how to do well in this course and what attitude you should have. First advice is you should come to the lectures. It's not self-serving. It's not so much for my benefit. I think there's something useful about hearing the subject presented once orally. Secondly, the book, you can see one of you had a book here. It's for 1100 pages, and when I learned physics, it was like 300 pages. Now, I look around this room. I don't see anybody whose head is three times bigger than mine. So I know that you cannot digest everything the books have. So I have to take out what I think is a really essential part and cover that in the lecture. So you come to class to find out what's in and what's not in. If you don't do that, it's a danger. You will learn something. You don't have to, and we don't want that. Okay, so that's why you come to class. Second thing, most important thing for doing well in physics is to do the homework. The 20% given to the homework is not a real measure of how important it is. Homework is when you really figure out how much you know and don't know. If you watch me do the thing on the blackboard, it looks very reasonable. It looks like you can do it, but the only way you're going to find out is when you actually deal with a problem. That's the only time you're going to find out. So I ask you to do the problems as and when they are posted. So if you're posted on Wednesday, it's to cover the material for that week, then you should attempt it as quickly as possible, because I'm going to assume you've done the problems when you come for the next few lectures. And in doing the homework, it is perfectly okay to work in groups. You don't have to do it by yourself. That's not how physics is done. I'm now writing a paper with two other people that are my experimental colleagues who write papers with 400 other people, maybe even a thousand other people. When they do the big collider experiments in Geneva and Fermilab, collaborations can run into hundreds. So it's perfectly okay to be a part of a collaboration. But you've got to make sure that you're pulling your weight. You've got to make sure that if you explain to others how to do this problem, then somebody else contributed something else. But you know what everybody contributed in the end. So the game is not just to somehow or other get the solution to the problem set, but to fully understand how it's done. And the TAs will be there to help you. Every day there's going to be a TA in the undergraduate lounge. I would urge you to use that. That's a beautiful new lounge that the Provost's office allowed us to build for physicists and chemists who whoever happens to be in the building. If you go there on the third floor of Sloan, you may run into other people like you who are trying to work on problems. You may run into upper class students. Students are more advanced. You will run into your TA. So that's a good climate that our coffee machine is, and that allows so far as in everything else. There are computers, there are printers. So it's a good lounge. And I think if you go there one day a week to do your problem sets or more often, that's a good meeting place. I recommend that. The final piece of advice, this is very important, so please pay attention to this, which is I ask you not to talk to your neighbors during lecture. Now this looks like a very innocuous thing, but you will find out it's the only thing that really gets my back up. Most of the time I don't really care. I'm very liberal. But this disturbs me because I'm looking at you, I'm trying to see from your reaction how much of my lecture you're following, and then it's very distracting when people are talking. So please don't do that. If you talk, I'm going to assume you're talking about me. If you laugh, I'm going to assume you're laughing at me. Let's make that, that's not really what I think, but that's how disturbing it is when people talk. And very nice students who do not realize this often disrupt my straight, my line of thinking. So I ask you to keep that to a minimum. Once in a while you'll have to talk to your neighbor and say, can you please pass me my pacemaker that fell down? That's fine. Then you go back to your business, but don't do too much of that. Finally, there is this ancient issue about sleeping in the class. Now my view is it's just fine. I know you guys need the rest. And interestingly, the best sleepers are in the first couple of rows. I haven't met you guys. It's not personal. I have found. Some people really have to come to the first or second row because they claim that if they don't hear me, they cannot really go to sleep. Now that was true in Sloan, but I think Luce has got very good acoustics so you can stretch out in the back. But my only criterion is if you talk in your sleep, that's not allowed because talking is not allowed. Next, if you're going to sleep, I ask you to sit between two non-sleepers. Sometimes what happens, the whole row will topple over. We don't want the domino effect. Now it's going to be captured on tape, and that's going to be really bad for my reputation. So spread yourself around the other people. All right, so that's it in terms of class, logistics and everything. I'm going to start going into the physics proper. I will try to finish every lecture on time, but sometimes I'm in the middle of a sentence or in the middle of a derivation, and I have to go over by a couple of minutes. There's no need to shuffle your feet and move stuff around. I know what time it is. I also want to get out like you guys, but let me finish something. Other days I may finish a few minutes before time. That's because the ideas of physics don't fall into 75-minute segments, and sometimes they spill over a little bit. Also, I'm used to teaching this course three times a week, and now it's suddenly twice a week. And so things that fell into nice 50-minute units are now being snipped up different ways, so it's pretty difficult. So even for me, some of it will be new, and the timing may not be just right. Okay, anybody have any? I should tell you, first of all, that in this class, the taping is not going to affect you, because the camera is going to be behind your head. I mentioned to you in the website that this is not the big opportunity you've been looking for to be a star. Only the back of your head will be seen. In some cases, the back of the head could be more expressive than the front, in which case this is your opportunity, and I wish you luck. But otherwise, just don't worry about it, because you will be only heard. You may not even be heard. So I've been asked that if your question is not very clear, I should repeat it, so that people listening to this later will know what the question was. But I would ask you, let me make one thing very clear. That is, I'm not in favor of you or talking to each other because of distracting. You're stopping me any time is just fine. I welcome that because I've seen this subject for God knows how many years. The only thing that makes it different for me is the questions that you people have. So you can stop me any time, and you should not feel somehow you're stopping the progress of the class. There is no fixed syllabus. We can move things around, and it's far more exciting for me to answer your questions than to have a monologue. So don't worry about that. So stop me any time you don't follow something, and don't assume that you're not following something because there's something wrong with your level of comprehension. Quite often, you guys come up with questions that never crossed my mind. It's very interesting. And things we've been repeating year after year after year because they sound so reasonable, suddenly sound unreasonable when some of you point out some aspect of it that you didn't follow. So it could be very interesting for all of us to have issues with this discussion class. And quite often, some questions are very common, and your classmates will be grateful to you that you brought it up. Otherwise, you know, you get 10 E-mails all with the same question. Okay, so I'm going to start now. Anybody have any questions about the class, the format, the midterm, the exams? All right. Yes? Oh, you mean my office hours? No, the discussion sections are Tuesday afternoon from 1 to 2, and Tuesday night from 8 to 10. And the website has got all the details on when and where. Yes? Yeah, there are many, many lab times. And you have to go to the website for the lab. And by the way, that reminds me, I got here lots of flyers given to me by the director of the laboratories, which will tell you which lab is the right lab for you. And they're offered many times a week. Yes? As far as notion of material, just from your class, how important is taking the lab in terms of the class? I think it's a good idea to take the lab because it's particularly in this particular class because I don't have any demonstrations. They're all in the other building. So this will remind you that physics is, after all, an experimental science. And you will be able to see where all the laws of physics come from. So if you're going to take it, you should take it at the same time. Yes? Very good. This is a calculus-based class, and I expect everyone to know at least the rudiments of differential calculus. What's a function? What's a derivative? What's a second derivative? How do you take derivatives of elementary functions? How to do elementary integrals? Sometime later, I will deal with functions of more than one variable, which I will briefly introduce to you, because that may not be prerequisite, but certainly something you will learn and you may use on and off. But there are different ways of doing physics. Mine is to demonstrate over and over how little mathematics you need to get the job done. There are others who like to show you how much mathematics you could somehow insinuate into the process. Okay, there are different ways of playing the game, and some of us find great pride in finding the most simple way to understand something. That's certainly my trademark. That's how I do my research also. So if you feel there's not enough math used, I guarantee you that I certainly know enough eventually to snow the whole class, but that's not the point. I will use it in moderation and use it to the best effect possible rather than use it because it is there. Okay, so I don't know your mathematical background, but the textbook has an appendix, which is a reasonable measure of how much math you should know. You've got to know your trigonometry, you've got to know what's a sine and what's a cosine. You cannot say, I will look it up. Your birthday and social security number is what you look up. Trigonometric functions you know all the time. Okay, I will assume you do. And of course, trigonometric identities you know from high school, pages and pages of them. So no one expects you to know all those identities, but there are a few popular ones we will use. All right, anything else? Yes? I was saying that earlier, did you have one midterm? Yeah, midterm will be sometime around 20th of October. I have to find out exactly the right time. I have 24 lectures for this class and the first 12 roughly will be part of the midterm. But after the 12th lecture, I may wait a week so that you have time to do the problems and get the solutions. Then I will give you the midterm. Yes? Is there going to be more to that first one? Yeah, this tells you in detail. This flyer answers exactly that. Okay, there's one more question for somebody. Yes? A few people I talked to have recommended that we start taking the lab second semester and the first. Is that the advice or should we take both? I don't have a strong view. I think you should take the lab sometime. But I don't know how many semesters of it you have to take. But I would say the advice of your predecessors is very important. If they tell you this is what works, that's better than what somebody like me can tell you. Also, you should talk to Stephen Irons, who is the director of the labs. He has seen every possible situation. He will give you good advice. Let's start now. Okay, so we are going to be studying in the beginning what's called Newtonian mechanics. It's pretty remarkable that the whole edifice is set up by just one person, Newton. He set us on the road to understanding all the natural phenomena till the year 1800 and something when Maxwell invented the laws of electromagnetism and wrote down the famous Maxwell equation. But except for electromagnetism, the basics of mechanics, which is the motion of billiard balls and trucks and marbles and whatnot, was set up by Newton. So that's what we are going to focus on. And you will find out that the laws of physics for this entire semester certainly can be written on one of those blackboards or even half of those blackboards. And the purpose of this course is to show you over and over and over again that starting with those one or two laws, you can deduce everything. And I would encourage you to think the same way. In fact, I would encourage you to think the way physicists do, even if you don't plan to be a physicist, because that's the easiest way to do the subject. And that is to follow the reasoning behind everything I give you. And my purpose will be not to say something as a postulate, but to show you where everything comes from. And it's best for you if you try to follow the logic. That way you don't have to store too many things in your head. In the early days, when there were four or five formulas, you can memorize all of them. And you can try each one of them until something works. But after a couple of weeks, you will have 100 formulas, and you cannot memorize all of them. You cannot resort to trial and error. You have to know the logic. So the logical way is not just the way the physicists do it. It's the easier way to do it. If there is another way that will work for non-physicists, I won't hesitate to teach it to you that way. But that turns out to be the best way. So try to follow the logic of everything. So Newtonian mechanics is our first topic. So Newtonian mechanics has two parts. All of physics consists of two, is a two-part program. The plan every time is to predict the future given the present. That's what we always do. When we do that right, we are satisfied. The question is, what do you mean by predict the future? What do you mean by the future? What do you mean by the present? By present, we mean we will pick some part of the universe we want to study, and we will ask, what