# 1. Course Introduction and Newtonian Mechanics

be meters The unit for time will be a 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, pounds per square foot, whatever it is You’ve got to learn to transform them, but I won’t do them I think that’s pretty elementary stuff But sometimes I might not write the units but I’ve earned the right to do that and you guys haven’t so you’ll 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 would be something like this You’ve got to learn how to read this graph I’m assuming everyone knows how to read it [draws a graph of x versus t] This doesn’t mean the object is bobbing up and down I hope you realize that Even though the graph is 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 is going to the left of the origin Now, at 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 a 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 = t, x = t^(2), you’re going to have x = sin t and cos 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 x (t) So you should know when you look at a graph like this what it’s doing So, the two most elementary ideas you learn are what is the average velocity of an object, as then ordered by the symbol v-bar So, the average is found by taking two instants in time, say t_1 and later t_2, and you find out where it was at t_2 minus where it was at t_1 and divide by the 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 would be zero, because you start and end at the same value of x, you get something; 0 over time 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 So the average is what it is It’s an average, it doesn’t 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 But before you even do that, I want to define for you an important concept, which is the velocity at a given time, v (t) So this is the central idea of calculus, right? I am 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 I will draw the famous picture of some particle moving and it’s here at t of some value of x A little later, which is t + Δt So Δt is going to stand always for a small finite integral of time; infinitesimal interval of time not yet 0 So, during that time, the particle has gone from here to there, that is x + Δx, and the average velocity in that interval is Δ x/ Δt Graphically, this guy is Δ x and this guy is Δt, and Δx over Δt is a 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 your 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 is to know that by looking at the position now, the position slightly later and taking the ratio and bringing later as close as possible to right now, we define a quantity that we can say is the velocity at this instant So v of t, v(t) is the limit, Δt goes to 0 of Δx over Δt and we use the symbol dx/dt for velocity So technically, if you ask what does the velocity stand for–Let me draw a general situation If a particle goes from here to here, Δx over Δt, 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, the straight line would become tangent to the curve So the velocity at any part of the curve is tangent to the curve at that point The tangent of, this angle, this θ, is then Δx over Δt 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 first two derivatives have a name The first one is velocity, the 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 = ma The a is this fellow 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, if x(t) is t^(n), you’re supposed to know dx/dt is nt^(n-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 At 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 of 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 A 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 of why we are interested in that Does anybody know why so much time is spent on that? Yes? Student: [inaudible] Professor Ramamurti Shankar: Pardon me? Student: [inaudible] Professor Ramamurti Shankar: Right 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 constant is called g, and that’s 9.8 meters/second^(2) So that’s a very typical problem When you’re falling to 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 So, the question we are going to ask is the following, “If I tell you that a 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’re only given the particle has acceleration a, and you are asked to find out what is x? In other words,