information do I need to know about that system at the initial time, like right now, in order to be able to predict the future? So, for example, if you were trying to study the motion of some object, here is one example here. See, that's an example of Newtonian mechanics. I'll give you one more demonstration. Let's see who can catch this one. That's a good example. So that was Newtonian mechanics at work, because what did I do? I released a piece of candy through it from my hand, and the initial conditions had to do with where did I release it, and with what velocity. That's what he sees with his eyes. Then that's all you really need to know. Then he knows it's going to go up, it's going to curve, follow some kind of parabola, then his hands go there to receive it. That is verification of a prediction. His prediction was the candy is going to land here, and he put his hand there. He also knew where the candy is going to land, but he couldn't get his hand there in time. But we can always make predictions. But this is a good example of what you need to know. What is it that he had to know about this object that was thrown? I claim it's the initial location of the object and the initial velocity. The fact that it was blue or red, not relevant. And if I threw a gorilla at him, it doesn't matter what the color of the gorilla is, what mood it is in. These are things we don't deal with in physics. Or if there is a tall building, a standard physics problem, an object falls off a tall building. Object could be a person. So we don't ask, why is this guy ending it all today? That's not... we don't know. And we cannot deal with that. So we don't answer everything. We just want to know when he's going to hit the pavement. And with what speed? So we ask very limited questions, which is why we brag about how accurately we can predict the future. So we only ask limited goals and we are very successful and satisfying them. So we are basically dealing with inanimate objects. So the project of Newtonian mechanics, predicting the future given the present, has got two parts. And one is called kinematics, and the other is called dynamics. So kinematics is a complete description of the present. It's a list of what you have to know about a system right now. For example, if you're talking about the chalk, if I throw the chalk, you will have to know where it is and how fast it's moving. The dynamics doesn't... Dynamics then tells you why the object goes up, why the object goes down, why is it pulled down, and so on. That's dynamics. The reason it comes down is gravity is pulling it. In kinematics, you don't ask the reason behind anything. You simply want to describe things the way they are. And then dynamics tells you how they change and why they change. So I'm going to illustrate the idea of kinematics by taking the simplest possible example. That's going to be the way I'm going to do everything in this course. I'm going to start with the simplest example and slowly add on bells and whistles and make it more and more complicated. So some of you might say, well, I've seen this before, so maybe there's nothing new here. That may well be. I don't know how much you've seen, but quite often the way you learned physics earlier on in high school is probably different from the way professional physicists think about it. The sense of values we have, the things that we get excited about are different. Problems may be more difficult. But I want to start in every example, in every situation that I explain to you, the simplest example, and then slowly add on things. So what we are going to study now is a non-living object, and we are going to pick it to be a mathematical point. So the object is a mathematical point, has no size. If you rotate it, you won't know. It's not like a potato. You take a potato, you turn it around, it looks different. So it's not enough to say the potato is here. You've got to say which way the nose is pointing and so on. So we don't want to deal with that now. That comes later when we study what are called rigid bodies. Right now we want to study an entity which has no spatial extent. So just a dot. And the dot can move around all over space. So we're going to simplify that too. We're going to take an entity that lives along the x-axis. It moves along a line. So you can imagine a bead with a wire going through it, and the bead can only slide back and forth. So this is about the simplest thing. I cannot reduce the number of dimensions. One is the lowest dimension. I cannot make the object simpler than being just a mathematical point. So then you've got to say, what do I have to know about this object at the initial time? What constitutes the present? Or what constitutes maximal information about the present? So what we do is we pick an origin, call it 0. We put some markers there to measure distance. And we say this guy is sitting at 1, 2, 3, 4, 5. This is sitting at x equal to 5. Now, of course, we've got to have units, and the units for length are going to be meters. The units for time will be second. And time will be measured in seconds. Then we'll come to other units. Right now, in kinematics, this is all you need. Now, there are some tricky problems in the book. Sometimes they give you the speed in miles per hour, kilometers per year, poundles per square foot, whatever it is. You've got to learn to transform them. I won't do that. I think that's pretty elementary stuff. But sometimes I may not write units, but I've earned the right to do that, and you guys haven't. So you still have to keep track of your units. Everything's got to be in the right units. If you don't have the units, then if you say the answer is 19, then we don't know what it means. Okay, so here's an object. At a given instant, it's got a location. So what we would like to do is to describe what the object does by drawing a graph of time versus space, and the graph could be something like this. You've got to learn how to read this graph. I'm assuming everyone knows how to read it. This doesn't mean the object is bobbing up and down. I hope you realize that. If you're going up and down, the object is moving from left to right. So for example, when it does this, it's crossed the origin and it's going to the left of the origin. Now, which is the left of the origin, it turns around and starts coming to the origin and going to the right. That is x versus t. So in the language of calculus, x is the function of time, and this is a particular function. This function doesn't have a name. There are other functions which have a name. For example, this is x equals t, x equals t squared. You can have x equals sine t and cosine t and log t. So some functions have a name, some functions don't have a name. What a particle tries to do generally is some crazy thing which doesn't have a name, but it's a function except t. So you should know when you look at a graph like this what it's doing. The two most elementary ideas you learn are what is the average velocity of an object. It's denoted by the symbol d bar. So the average is found by taking two instance in time, say t1 and later t2. So you can figure out where it was at t2 minus where it was at t1 and divide by time. So the average velocity may not tell you the whole story. For example, if you started here and you did all this and you came back here, the average velocity will be 0 because you start and end at the same value of x. So if you do something 0 divided by time, it will still be 0. So you cannot tell from the average everything that happened because another way to get the same 0 is to just not move at all. The average is what it is. An average does not give you enough detail. So it's useful to have the average velocity. It's useful to have the average acceleration, which you can find by taking similar differences of velocities. Before you even do that, I want to define for you an important concept, which is the velocity at a given time. So this is the central idea of calculus. I'm hoping that if you learned your calculus, you learned about derivatives and so on by looking at x versus t. So I will remind you, again, this is not a course in calculus. I don't have to do it in any detail. So this is another famous picture of some particle moving, and it's here at time t at some value of x, a little later, which is time t plus delta t. So delta t is going to stand always for a small finite interval of time, infinitesimal interval of time, not yet 0. So during that time, the particle has gone from here to there, and that is x plus delta x. And the average velocity in that interval is delta x divided by delta t. Graphically, this guy is delta x, and this guy is delta t, and delta x over delta t is the ratio. So in calculus, what you want to do is to get the notion of the velocity right now. We all have an intuitive notion of velocity right now. When you're driving in your car, there's a needle, and the needle says 60, that's the velocity at this instant. It's very interesting because velocity seems to require two different times to define it, the initial time and the final time. And yet you want to talk about the velocity right now, that is the whole triumph of calculus. It's to know that by looking at the position now, the position slightly later, take the ratio, and bringing later, as close as possible to right now, we define a quantity that we can say is a velocity at this instant. So v of t is the limit, delta t goes to 0 of delta x over delta t, and we use the symbol dx dt for velocity. So technically, if you ask what does the velocity stand for, it stands for, let me draw a general situation. If a particle goes from here to here, delta x over delta t, I don't know how well you can see it in this figure here, is the slope of a straight line connecting these two points. And as the points come closer and closer, straight line will become tangent to the curve. The velocity at any part of the curve is a tangent to the curve at that point. It's tangent of that theta. This angle is theta, then delta x over delta t, by trigonometry, stands theta. Okay, once you can take one derivative, you can take any number of derivatives, and the derivative of the velocity is called the acceleration, and we write it as the second derivative of position. So I'm hoping you guys are comfortable with the notion of taking one or two or any number of derivatives. Interestingly, the only, the first two derivatives have a name. First one is velocity, second one is acceleration. The third derivative, unfortunately, was never given a name, and I don't know why. I think the main reason is that there are no equations that involve the third derivative explicitly. F equals ma, the a is the disfeller here, and nothing else is given an independent name. Of course, you can take a function and take derivatives any number of times. So you are supposed to know, for example, x of t is t to the n, you are supposed to know dx dt is n to the t to the n minus 1. Then you're supposed to know derivatives of simple functions like sines and cosines. So if you don't know that, then, of course, you have to work harder than other people. If you know that, that may be enough for quite some time. Okay, so what I've said so far is a particle moving in time from point to point can be represented by a graph, x versus t, or any point on the graph, you can take the derivative, which will be tangent to the curve at each point, and its numerical value will be what you can call the instantaneous velocity at that point, and you can take the derivative over the derivative and call it the acceleration. So we are going to specialize to a very limited class of problems in the rest of this class. The limited class of problems is one in which the acceleration is just a constant. Now that is not the most general thing, but I'm sure you guys have some idea why we are interested in that. Anybody know why that's such a big, so much time is spent on that, yes? Pardon me? The most famous example is that when things fall near the surface of the earth, they all have the same acceleration, and the acceleration that's most common is called g, and that's 9.8 meters per second square. So that's a very typical problem. When you fall near the surface of the earth, you are describing a problem of constant acceleration. That's why there's a lot of emphasis on sharpening your teeth by doing this class of problems. The question we are going to ask is the following. If I tell you that the particle has a constant acceleration, A, can you tell me what the position x is? Normally I will give you a function and tell you to take any number of derivatives. That's very easy. This is the backwards problem. You are only given the particle as acceleration A, and you are asked to find out what is x. In other words, your job is to guess a function whose second derivative is A, and this is called integration, which is the opposite of differentiation. Integration is just guessing. Integration is not an algorithmic process like differentiation. If I give you a function, you know how to take the derivative, change the independent variable, find the change in the function, take the ratio, and that's the derivative. The opposite is being asked here. I tell you something about the second derivative of a function and ask you what is the function. The way we do that is we guess, and the guessing has been going on for 300 years, so we sort of know how to guess. So, let me think aloud and ask how I will guess in this problem. I will say, okay, this guy wants me to find a function which reduces to the number A when I take two derivatives, and I wrote somewhere here this result, which says that when I take a derivative, I lose a power of t. In the end, I don't want any powers of t. It's very clear I got to start with a function that looks like t squared. This way, when I take two derivatives, there will be no t left. Unfortunately, we know this is not the right answer, because if I take the first derivative, I get 2t, and I take the second derivative, I get 2, but I want to get A and not 2. Then it's very clear the way you patch that up is you multiply it by this constant, and now we are all set. This function will have the right second derivative. This certainly describes a particle whose acceleration is A, where A is not dependent on time. But the question is, is this the most general answer, or is it just one answer? I think you all know that this is not the most general answer. But I can add to this some number like 96. They'll still have the property that if you take two derivatives, you're going to get the same acceleration. So 96 now is a typical constant, so I'm going to give the name C the constant. Everyone knows from calculus that if you're trying to find a function about which you know only the derivative, you can always add a constant to one person's answer without changing anything. But I think here you know you can do more. You can add something else to the answer without invalidating it, and that is anything with one power of t in it. Because if you take one derivative, it'll survive, but if you take two derivatives, it'll get wiped out. So it's not obvious, but it is true that you cannot add to this anymore. Basic idea in solving these equations, in integrating, is you find one answer, so that when you take enough derivatives, the function does what it is supposed to do. But then having found one answer, you can add to it anything that gets killed by the act of taking derivatives. If you take only one derivative, you can add a constant. If you're taking two derivatives, you can add a constant and something linear in t. If you knew only the third derivative of the function, you can add something quadratic in t without changing the outcome. So this is the most general position for a particle of constant acceleration, A. Now you must remember that this describes a particle going side to side. I can also describe a particle going up and down. If I do that, I would like to call that coordinate y, then I will write the same thing. You've got to realize that in calculus, the symbols that you call x and y are completely arbitrary. If you know the second derivative of y to be A, then the answer looks like this. If you knew the second derivative of x, the answer looks like that. Now we have to ask, what are these numbers, B and C? Let me go back now to this expression. x of t equals At square over 2 plus C plus Bt. It is true mathematically you can add two numbers, but you've got to ask yourself, what am I doing as a physicist when I add these two numbers? What am I supposed to do with A and B? I mean with this B and C, what value should I pick? The answer is that simply knowing the particle has that acceleration is not enough to tell you where the particle will be. For example, let's take the case where the particle is falling under gravity. Then you guys know, you just told me acceleration is minus 9.8, where G is minus 9.8. We call it minus because it's accelerating down and up is taken to be the positive direction. In that case, y of t will be minus 1 half gt square plus C plus Bt. The point is every object falling under gravity is given by the same formula, but there are many, many objects that can have many histories all falling under gravity. What's different from one object than the other object is, when was it dropped from what height and with what initial speed? That's what these numbers are going to tell us and we can verify that as follows. If you want to know what the number C is, you say let's put time t equal to 0. In fact, let me go back to this equation here. You put time t equal to 0, x at 0, it doesn't have this term, and it is C. So I realize that the constant C is the initial location of the object, and it's very common to denote that by x0. So the meaning of the constant is where was the object at the initial time? It could have been anywhere, simply knowing the acceleration is not enough to tell you where it was at the initial time. You get to pick where it was at the initial time. Then to find the meaning of B, we take one derivative of this, d x d t, that's velocity as a function of time, and if you took the derivative of this guy, you will find it's at plus B. That's the velocity of the object. Then you can then understand that v of 0 is what B is, which we write as v0. So the final answer is that x of t looks like x0 plus v0 t plus 1 half at square. So what I'm saying here is we are specializing to limited class of motion, because the particle has a definite acceleration A. Then in every situation where the body has an acceleration A, the location has to have this form, where this number is where it was initially, this was the initial velocity of the object. So when I threw that thing up and you caught it, what you were doing mentally was immediately figuring out where it started and at what speed. That is your initial data. Then in your mind, without realizing it, you found the trajectory at all future times. Now, that is one other celebrated formula that goes with this. I'm going to find that, then I'll give you an example. Now I'm fully aware that this is not the flashiest example in physics, but I'm worried about that right now. You'll see enough things that will confound you. But right now, I want to demonstrate a simple paradigm of what it means to know the present and what it means to say this is what the future behavior will be. We want to do that in the simplest context. Then we can make the example more and more complicated, but the phenomenon will be the same. So what we have found out so far, I'm purposely going from x to y, to know that the unknown variable can be called an x or can be called a y. It doesn't matter. As long as the second derivative is a, that's the answer. Now that's the second formula one derives from this. You guys probably know that too from your days of the daycare, but I want to derive that formula and put it up. Then we'll see how to use that. The second formula tries to relate the final velocity at some time t at the initial velocity, and the distance traveled with no reference to time. So the trick is to eliminate time from this equation. So let's see how we can eliminate time. You know that if you took a derivative of this, you'll find v of t is v0 plus at. What that means is if you know the velocity at a given time and you know the initial velocity, you know what time it is. The time in fact is v minus v0 over a. If I don't show you any argument for v, it means v at time t, and the subscript 0 means v of 0. So what this says is you can measure time by having your own clock. The clock tells you what time it is. But you can also say what time it is by seeing how fast the particle is moving, because you know it started with some speed. It's gaining speed at some rate a. So if the speed was so and so now, then the time had to be this. The time can be indirectly inferred from these quantities. Then you take that formula here and you put it here. Wherever you see a time t, you put this expression. So what will you get? We'll get an expression in which there is no t. t has been banished in favor of this. I'm not going to waste your time by asking what happens if you put it in. I will just tell you what happens. So I'm going to find that v squared is equal to v0 squared plus 2a times x minus x0. How many people have seen this thing before? Okay, that's a lot. Look, I know you've seen this. At the moment, I have to go through some of the more standard material before we go to the more non-standard material. This part's very easy for you. There's not much I can do right now. So let me draw a box. Just drawing a box, you guys mean important. These are the two important things. I claim now, remember, I want you to understand one thing. How much of this should you memorize? Suppose you've never seen this in high school. How much are you supposed to memorize? I would say keep that to a minimum. Because what the first formula tells you should be so intuitive that you don't have to cram this. We're talking about particles of constant acceleration. That means when I take two derivatives, I want to get a. Then you should know enough calculus. It has to be something like at squared. And the half comes from taking two derivatives. The other two you know are stuff you can add. And you know why you're adding those things. Because the particle has a head start. It's got an initial position, even at t equal to 0. It has an initial velocity. So even without any acceleration, it will be moving from y0 to y0 plus vt. The acceleration gives you an extra stuff quadratic in time. Once you've got that, one derivative will give you the velocity. Then in a crunch, you can eliminate t and put it into this formula. But most people end up memorizing these two because you use it so many times. Eventually it sticks in you. But you shouldn't try to memorize everything. So we are now going to do one standard problem where we will convince ourselves we can apply this formula and predict the future given the present. So the problem I want to do, there are many things you could do, but I just picked one. This is the one with the round numbers so I can do it without a calculator. So the problem, that is this building and it's going to be 15 meters high. And I'm going to throw something and it's going to go up and come down. Something I throw up has an initial speed of 10 meters per second. So we have to ask now, now my claim is you can ask me any question you want about this particle. And I can answer you. You can ask me where it will be, 9 seconds from now, 8 seconds from now, how fast will it be moving? I can answer anything at all. But what I needed to do it to do this problem was to find these two unknowns. So you've got to get used to the notion of what will be given in general and what is Taylor made to the occasion. So we know in this example the initial height should be 15 meters and the initial velocity should be 10. And for acceleration I'm going to use minus g and to keep life simple I'm going to call it minus 10. As you know the correct answer is 9.8 but we don't want to use the calculator now so we'll call it minus 10. So, for this object the position y at any time t is known to be 15 plus 10t minus 5t square. That is the full story of this object. Of course you've got to be a little careful when you use it. For example, let's put t equal to 10,000 years. What are you going to get? If you put t equal to 10,000 years or 10,000 seconds you're going to find y is some huge negative number. Is that right or what's wrong with that reasoning? So you cannot use a formula once it hits the ground because once it hits the ground the fundamental premise that A was a constant of minus 9.8 or minus 10 is wrong. So that's another thing to remember. Once you get a formula you've got to always remember the terms under which the formula was derived. If you blindly use it beyond the validity you will get results which don't make any sense. Conversely if you get an answer and it doesn't seem to make sense then you've got to go back and ask, am I violating some of the assumptions? The assumption that the particle had that acceleration is true as long as it's freely falling under gravity but not when you hit the ground. Now if you dug a hole here till there then of course it may work till that happens. But you've got to know every time when you, this is so obvious in this problem but when you see a more complicated formula you may not know all the assumptions that went into derivation but more often you will be using it when you should. Alright see this you agree is a complete solution to this miniature tiny Mickey Mouse problem. Give me the time and I'll tell you where it is. If you want to know how fast it's moving at a given time, if you want another velocity I just take the derivative of this answer which is 10 minus 10t. So let me pick a couple of trivial questions one can ask. One can ask the following question, how high does it go? How high will it rise? To what height will it rise? So we know it's going to go up and turn around and come down. We're trying to see how high that is. So that is a tricky problem to begin with because if you take this formula here you can ask us why if you know t but now we're not saying that. We don't know the time and we don't know how high it's rising so you can ask how am I supposed to deal with this problem. Then you put something else that you know in your mind which is that the highest point is the point when it's neither going up nor coming down. If it's going up that's not the highest point. If it's coming down that's not the highest point. So at the highest point it cannot go up and it cannot go down. So that's the point where velocity is 0. If you do that, let's call that particular time t star, then 10 t star minus 10 is equal to 0 or t star is 1 second. So we know that it'll go up for 1 second then it'll turn around and come back. Now we are done because now you can ask how high does it go? So you go back to your y of 1 and y of 1 is 15 plus 10 minus 5 which is what, 20 meters. By the way, you will find that I make quite a lot of mistakes on the blackboard. You're going to find out, you know, one of these years when you start teaching that when you get really close to the blackboard you just cannot think. It's an inverse correlation between your level of thinking and the proximity to the blackboard. So if you find me making a mistake, you've got to stop me. Why do you stop me? For two reasons. First of all, I'm very pleased when this happens. I'm pretty confident that I can do this under duress but I may not do it right every time. But if my students can catch me making a mistake it means they're following it and they're not hesitating to tell me. Secondly, as we go to the more advanced part of the course, we'll take a result from this part of the blackboard, stick it into the second part and keep manipulating. If I screw it up in the beginning and you guys keep quiet, we'll have to do the whole thing again. I would ask you when you follow this thing to do it actively, try to be one step ahead of me. For example, if I'm stuck by lightning, can you do anything? Can you guess what I'm going to say next year? Any idea where this is going? You should have a clue. If I die and you stop, that's not a good sign. You've got to keep going a little further because you should follow the logic. So for example, I'm going to calculate next when it hits the ground. You should have some idea how I'll do it. Because this is not a spectator sport. If you just watch me, you're going to learn nothing, like watching the US Open and thinking you're some kind of player. You will have to shed the tears and you've got to bang your head on the wall, go through your own private struggle. I cannot do that for you. I cannot even make it look hard because I've memorized this problem from childhood. So there is no way I can make this look difficult. That's your job. All right, so we know this point at one second is 20 meters. So let's just ask one other question, we'll stop. One other question may be, when does it hit the ground and at what speed? That's a typical physics question. So when does it hit the ground? Well, I think you must know now how to formulate that question. When does it hit the ground? When is y equal to 0? By the way, I didn't tell you this, but I think you knew that. I picked my origin to be here and measured y positively to be upwards, and I called that 15 meters. You can call that your origin. Your y0 will be 0, but ground will be called minus 15. So in the end, the physics is the same, but the numbers describing it can be different. You have to interpret the data differently. But the standard origin for everybody is the foot of the building. You can pick your origin here, some crazy spot. It doesn't matter, but some origins are more equal than others because there are some natural landmarks there. Here, foot of the building is what I call the origin. So in that notation, I want to ask when is y equal to 0? I ask when y is equal to 0, then I say 0 is 15 plus 10t minus 5t squared. Or I canceling the 5 everywhere and changing the sign here. I get t squared minus 2t minus 3 equal to 0. That's when it hits ground. So let's find out what the time is. So t is then 2 plus or minus 4 plus 12 over 2, which is 2 plus or minus 4 over 2, which is minus 1 or 3. So you get two answers when it hits the ground. So it's clear that we should pick 3. But you can ask why is it giving me a second solution? Anybody have an idea why it's yes? That's correct. So her answer was, if it was a full parabola, then we know it would have been at the ground before I set my clock to 0. First of all, negative time should not bother anybody. t equals 0 is when I set the clock. I measure time forward, but yesterday would be t equal to minus 1 day. So I'm going to be in trouble with negative times. So the point is this equation does not know about the building. It doesn't know the whole song and dance that you went to a building and you threw up a rock. It knows nothing. What does the mathematics know? It knows that this particle happened to have a height of 15 at time 0 and a velocity of 10 at time 0, and it is falling under gravity with an acceleration of minus 10. That's all it knows. If that's all it knows, then in that scenario, if there is no building at anything else, it continues the trajectory both forward and time and backward in time. And it says that whatever seconds, one second before you set your clock to 0, it would have been in the ground. What it means is if you release the rock at that location, one second before, with a certain speed that we can calculate, it would have ended up here with precisely the position and velocity it had at the beginning of our experiment. So sometimes the extra solution is very interesting, and you should always listen to the mathematics when you get extra solutions. In fact, when a very famous physicist called Dirac was looking for the energy of a particle in relativistic quantum mechanics, he found the energy of a particle is connected to its momentum. This P is what we call momentum, and its mass by this relation. So a particle of mass, m and momentum, P has its energy. So you solve for the energy, you get two answers. Now your temptation is to keep the first answer because you know energy is not going to be negative. Particle is moving, it's got some energy, and that's it. The mathematicians told Dirac, you cannot ignore the negative energy solution because it tells you that's the second solution, and you cannot throw them out. In terms of the second solution with negative energy was when the theory is telling you, hey, there are particles and there are antiparticles. And the negative energy, even properly interpreted, will describe antiparticles. So the equations are very smart. The way the physics works is you find some loss of motion in mathematical form. You put in the initial conditions or whatever, you solve the equations, and the answer that comes, you have no choice. You have to accept the answer. There are new answers besides the one you were looking for, you've got to think about what they mean. And that's one of the best things about physics because here's a person who is not looking for antiparticles. We're just trying to describe electrons, but the theory said there are two roots in the quadratic equation, and the second root is mathematically as interesting as the first one. It has to be part of the theory, and in trying to adjust it so it can be incorporated, you discovered antiparticles. So always amazing to us how we go into the problem, our eye or mind can see one class of solutions, but the math will tell you sometimes there are new solutions, and you've got to respect it and understand and interpret the unwanted solutions. And this is a simple example where you can follow what the meaning of the second solution is. It means that to the problem you post, there's more than the answers that you could imagine. Here it meant particle that was released from the ground earlier that it meant something much more interesting, namely antiparticles accompanying particles. They are going to accompany particles surely as every quadratic equation as two solutions. All right, so now in this problem, we can do something slightly different, and that's used this expression here, and if you do that, then I'll stop for today. If you were asking questions like, how high does it go? But you don't ask, when does it go to the highest point? Then you don't have to go through the whole process of finding the time at which it turned around, I don't know where that is, that disappeared on the blackboard, then putting the time equal to one second into this formula. If the question of time is not explicitly brought up, then you should know that you have to use this formula. So how do we get it here? Well, we say at the top of the loop, when the coefficient comes down, the velocity is 0. Therefore, you say 0 square equals initial velocity square plus 2 times minus g, that's my acceleration, times y minus y0. If you solve for that, you find y minus y0 equals v0 square over 2g, and if you put in the v0 I gave you, which is what? 10 is 100 over 20, which is 5 meters. So y equals y0 plus 5 meters, and that is the height to which it rises. So you have to find it somewhere else. We found the maximum height to be 20 meters. Another thing you can do is you can find the speed here. If you want to find the speed there, you put the equation v square equals v0 square plus 2 times minus g. What is y minus y0? The final y is 0, the initial y is minus, is 15. If you solve for that equation, then you will find the final velocity. So if time is not involved, you can do it that way. I want to derive the last result in another way, then I will stop, and that's pretty interesting because it tells you the use and abuse of calculus. I'm going to find for you this result using calculus in a different way. From the calculus, we know dv dt is equal to a. Multiply both sides by v. Now you have to know from elementary calculus that v times dv dt is really d by dt of v square over 2. You guys know that much calculus that when you take a derivative of a function of a function, namely v square over 2 is a function of v, and v itself is a function of t, then the rule for taking the derivative is first take the v derivative of this object, then take the d by dt of v, which is this one. On the right-hand side, I'm going to write as a dx dt. It's not going to be much as standard, but now here is what I'm going to do something with, somehow, you are told never, ever to do, which is to just cancel the dT's. You all know that when you do dy dx, you're not supposed to cancel that d. That's actually correct. You don't want to cancel the d in the derivative. So I'm going to assume it's true, and I'll maybe take a second to explain why it's related to me. What this really means is, in a given time delta t, the change in this quantity is 8 times the change in this quantity. Therefore, you can multiply both sides by the delta t, but the only thing you should understand is delta t, as long as it's small and finite, will lead to some small and finite errors in the formula, because the formula is really the limit in which delta x and delta t both go to 0. So what you have to do is multiply both sides by delta t, but remember it's got to be in the end, made vanishingly small. As long as we understand that, we can do this cancellation, and this says on the left-hand side, the change in the quantity v square over 2 is the change, 8 times the change in the quantity x. So add up all the changes, or what you mean by integral, same thing, add up all the changes. The change in v square over 2 will be the final v square over 2 minus the initial v square over 2, and the other side will be 8 times the change in x, x minus x naught. And that's the formula I wrote for you, v square is v0 square plus 2a x minus x naught. So the point is whenever you have derivatives with something over dt, do not hesitate to cancel the dt's and think of them as delta v square over 2 is equal to 8 times delta of x. This will be actually true as long as both quantities are vanishingly small. They will become more and more true as delta x and delta v square become vanishingly small. In the limit in which they're approaching 0, the 2 will be in fact equal. If delta x is a finite amount like 1 second, this will not be true because in the starting equation, delta x and delta t and delta v square are all assumed to be infinitesimal. So don't hesitate to do manipulations of this type, and I will do them quite often. You've got to understand when it's okay and when it's not okay. What this means is in a time delta t, if this quantity changes by some amount, and the same time delta t, that quantity changes by some amount, then keeping the delta t equal to some number, we may equate the changes in the two quantities. Provided it is understood that delta v square over 2 is a change in v square over 2 in the same time in which the particle moved a distance delta x. Then by adding the differences, we eliminate time and we get this final result. All right, so if you go to your website today, you will find I have assigned some problems, and you should try to do them, they apply to this chapter. Then next week, we'll do more complicated problems that involve motion in higher dimensions, how to go to two dimensions or three dimensions.
 This is a first part of the year-long course introducing you  to all the major ideas in physics, starting from Galileo and  Newton, right up to the big revolutions of the last century,  which was on relativity and quantum mechanics.  The target audience for this course is really very broad.  I have always been surprised at how broad the representation is.  So I don't know what your major is.  I don't know what you're going to do later.  So I picked the topics that all of us in physics find fascinating.  Some may or may not be useful, but you just don't know.  Some of you are probably going to be doctors,  and you don't know why I'm going to do special activity or quantum mechanics.  But you don't know when it'll come in handy.  If you're a doctor and you've got a patient who's running away from you  at the speed of light, you'll know what to do.  Or if you're a pediatrician with a really small patient who will not sit still,  it's because the loss of quantum mechanics don't allow  an object to have a definite, cannot have a definite position in momentum.  So these are all things you just don't know when they will come in handy.  And I tease them because these are the things that turned me on  and got me going into physics, and whether or not you go into physics,  you should certainly learn about the biggest and most interesting revolutions  right up to present-day physics.  All right, so that's what the subject matter is going to be.  And I'm going to tell you a little bit about how the course is organized.  First thing is this year it's going to be taped.  You can see some people in the back with cameras.  It's part of an experimental pilot program funded by the Hewlett Foundation.  And at some point they will decide what they will do with these lectures.  Most probably they'll post them somewhere so people elsewhere  can have the benefit of what you have sitting in the Yale classroom.  So I've been told that from now on we just ignore the camera and do business as usual.  Nothing's going to be changed.  I try to negotiate a laugh track so that the jokes don't work.  We can superpose some laughter.  I was told, no, I just got to deal with it as it happens.  So it's going to be, it's like one of the reality shows where things are going to be as they are.  And hopefully after a while we'll learn to act and behave normally  and not worry about this presence.  Then, coming to the rest of the details of the course,  by the way, there are more details on the website that I posted that was given to me by the university  if you want to know more about what all this is about.  The course organization is fairly simple.  We're going to meet Monday and Wednesday in this room, 11.30 to 12.45.  I will give you some problems to do on Wednesday and I'll post them on the website.  You guys should get used to going to the classes website.  I'm really dependent on that now.  I finally learned how to use it.  I will use that to post information.  Maybe once in a while send email to the whole class.  If you want to get those emails, you've got to sign up for the course  because I push a button and it goes to anybody who's signed up there.  The homework will be given on Wednesday and it's due before class the following Wednesday.  And let me introduce you to our head TA, Mara Daniel, who's recently Mara Barabas.  Right.  She's, will you come a little, okay, so Mara's going to be the person who will see you after class  and she will take the problem sets that you have submitted before class  and she'll give you the graded ones from after class.  Okay, that will be sorted out.  It will be up there.  So you should drop the homework before you come into class  rather than furiously working on it during class.  And the solutions will be posted the same afternoon.  So there is not much point in giving a homework that's late.  But once in a while, you know, you come up with a reason that I just cannot argue with.  You got married, getting a transplant, whatever it is, that's fine.  You got a transplant, I want to see the old body part.  You got married, I want to see your spouse.  If something happened to a grandparent, I'm counting, and up to four, I don't get suspicious.  Go five, six, seven, eight.  I will have to look into the family tree.  But you know, look, any reasonable excuse will be entertained.  And the relative importance given to these different things is 20% for your homework,  30% for the midterm, which will be sometime in October, and 50% for the final.  That'll be the weighted average.  But I have another plan called the amnesty plan in which I also compare just your final grade,  what you did in the final exam, and whichever is higher up the 2 is what I will take to determine your overall course grade.  This is something I used to announce near the end,  but then some people felt that it's not fair not to know this from the beginning.  So I'm telling you from the beginning, but don't dream and think that somehow the final is going to be so much different from your regular day-to-day performance,  but to give you some reason to live after the midterm.  You feel that is hope, I can change everything overnight.  It does happen.  I put that in for a reason because sometimes some of you have not taken a physics course,  you don't know how to do well in physics, and slowly you catch on, and by the time it's final exam,  you crack the code, you know how to do well.  As far as I'm concerned, that's just fine.  If at the end of the semester you take a three-hour exam in a close environment and you answer everything,  I don't care what you did in your homework or your midterm, that's not relevant.  So that's how the grading will be done.  We have Mara's group of TAs.  She's the head TA and she's the one you should write to whenever you have a problem.  Then we also have two faculty members.  One is our post-doctoral fellow, Mark Caprio.  Mark, okay.  So he will have a discussion section on Tuesdays between one and two in Sloan Lab.  And Steve Fulanetto, I don't know if Steve is here or not.  There's Steve, our new assistant professor.  He will have a section on Tuesday night in Dunham Lab, Room 220.  Tuesday night is a night when you people realize homework is due on Wednesday.  So we know that, so he will be there to comfort you and give you whatever help you need.  All right, my own office hours, I'm not determined yet.  I'll have to find out when it is good for you.  I used to, you know, I live and work out of Sloan Lab up on the hill,  and it was easy to have office hours before or after class,  but now you have to make a special trip.  So just give me a little bit of time to find out maybe by soliciting email responses from you.  What would be a good time for my office hours?  But for any procedural things like, you know, this problem set was not great or properly and so on,  there's no point emailing me because I'm going to send it to Mara anyway.  So directly deal with the powers that be.  Okay, finally, I want to give you some tips on how to do well in this course  and what attitude you should have.  First advice is you should come to the lectures.  It's not self-serving.  It's not so much for my benefit.  I think there's something useful about hearing the subject presented once orally.  Secondly, the book, you can see one of you had a book here.  It's for 1100 pages, and when I learned physics, it was like 300 pages.  Now, I look around this room.  I don't see anybody whose head is three times bigger than mine.  So I know that you cannot digest everything the books have.  So I have to take out what I think is a really essential part and cover that in the lecture.  So you come to class to find out what's in and what's not in.  If you don't do that, it's a danger.  You will learn something.  You don't have to, and we don't want that.  Okay, so that's why you come to class.  Second thing, most important thing for doing well in physics is to do the homework.  The 20% given to the homework is not a real measure of how important it is.  Homework is when you really figure out how much you know and don't know.  If you watch me do the thing on the blackboard, it looks very reasonable.  It looks like you can do it, but the only way you're going to find out is when you actually deal with a problem.  That's the only time you're going to find out.  So I ask you to do the problems as and when they are posted.  So if you're posted on Wednesday, it's to cover the material for that week,  then you should attempt it as quickly as possible,  because I'm going to assume you've done the problems when you come for the next few lectures.  And in doing the homework, it is perfectly okay to work in groups.  You don't have to do it by yourself.  That's not how physics is done.  I'm now writing a paper with two other people that are my experimental colleagues  who write papers with 400 other people, maybe even a thousand other people.  When they do the big collider experiments in Geneva and Fermilab,  collaborations can run into hundreds.  So it's perfectly okay to be a part of a collaboration.  But you've got to make sure that you're pulling your weight.  You've got to make sure that if you explain to others how to do this problem,  then somebody else contributed something else.  But you know what everybody contributed in the end.  So the game is not just to somehow or other get the solution to the problem set,  but to fully understand how it's done.  And the TAs will be there to help you.  Every day there's going to be a TA in the undergraduate lounge.  I would urge you to use that.  That's a beautiful new lounge that the Provost's office allowed us to build  for physicists and chemists who whoever happens to be in the building.  If you go there on the third floor of Sloan,  you may run into other people like you who are trying to work on problems.  You may run into upper class students.  Students are more advanced.  You will run into your TA.  So that's a good climate that our coffee machine is,  and that allows so far as in everything else.  There are computers, there are printers.  So it's a good lounge.  And I think if you go there one day a week to do your problem sets  or more often, that's a good meeting place.  I recommend that.  The final piece of advice, this is very important,  so please pay attention to this,  which is I ask you not to talk to your neighbors during lecture.  Now this looks like a very innocuous thing,  but you will find out it's the only thing that really gets my back up.  Most of the time I don't really care.  I'm very liberal.  But this disturbs me because I'm looking at you,  I'm trying to see from your reaction how much of my lecture you're following,  and then it's very distracting when people are talking.  So please don't do that.  If you talk, I'm going to assume you're talking about me.  If you laugh, I'm going to assume you're laughing at me.  Let's make that, that's not really what I think,  but that's how disturbing it is when people talk.  And very nice students who do not realize this often disrupt my straight,  my line of thinking.  So I ask you to keep that to a minimum.  Once in a while you'll have to talk to your neighbor and say,  can you please pass me my pacemaker that fell down?  That's fine.  Then you go back to your business, but don't do too much of that.  Finally, there is this ancient issue about sleeping in the class.  Now my view is it's just fine.  I know you guys need the rest.  And interestingly, the best sleepers are in the first couple of rows.  I haven't met you guys.  It's not personal.  I have found.  Some people really have to come to the first or second row  because they claim that if they don't hear me, they cannot really go to sleep.  Now that was true in Sloan,  but I think Luce has got very good acoustics so you can stretch out in the back.  But my only criterion is if you talk in your sleep,  that's not allowed because talking is not allowed.  Next, if you're going to sleep,  I ask you to sit between two non-sleepers.  Sometimes what happens, the whole row will topple over.  We don't want the domino effect.  Now it's going to be captured on tape,  and that's going to be really bad for my reputation.  So spread yourself around the other people.  All right, so that's it in terms of class, logistics and everything.  I'm going to start going into the physics proper.  I will try to finish every lecture on time,  but sometimes I'm in the middle of a sentence or in the middle of a derivation,  and I have to go over by a couple of minutes.  There's no need to shuffle your feet and move stuff around.  I know what time it is.  I also want to get out like you guys, but let me finish something.  Other days I may finish a few minutes before time.  That's because the ideas of physics don't fall into 75-minute segments,  and sometimes they spill over a little bit.  Also, I'm used to teaching this course three times a week,  and now it's suddenly twice a week.  And so things that fell into nice 50-minute units are now being snipped up different ways,  so it's pretty difficult.  So even for me, some of it will be new, and the timing may not be just right.  Okay, anybody have any?  I should tell you, first of all, that in this class,  the taping is not going to affect you,  because the camera is going to be behind your head.  I mentioned to you in the website that this is not the big opportunity  you've been looking for to be a star.  Only the back of your head will be seen.  In some cases, the back of the head could be more expressive than the front,  in which case this is your opportunity, and I wish you luck.  But otherwise, just don't worry about it,  because you will be only heard.  You may not even be heard.  So I've been asked that if your question is not very clear,  I should repeat it, so that people listening to this later  will know what the question was.  But I would ask you, let me make one thing very clear.  That is, I'm not in favor of you or talking to each other because of distracting.  You're stopping me any time is just fine.  I welcome that because I've seen this subject for God knows how many years.  The only thing that makes it different for me is the questions that you people have.  So you can stop me any time,  and you should not feel somehow you're stopping the progress of the class.  There is no fixed syllabus.  We can move things around, and it's far more exciting for me  to answer your questions than to have a monologue.  So don't worry about that.  So stop me any time you don't follow something,  and don't assume that you're not following something  because there's something wrong with your level of comprehension.  Quite often, you guys come up with questions that never crossed my mind.  It's very interesting.  And things we've been repeating year after year after year  because they sound so reasonable,  suddenly sound unreasonable when some of you point out some aspect of it  that you didn't follow.  So it could be very interesting for all of us to have issues with this discussion class.  And quite often, some questions are very common,  and your classmates will be grateful to you that you brought it up.  Otherwise, you know, you get 10 E-mails all with the same question.  Okay, so I'm going to start now.  Anybody have any questions about the class, the format, the midterm, the exams?  All right. Yes?  Oh, you mean my office hours?  No, the discussion sections are Tuesday afternoon from 1 to 2,  and Tuesday night from 8 to 10.  And the website has got all the details on when and where. Yes?  Yeah, there are many, many lab times.  And you have to go to the website for the lab.  And by the way, that reminds me, I got here lots of flyers  given to me by the director of the laboratories,  which will tell you which lab is the right lab for you.  And they're offered many times a week. Yes?  As far as notion of material, just from your class,  how important is taking the lab in terms of the class?  I think it's a good idea to take the lab  because it's particularly in this particular class  because I don't have any demonstrations.  They're all in the other building.  So this will remind you that physics is, after all, an experimental science.  And you will be able to see where all the laws of physics come from.  So if you're going to take it, you should take it at the same time.  Yes?  Very good. This is a calculus-based class, and I expect everyone to know  at least the rudiments of differential calculus.  What's a function? What's a derivative?  What's a second derivative? How do you take derivatives of elementary functions?  How to do elementary integrals?  Sometime later, I will deal with functions of more than one variable,  which I will briefly introduce to you,  because that may not be prerequisite,  but certainly something you will learn and you may use on and off.  But there are different ways of doing physics.  Mine is to demonstrate over and over how little mathematics you need to get the job done.  There are others who like to show you how much mathematics you could somehow  insinuate into the process.  Okay, there are different ways of playing the game,  and some of us find great pride in finding the most simple way to understand something.  That's certainly my trademark. That's how I do my research also.  So if you feel there's not enough math used, I guarantee you that I certainly know enough  eventually to snow the whole class, but that's not the point.  I will use it in moderation and use it to the best effect possible  rather than use it because it is there.  Okay, so I don't know your mathematical background,  but the textbook has an appendix, which is a reasonable measure of how much math you should know.  You've got to know your trigonometry, you've got to know what's a sine and what's a cosine.  You cannot say, I will look it up.  Your birthday and social security number is what you look up.  Trigonometric functions you know all the time.  Okay, I will assume you do.  And of course, trigonometric identities you know from high school, pages and pages of them.  So no one expects you to know all those identities, but there are a few popular ones we will use.  All right, anything else? Yes?  I was saying that earlier, did you have one midterm?  Yeah, midterm will be sometime around 20th of October.  I have to find out exactly the right time.  I have 24 lectures for this class and the first 12 roughly will be part of the midterm.  But after the 12th lecture, I may wait a week so that you have time to do the problems and get the solutions.  Then I will give you the midterm.  Yes?  Is there going to be more to that first one?  Yeah, this tells you in detail.  This flyer answers exactly that.  Okay, there's one more question for somebody. Yes?  A few people I talked to have recommended that we start taking the lab second semester and the first.  Is that the advice or should we take both?  I don't have a strong view. I think you should take the lab sometime.  But I don't know how many semesters of it you have to take.  But I would say the advice of your predecessors is very important.  If they tell you this is what works, that's better than what somebody like me can tell you.  Also, you should talk to Stephen Irons, who is the director of the labs.  He has seen every possible situation. He will give you good advice.  Let's start now.  Okay, so we are going to be studying in the beginning what's called Newtonian mechanics.  It's pretty remarkable that the whole edifice is set up by just one person, Newton.  He set us on the road to understanding all the natural phenomena  till the year 1800 and something when Maxwell invented the laws of electromagnetism  and wrote down the famous Maxwell equation.  But except for electromagnetism, the basics of mechanics,  which is the motion of billiard balls and trucks and marbles and whatnot,  was set up by Newton.  So that's what we are going to focus on.  And you will find out that the laws of physics for this entire semester  certainly can be written on one of those blackboards or even half of those blackboards.  And the purpose of this course is to show you over and over and over again  that starting with those one or two laws, you can deduce everything.  And I would encourage you to think the same way.  In fact, I would encourage you to think the way physicists do,  even if you don't plan to be a physicist,  because that's the easiest way to do the subject.  And that is to follow the reasoning behind everything I give you.  And my purpose will be not to say something as a postulate,  but to show you where everything comes from.  And it's best for you if you try to follow the logic.  That way you don't have to store too many things in your head.  In the early days, when there were four or five formulas, you can memorize all of them.  And you can try each one of them until something works.  But after a couple of weeks, you will have 100 formulas,  and you cannot memorize all of them.  You cannot resort to trial and error.  You have to know the logic.  So the logical way is not just the way the physicists do it.  It's the easier way to do it.  If there is another way that will work for non-physicists,  I won't hesitate to teach it to you that way.  But that turns out to be the best way.  So try to follow the logic of everything.  So Newtonian mechanics is our first topic.  So Newtonian mechanics has two parts.  All of physics consists of two, is a two-part program.  The plan every time is to predict the future given the present.  That's what we always do.  When we do that right, we are satisfied.  The question is, what do you mean by predict the future?  What do you mean by the future?  What do you mean by the present?  By present, we mean we will pick some part of the universe we want to study,  and we will ask, what information do I need to know about that system  at the initial time, like right now,  in order to be able to predict the future?  So, for example, if you were trying to study the motion of some object,  here is one example here.  See, that's an example of Newtonian mechanics.  I'll give you one more demonstration.  Let's see who can catch this one.  That's a good example.  So that was Newtonian mechanics at work, because what did I do?  I released a piece of candy through it from my hand,  and the initial conditions had to do with where did I release it,  and with what velocity.  That's what he sees with his eyes.  Then that's all you really need to know.  Then he knows it's going to go up, it's going to curve,  follow some kind of parabola, then his hands go there to receive it.  That is verification of a prediction.  His prediction was the candy is going to land here, and he put his hand there.  He also knew where the candy is going to land,  but he couldn't get his hand there in time.  But we can always make predictions.  But this is a good example of what you need to know.  What is it that he had to know about this object that was thrown?  I claim it's the initial location of the object and the initial velocity.  The fact that it was blue or red, not relevant.  And if I threw a gorilla at him, it doesn't matter what the color of the gorilla is,  what mood it is in.  These are things we don't deal with in physics.  Or if there is a tall building, a standard physics problem,  an object falls off a tall building.  Object could be a person.  So we don't ask, why is this guy ending it all today?  That's not... we don't know.  And we cannot deal with that.  So we don't answer everything.  We just want to know when he's going to hit the pavement.  And with what speed?  So we ask very limited questions, which is why we brag about  how accurately we can predict the future.  So we only ask limited goals and we are very successful and satisfying them.  So we are basically dealing with inanimate objects.  So the project of Newtonian mechanics,  predicting the future given the present, has got two parts.  And one is called kinematics,  and the other is called dynamics.  So kinematics is a complete description of the present.  It's a list of what you have to know about a system right now.  For example, if you're talking about the chalk,  if I throw the chalk, you will have to know where it is  and how fast it's moving.  The dynamics doesn't...  Dynamics then tells you why the object goes up,  why the object goes down, why is it pulled down, and so on.  That's dynamics.  The reason it comes down is gravity is pulling it.  In kinematics, you don't ask the reason behind anything.  You simply want to describe things the way they are.  And then dynamics tells you how they change and why they change.  So I'm going to illustrate the idea of kinematics  by taking the simplest possible example.  That's going to be the way I'm going to do everything in this course.  I'm going to start with the simplest example  and slowly add on bells and whistles  and make it more and more complicated.  So some of you might say,  well, I've seen this before, so maybe there's nothing new here.  That may well be.  I don't know how much you've seen,  but quite often the way you learned physics earlier on in high school  is probably different from the way professional physicists think about it.  The sense of values we have, the things that we get excited about are different.  Problems may be more difficult.  But I want to start in every example,  in every situation that I explain to you, the simplest example,  and then slowly add on things.  So what we are going to study now is a non-living object,  and we are going to pick it to be a mathematical point.  So the object is a mathematical point, has no size.  If you rotate it, you won't know.  It's not like a potato.  You take a potato, you turn it around, it looks different.  So it's not enough to say the potato is here.  You've got to say which way the nose is pointing and so on.  So we don't want to deal with that now.  That comes later when we study what are called rigid bodies.  Right now we want to study an entity which has no spatial extent.  So just a dot.  And the dot can move around all over space.  So we're going to simplify that too.  We're going to take an entity that lives along the x-axis.  It moves along a line.  So you can imagine a bead with a wire going through it,  and the bead can only slide back and forth.  So this is about the simplest thing.  I cannot reduce the number of dimensions.  One is the lowest dimension.  I cannot make the object simpler than being just a mathematical point.  So then you've got to say, what do I have to know about this object  at the initial time?  What constitutes the present?  Or what constitutes maximal information about the present?  So what we do is we pick an origin, call it 0.  We put some markers there to measure distance.  And we say this guy is sitting at 1, 2, 3, 4, 5.  This is sitting at x equal to 5.  Now, of course, we've got to have units,  and the units for length are going to be meters.  The units for time will be second.  And time will be measured in seconds.  Then we'll come to other units.  Right now, in kinematics, this is all you need.  Now, there are some tricky problems in the book.  Sometimes they give you the speed in miles per hour,  kilometers per year, poundles per square foot, whatever it is.  You've got to learn to transform them.  I won't do that.  I think that's pretty elementary stuff.  But sometimes I may not write units,  but I've earned the right to do that, and you guys haven't.  So you still have to keep track of your units.  Everything's got to be in the right units.  If you don't have the units, then if you say the answer is 19,  then we don't know what it means.  Okay, so here's an object.  At a given instant, it's got a location.  So what we would like to do is to describe what the object does  by drawing a graph of time versus space,  and the graph could be something like this.  You've got to learn how to read this graph.  I'm assuming everyone knows how to read it.  This doesn't mean the object is bobbing up and down.  I hope you realize that.  If you're going up and down,  the object is moving from left to right.  So for example, when it does this,  it's crossed the origin and it's going to the left of the origin.  Now, which is the left of the origin,  it turns around and starts coming to the origin  and going to the right.  That is x versus t.  So in the language of calculus, x is the function of time,  and this is a particular function.  This function doesn't have a name.  There are other functions which have a name.  For example, this is x equals t,  x equals t squared.  You can have x equals sine t and cosine t and log t.  So some functions have a name, some functions don't have a name.  What a particle tries to do generally is some crazy thing  which doesn't have a name, but it's a function except t.  So you should know when you look at a graph like this what it's doing.  The two most elementary ideas you learn are what is the average velocity of an object.  It's denoted by the symbol d bar.  So the average is found by taking two instance in time,  say t1 and later t2.  So you can figure out where it was at t2 minus where it was at t1  and divide by time.  So the average velocity may not tell you the whole story.  For example, if you started here and you did all this and you came back here,  the average velocity will be 0 because you start and end at the same value of x.  So if you do something 0 divided by time, it will still be 0.  So you cannot tell from the average everything that happened  because another way to get the same 0 is to just not move at all.  The average is what it is.  An average does not give you enough detail.  So it's useful to have the average velocity.  It's useful to have the average acceleration,  which you can find by taking similar differences of velocities.  Before you even do that, I want to define for you an important concept,  which is the velocity at a given time.  So this is the central idea of calculus.  I'm hoping that if you learned your calculus, you learned about derivatives and so on  by looking at x versus t.  So I will remind you, again, this is not a course in calculus.  I don't have to do it in any detail.  So this is another famous picture of some particle moving,  and it's here at time t at some value of x,  a little later, which is time t plus delta t.  So delta t is going to stand always for a small finite interval of time,  infinitesimal interval of time, not yet 0.  So during that time, the particle has gone from here to there,  and that is x plus delta x.  And the average velocity in that interval is delta x divided by delta t.  Graphically, this guy is delta x,  and this guy is delta t, and delta x over delta t is the ratio.  So in calculus, what you want to do is to get the notion of the velocity right now.  We all have an intuitive notion of velocity right now.  When you're driving in your car, there's a needle,  and the needle says 60, that's the velocity at this instant.  It's very interesting because velocity seems to require two different times to define it,  the initial time and the final time.  And yet you want to talk about the velocity right now,  that is the whole triumph of calculus.  It's to know that by looking at the position now,  the position slightly later, take the ratio, and bringing later,  as close as possible to right now,  we define a quantity that we can say is a velocity at this instant.  So v of t is the limit,  delta t goes to 0 of delta x over delta t,  and we use the symbol dx dt for velocity.  So technically, if you ask what does the velocity stand for,  it stands for, let me draw a general situation.  If a particle goes from here to here,  delta x over delta t, I don't know how well you can see it in this figure here,  is the slope of a straight line connecting these two points.  And as the points come closer and closer,  straight line will become tangent to the curve.  The velocity at any part of the curve is a tangent to the curve at that point.  It's tangent of that theta.  This angle is theta, then delta x over delta t,  by trigonometry, stands theta.  Okay, once you can take one derivative,  you can take any number of derivatives,  and the derivative of the velocity is called the acceleration,  and we write it as the second derivative of position.  So I'm hoping you guys are comfortable with the notion of taking one or two  or any number of derivatives.  Interestingly, the only, the first two derivatives have a name.  First one is velocity, second one is acceleration.  The third derivative, unfortunately, was never given a name,  and I don't know why.  I think the main reason is that there are no equations  that involve the third derivative explicitly.  F equals ma, the a is the disfeller here,  and nothing else is given an independent name.  Of course, you can take a function and take derivatives any number of times.  So you are supposed to know, for example,  x of t is t to the n,  you are supposed to know dx dt is n to the t to the n minus 1.  Then you're supposed to know derivatives of simple functions like  sines and cosines.  So if you don't know that, then, of course, you have to work harder than other people.  If you know that, that may be enough for quite some time.  Okay, so what I've said so far is a particle moving in time from point to point  can be represented by a graph, x versus t,  or any point on the graph, you can take the derivative,  which will be tangent to the curve at each point,  and its numerical value will be what you can call the instantaneous velocity at that point,  and you can take the derivative over the derivative and call it the acceleration.  So we are going to specialize to a very limited class of problems in the rest of this class.  The limited class of problems is one in which the acceleration is just a constant.  Now that is not the most general thing, but I'm sure you guys have some idea why we are interested in that.  Anybody know why that's such a big, so much time is spent on that, yes?  Pardon me?  The most famous example is that when things fall near the surface of the earth,  they all have the same acceleration, and the acceleration that's most common is called g,  and that's 9.8 meters per second square.  So that's a very typical problem.  When you fall near the surface of the earth, you are describing a problem of constant acceleration.  That's why there's a lot of emphasis on sharpening your teeth by doing this class of problems.  The question we are going to ask is the following.  If I tell you that the particle has a constant acceleration, A,  can you tell me what the position x is?  Normally I will give you a function and tell you to take any number of derivatives.  That's very easy.  This is the backwards problem.  You are only given the particle as acceleration A, and you are asked to find out what is x.  In other words, your job is to guess a function whose second derivative is A,  and this is called integration, which is the opposite of differentiation.  Integration is just guessing.  Integration is not an algorithmic process like differentiation.  If I give you a function, you know how to take the derivative,  change the independent variable, find the change in the function,  take the ratio, and that's the derivative.  The opposite is being asked here.  I tell you something about the second derivative of a function and ask you what is the function.  The way we do that is we guess, and the guessing has been going on for 300 years,  so we sort of know how to guess.  So, let me think aloud and ask how I will guess in this problem.  I will say, okay, this guy wants me to find a function which reduces to the number A  when I take two derivatives, and I wrote somewhere here this result,  which says that when I take a derivative, I lose a power of t.  In the end, I don't want any powers of t.  It's very clear I got to start with a function that looks like t squared.  This way, when I take two derivatives, there will be no t left.  Unfortunately, we know this is not the right answer,  because if I take the first derivative, I get 2t,  and I take the second derivative, I get 2,  but I want to get A and not 2.  Then it's very clear the way you patch that up is you multiply it by this constant,  and now we are all set.  This function will have the right second derivative.  This certainly describes a particle whose acceleration is A,  where A is not dependent on time.  But the question is, is this the most general answer,  or is it just one answer?  I think you all know that this is not the most general answer.  But I can add to this some number like 96.  They'll still have the property that if you take two derivatives,  you're going to get the same acceleration.  So 96 now is a typical constant,  so I'm going to give the name C the constant.  Everyone knows from calculus that if you're trying to find a function  about which you know only the derivative,  you can always add a constant to one person's answer without changing anything.  But I think here you know you can do more.  You can add something else to the answer without invalidating it,  and that is anything with one power of t in it.  Because if you take one derivative, it'll survive,  but if you take two derivatives, it'll get wiped out.  So it's not obvious, but it is true that you cannot add to this anymore.  Basic idea in solving these equations, in integrating,  is you find one answer, so that when you take enough derivatives,  the function does what it is supposed to do.  But then having found one answer, you can add to it anything  that gets killed by the act of taking derivatives.  If you take only one derivative, you can add a constant.  If you're taking two derivatives, you can add a constant and something linear in t.  If you knew only the third derivative of the function,  you can add something quadratic in t without changing the outcome.  So this is the most general position for a particle of constant acceleration, A.  Now you must remember that this describes a particle going side to side.  I can also describe a particle going up and down.  If I do that, I would like to call that coordinate y,  then I will write the same thing.  You've got to realize that in calculus, the symbols that you call x and y  are completely arbitrary.  If you know the second derivative of y to be A, then the answer looks like this.  If you knew the second derivative of x, the answer looks like that.  Now we have to ask, what are these numbers, B and C?  Let me go back now to this expression.  x of t equals At square over 2 plus C plus Bt.  It is true mathematically you can add two numbers,  but you've got to ask yourself, what am I doing as a physicist when I add these two numbers?  What am I supposed to do with A and B?  I mean with this B and C, what value should I pick?  The answer is that simply knowing the particle has that acceleration is not enough to tell you  where the particle will be.  For example, let's take the case where the particle is falling under gravity.  Then you guys know, you just told me acceleration is minus 9.8,  where G is minus 9.8.  We call it minus because it's accelerating down and up is taken to be the positive direction.  In that case, y of t will be minus 1 half gt square plus C plus Bt.  The point is every object falling under gravity is given by the same formula,  but there are many, many objects that can have many histories all falling under gravity.  What's different from one object than the other object is,  when was it dropped from what height and with what initial speed?  That's what these numbers are going to tell us and we can verify that as follows.  If you want to know what the number C is, you say let's put time t equal to 0.  In fact, let me go back to this equation here.  You put time t equal to 0, x at 0, it doesn't have this term,  and it is C. So I realize that the constant C is the initial location of the object,  and it's very common to denote that by x0.  So the meaning of the constant is where was the object at the initial time?  It could have been anywhere, simply knowing the acceleration is not enough to tell you  where it was at the initial time.  You get to pick where it was at the initial time.  Then to find the meaning of B, we take one derivative of this, d x d t,  that's velocity as a function of time, and if you took the derivative of this guy,  you will find it's at plus B. That's the velocity of the object.  Then you can then understand that v of 0 is what B is, which we write as v0.  So the final answer is that x of t looks like x0 plus v0 t plus 1 half at square.  So what I'm saying here is we are specializing to limited class of motion,  because the particle has a definite acceleration A.  Then in every situation where the body has an acceleration A,  the location has to have this form, where this number is where it was initially,  this was the initial velocity of the object.  So when I threw that thing up and you caught it,  what you were doing mentally was immediately figuring out where it started and at what speed.  That is your initial data.  Then in your mind, without realizing it, you found the trajectory at all future times.  Now, that is one other celebrated formula that goes with this.  I'm going to find that, then I'll give you an example.  Now I'm fully aware that this is not the flashiest example in physics,  but I'm worried about that right now.  You'll see enough things that will confound you.  But right now, I want to demonstrate a simple paradigm of what it means to know the present  and what it means to say this is what the future behavior will be.  We want to do that in the simplest context.  Then we can make the example more and more complicated,  but the phenomenon will be the same.  So what we have found out so far, I'm purposely going from x to y,  to know that the unknown variable can be called an x or can be called a y.  It doesn't matter.  As long as the second derivative is a, that's the answer.  Now that's the second formula one derives from this.  You guys probably know that too from your days of the daycare,  but I want to derive that formula and put it up.  Then we'll see how to use that.  The second formula tries to relate the final velocity at some time t  at the initial velocity, and the distance traveled with no reference to time.  So the trick is to eliminate time from this equation.  So let's see how we can eliminate time.  You know that if you took a derivative of this, you'll find v of t is v0 plus at.  What that means is if you know the velocity at a given time  and you know the initial velocity, you know what time it is.  The time in fact is v minus v0 over a.  If I don't show you any argument for v, it means v at time t,  and the subscript 0 means v of 0.  So what this says is you can measure time by having your own clock.  The clock tells you what time it is.  But you can also say what time it is by seeing how fast the particle is moving,  because you know it started with some speed.  It's gaining speed at some rate a.  So if the speed was so and so now, then the time had to be this.  The time can be indirectly inferred from these quantities.  Then you take that formula here and you put it here.  Wherever you see a time t, you put this expression.  So what will you get?  We'll get an expression in which there is no t.  t has been banished in favor of this.  I'm not going to waste your time by asking what happens if you put it in.  I will just tell you what happens.  So I'm going to find that v squared is equal to v0 squared  plus 2a times x minus x0.  How many people have seen this thing before?  Okay, that's a lot.  Look, I know you've seen this.  At the moment, I have to go through some of the more standard material  before we go to the more non-standard material.  This part's very easy for you.  There's not much I can do right now.  So let me draw a box.  Just drawing a box, you guys mean important.  These are the two important things.  I claim now, remember, I want you to understand one thing.  How much of this should you memorize?  Suppose you've never seen this in high school.  How much are you supposed to memorize?  I would say keep that to a minimum.  Because what the first formula tells you should be so intuitive  that you don't have to cram this.  We're talking about particles of constant acceleration.  That means when I take two derivatives, I want to get a.  Then you should know enough calculus.  It has to be something like at squared.  And the half comes from taking two derivatives.  The other two you know are stuff you can add.  And you know why you're adding those things.  Because the particle has a head start.  It's got an initial position, even at t equal to 0.  It has an initial velocity.  So even without any acceleration, it will be moving from y0 to y0 plus vt.  The acceleration gives you an extra stuff quadratic in time.  Once you've got that, one derivative will give you the velocity.  Then in a crunch, you can eliminate t and put it into this formula.  But most people end up memorizing these two because you use it so many times.  Eventually it sticks in you.  But you shouldn't try to memorize everything.  So we are now going to do one standard problem where we will convince ourselves  we can apply this formula and predict the future given the present.  So the problem I want to do, there are many things you could do, but I just picked one.  This is the one with the round numbers so I can do it without a calculator.  So the problem, that is this building and it's going to be 15 meters high.  And I'm going to throw something and it's going to go up and come down.  Something I throw up has an initial speed of 10 meters per second.  So we have to ask now, now my claim is you can ask me any question you want about this particle.  And I can answer you.  You can ask me where it will be, 9 seconds from now, 8 seconds from now, how fast will it be moving?  I can answer anything at all.  But what I needed to do it to do this problem was to find these two unknowns.  So you've got to get used to the notion of what will be given in general  and what is Taylor made to the occasion.  So we know in this example the initial height should be 15 meters  and the initial velocity should be 10.  And for acceleration I'm going to use minus g and to keep life simple I'm going to call it minus 10.  As you know the correct answer is 9.8 but we don't want to use the calculator now so we'll call it minus 10.  So, for this object the position y at any time t is known to be 15 plus 10t minus 5t square.  That is the full story of this object.  Of course you've got to be a little careful when you use it.  For example, let's put t equal to 10,000 years.  What are you going to get?  If you put t equal to 10,000 years or 10,000 seconds you're going to find y is some huge negative number.  Is that right or what's wrong with that reasoning?  So you cannot use a formula once it hits the ground because once it hits the ground the fundamental premise  that A was a constant of minus 9.8 or minus 10 is wrong.  So that's another thing to remember.  Once you get a formula you've got to always remember the terms under which the formula was derived.  If you blindly use it beyond the validity you will get results which don't make any sense.  Conversely if you get an answer and it doesn't seem to make sense then you've got to go back and ask,  am I violating some of the assumptions?  The assumption that the particle had that acceleration is true as long as it's freely falling under gravity  but not when you hit the ground.  Now if you dug a hole here till there then of course it may work till that happens.  But you've got to know every time when you, this is so obvious in this problem  but when you see a more complicated formula you may not know all the assumptions that went into derivation  but more often you will be using it when you should.  Alright see this you agree is a complete solution to this miniature tiny Mickey Mouse problem.  Give me the time and I'll tell you where it is.  If you want to know how fast it's moving at a given time, if you want another velocity  I just take the derivative of this answer which is 10 minus 10t.  So let me pick a couple of trivial questions one can ask.  One can ask the following question, how high does it go?  How high will it rise?  To what height will it rise?  So we know it's going to go up and turn around and come down.  We're trying to see how high that is.  So that is a tricky problem to begin with because if you take this formula here  you can ask us why if you know t but now we're not saying that.  We don't know the time and we don't know how high it's rising  so you can ask how am I supposed to deal with this problem.  Then you put something else that you know in your mind which is that the highest point  is the point when it's neither going up nor coming down.  If it's going up that's not the highest point.  If it's coming down that's not the highest point.  So at the highest point it cannot go up and it cannot go down.  So that's the point where velocity is 0.  If you do that, let's call that particular time t star,  then 10 t star minus 10 is equal to 0 or t star is 1 second.  So we know that it'll go up for 1 second then it'll turn around and come back.  Now we are done because now you can ask how high does it go?  So you go back to your y of 1 and y of 1 is 15 plus 10 minus 5 which is what, 20 meters.  By the way, you will find that I make quite a lot of mistakes on the blackboard.  You're going to find out, you know, one of these years when you start teaching  that when you get really close to the blackboard you just cannot think.  It's an inverse correlation between your level of thinking and the proximity to the blackboard.  So if you find me making a mistake, you've got to stop me.  Why do you stop me?  For two reasons.  First of all, I'm very pleased when this happens.  I'm pretty confident that I can do this under duress but I may not do it right every time.  But if my students can catch me making a mistake it means they're following it  and they're not hesitating to tell me.  Secondly, as we go to the more advanced part of the course, we'll take a result from this part of the blackboard,  stick it into the second part and keep manipulating.  If I screw it up in the beginning and you guys keep quiet, we'll have to do the whole thing again.  I would ask you when you follow this thing to do it actively, try to be one step ahead of me.  For example, if I'm stuck by lightning, can you do anything?  Can you guess what I'm going to say next year? Any idea where this is going?  You should have a clue.  If I die and you stop, that's not a good sign.  You've got to keep going a little further because you should follow the logic.  So for example, I'm going to calculate next when it hits the ground.  You should have some idea how I'll do it.  Because this is not a spectator sport.  If you just watch me, you're going to learn nothing,  like watching the US Open and thinking you're some kind of player.  You will have to shed the tears and you've got to bang your head on the wall,  go through your own private struggle.  I cannot do that for you.  I cannot even make it look hard because I've memorized this problem from childhood.  So there is no way I can make this look difficult.  That's your job.  All right, so we know this point at one second is 20 meters.  So let's just ask one other question, we'll stop.  One other question may be, when does it hit the ground and at what speed?  That's a typical physics question.  So when does it hit the ground?  Well, I think you must know now how to formulate that question.  When does it hit the ground?  When is y equal to 0?  By the way, I didn't tell you this, but I think you knew that.  I picked my origin to be here and measured y positively to be upwards,  and I called that 15 meters.  You can call that your origin.  Your y0 will be 0, but ground will be called minus 15.  So in the end, the physics is the same, but the numbers describing it can be different.  You have to interpret the data differently.  But the standard origin for everybody is the foot of the building.  You can pick your origin here, some crazy spot.  It doesn't matter, but some origins are more equal than others  because there are some natural landmarks there.  Here, foot of the building is what I call the origin.  So in that notation, I want to ask when is y equal to 0?  I ask when y is equal to 0, then I say 0 is 15 plus 10t minus 5t squared.  Or I canceling the 5 everywhere and changing the sign here.  I get t squared minus 2t minus 3 equal to 0.  That's when it hits ground.  So let's find out what the time is.  So t is then 2 plus or minus 4 plus 12 over 2,  which is 2 plus or minus 4 over 2, which is minus 1 or 3.  So you get two answers when it hits the ground.  So it's clear that we should pick 3.  But you can ask why is it giving me a second solution?  Anybody have an idea why it's yes?  That's correct.  So her answer was, if it was a full parabola,  then we know it would have been at the ground before I set my clock to 0.  First of all, negative time should not bother anybody.  t equals 0 is when I set the clock.  I measure time forward, but yesterday would be t equal to minus 1 day.  So I'm going to be in trouble with negative times.  So the point is this equation does not know about the building.  It doesn't know the whole song and dance that you went to a building  and you threw up a rock.  It knows nothing.  What does the mathematics know?  It knows that this particle happened to have a height of 15 at time 0  and a velocity of 10 at time 0,  and it is falling under gravity with an acceleration of minus 10.  That's all it knows.  If that's all it knows, then in that scenario,  if there is no building at anything else,  it continues the trajectory both forward and time and backward in time.  And it says that whatever seconds,  one second before you set your clock to 0,  it would have been in the ground.  What it means is if you release the rock at that location,  one second before, with a certain speed that we can calculate,  it would have ended up here with precisely the position and velocity  it had at the beginning of our experiment.  So sometimes the extra solution is very interesting,  and you should always listen to the mathematics when you get extra solutions.  In fact, when a very famous physicist called Dirac  was looking for the energy of a particle in relativistic quantum mechanics,  he found the energy of a particle is connected to its momentum.  This P is what we call momentum,  and its mass by this relation.  So a particle of mass, m and momentum, P has its energy.  So you solve for the energy, you get two answers.  Now your temptation is to keep the first answer  because you know energy is not going to be negative.  Particle is moving, it's got some energy, and that's it.  The mathematicians told Dirac,  you cannot ignore the negative energy solution  because it tells you that's the second solution,  and you cannot throw them out.  In terms of the second solution with negative energy  was when the theory is telling you,  hey, there are particles and there are antiparticles.  And the negative energy, even properly interpreted,  will describe antiparticles.  So the equations are very smart.  The way the physics works is you find some loss of motion in mathematical form.  You put in the initial conditions or whatever,  you solve the equations, and the answer that comes,  you have no choice.  You have to accept the answer.  There are new answers besides the one you were looking for,  you've got to think about what they mean.  And that's one of the best things about physics  because here's a person who is not looking for antiparticles.  We're just trying to describe electrons,  but the theory said there are two roots in the quadratic equation,  and the second root is mathematically as interesting as the first one.  It has to be part of the theory,  and in trying to adjust it so it can be incorporated,  you discovered antiparticles.  So always amazing to us how we go into the problem,  our eye or mind can see one class of solutions,  but the math will tell you sometimes there are new solutions,  and you've got to respect it and understand and interpret the unwanted solutions.  And this is a simple example where you can follow  what the meaning of the second solution is.  It means that to the problem you post,  there's more than the answers that you could imagine.  Here it meant particle that was released from the ground earlier  that it meant something much more interesting,  namely antiparticles accompanying particles.  They are going to accompany particles surely as every quadratic equation  as two solutions.  All right, so now in this problem,  we can do something slightly different,  and that's used this expression here,  and if you do that, then I'll stop for today.  If you were asking questions like,  how high does it go?  But you don't ask, when does it go to the highest point?  Then you don't have to go through the whole process  of finding the time at which it turned around,  I don't know where that is, that disappeared on the blackboard,  then putting the time equal to one second into this formula.  If the question of time is not explicitly brought up,  then you should know that you have to use this formula.  So how do we get it here?  Well, we say at the top of the loop,  when the coefficient comes down, the velocity is 0.  Therefore, you say 0 square equals initial velocity square  plus 2 times minus g, that's my acceleration,  times y minus y0.  If you solve for that, you find y minus y0  equals v0 square over 2g, and if you put in the v0  I gave you, which is what?  10 is 100 over 20, which is 5 meters.  So y equals y0 plus 5 meters, and that is the height to which  it rises.  So you have to find it somewhere else.  We found the maximum height to be 20 meters.  Another thing you can do is you can find the speed here.  If you want to find the speed there, you put the equation  v square equals v0 square plus 2 times minus g.  What is y minus y0?  The final y is 0, the initial y is minus, is 15.  If you solve for that equation, then you will find the final  velocity.  So if time is not involved, you can do it that way.  I want to derive the last result in another way,  then I will stop, and that's pretty interesting  because it tells you the use and abuse of calculus.  I'm going to find for you this result using calculus  in a different way.  From the calculus, we know dv dt is equal to a.  Multiply both sides by v.  Now you have to know from elementary calculus that v times  dv dt is really d by dt of v square over 2.  You guys know that much calculus that when you take a derivative  of a function of a function, namely v square over 2 is a  function of v, and v itself is a function of t,  then the rule for taking the derivative is first take the v  derivative of this object, then take the d by dt of v,  which is this one.  On the right-hand side, I'm going to write as a dx dt.  It's not going to be much as standard,  but now here is what I'm going to do something with,  somehow, you are told never, ever to do,  which is to just cancel the dT's.  You all know that when you do dy dx,  you're not supposed to cancel that d.  That's actually correct.  You don't want to cancel the d in the derivative.  So I'm going to assume it's true, and I'll maybe take a second  to explain why it's related to me.  What this really means is, in a given time delta t,  the change in this quantity is 8 times the change in this quantity.  Therefore, you can multiply both sides by the delta t,  but the only thing you should understand is delta t,  as long as it's small and finite, will lead to some small  and finite errors in the formula, because the formula is really  the limit in which delta x and delta t both go to 0.  So what you have to do is multiply both sides by delta t,  but remember it's got to be in the end, made vanishingly small.  As long as we understand that, we can do this cancellation,  and this says on the left-hand side,  the change in the quantity v square over 2 is the change,  8 times the change in the quantity x.  So add up all the changes, or what you mean by integral,  same thing, add up all the changes.  The change in v square over 2 will be the final v square over 2  minus the initial v square over 2,  and the other side will be 8 times the change in x,  x minus x naught.  And that's the formula I wrote for you,  v square is v0 square plus 2a x minus x naught.  So the point is whenever you have derivatives with something  over dt, do not hesitate to cancel the dt's  and think of them as delta v square over 2 is equal to  8 times delta of x.  This will be actually true as long as both quantities are  vanishingly small.  They will become more and more true as delta x and delta v  square become vanishingly small.  In the limit in which they're approaching 0,  the 2 will be in fact equal.  If delta x is a finite amount like 1 second,  this will not be true because in the starting equation,  delta x and delta t and delta v square  are all assumed to be infinitesimal.  So don't hesitate to do manipulations of this type,  and I will do them quite often.  You've got to understand when it's okay and when it's not okay.  What this means is in a time delta t,  if this quantity changes by some amount,  and the same time delta t, that quantity changes by some amount,  then keeping the delta t equal to some number,  we may equate the changes in the two quantities.  Provided it is understood that delta v square over 2  is a change in v square over 2 in the same time  in which the particle moved a distance delta x.  Then by adding the differences,  we eliminate time and we get this final result.  All right, so if you go to your website today,  you will find I have assigned some problems,  and you should try to do them, they apply to this chapter.  Then next week, we'll do more complicated problems  that involve motion in higher dimensions,  how to go to two dimensions or three dimensions.
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