Chapter 1Review of Newtonian MechanicsChapter 1IntroductionPhysics can be defined as the science of predicting the motion ofobjects. A science is supposed to be quantitative: the goal ofphysics is to predict the function x(t) that gives the positionof an object as a function of time. (More precisely, in athree-dimensional space we need the three functions x(t),y(t), and z(t) that determine the three coordinates of ourobject) The possibility of a physical science is obvious fromour daily observation of regularities in the motion of objects.We notice that an object under well-defined conditions willalways move in the same way. This knowledge is essential toour interactions with Nature. In fact, our brain is so good atpredicting motion that it makes many decisions automati-cally. Think about the complicated process of walking. Howmuch time do you spend planning the next step? Evenanimals can do an excellent job at predicting motion. Justwatch the fascinating TV program This is my dog. The claimthat physics is one of the most difficult subjects is thereforeridiculous.Elements of atheory of motionThe mere existence of a position function x(t) does notguarantee that the mathematical derivation of this functionwill be simple. It could actually be so complicated that themathematical description of motion might become impossi-ble. On the other hand, we notice that many types ofmotion in Nature are simple: we see circular waves in water,we find that the orbits of the planets are elliptical, we realizethat a flying stone describes a parabola in air. These aresimple geometrical forms; it is reasonable to expect that themathematical theory of motion, at least for these examples,will not be too complicated.The mathematics of motion prediction: KinematicsFrom a mathematical point of view, the problem of predict-ing motion can be formulated as follows: what do we need tonow at time t0 to calculate the function x(t) at a later timet1? A possible answer can be found in the well-known Taylorexpansion formulaphy 241 MenŽndez 2Chapter 1 x(t1) = x(t0) +dxdtt= t0(t1− t0) +12d2xdt2t= t0(t1− t0)2++13!d3xdt3t= t0(t1− t0)3+ ... +1n!dnxdtnt= t0(t1− t0)n+ ... (1)which provides exactly what we need: the value of a functionat time t1 based on information for time to only: the func-tion itself and all its time derivatives evaluated at time t0.There is a catch, however: first of all, the type of functionsthat can be expanded in a Taylor series is very limited,because they must have well-defined derivatives to all orders.We donÕt know yet whether the position function x(t) for allkinds of motion will satisfy this restrictive condition.Worse, we need an infinite amount of information topredict motion with Eq. (1), because we must know thefunction and all its derivatives at time to. LetÕs analyze thephysical meaning of those terms to see if we can understandthe origin of the problem.The first term in Eq. (1), x(to), is the position of our objectat time t0. That we need this information makes perfectsense: if we want to predict the motion of an object, weneed to know where it was at the initial time. If a tennis ballis served in Wimbledon, it will land in Wimbledon. If it isserved in Roland Garros, it will land in Roland Garros. Thereis no way we can tell the landing point if we donÕt know thestarting point, even if the trajectories of the two balls areidentical. The second term in Eq. (1), dxdtt= t0/ v(t0) is the velocity attime t0. That this quantity is also needed to predict motionfollows from our daily experience. We note that the trajecto-ry of a flying object can be changed by changing the initialvelocity (the science of ballistics depends on this fact). Wealso know that if an object is moving very fast it is easy toanticipate where the object is going to be an instant later.Hence from the present velocity, we know how to Òextrapo-phy 241 MenŽndez 3Chapter 1lateÓ into the future.The third term in Eq. (1) contains the factord 2 x d t 2 t = t 0 / d v d t t = t 0 / a ( t 0 ) which is the acceleration at theinitial time t0. Do we need this information? We certainlydo; however, there is a fundamental difference betweenposition and velocity on one side and acceleration on theother side. Let us consider the case of an object near thesurface of the Earth (a ball, a stone, etc.) We know that wecan change the trajectory of the object by changing its initialposition and velocity. However, the same is not true for theacceleration: no matter what acceleration the object haswhile on our hands, the acceleration becomes 9.8 m/s2(vertically down) the moment the object leaves our hands.The position and velocity of the ball immediately after it isreleased from our hands depends on the position andvelocity it had while still on our hands. Its acceleration,however, reverts ÒmagicallyÓ to 9.8 m/s2 no matter what itsvalue was while in our hands. Moreover, the accelerationremains at 9.8 m/s2 for the entire flight of our object. So itappears that Nature takes care of the acceleration complete-ly.If the acceleration of an object is determined by Nature atall times, then Eq. (1) is not needed: the solution to ourproblem is readily obtained by integrating the equationd v d t = a ( t ) , from which the velocity v(t) is given byv(t ) = v(t0) +tIt0a(t') dt' (2)Next we obtain the position fromx(t1) = v(t0) +t1It0v(t) dt (3)For example, in the case of a falling object, taking x as thevertical displacement (positive direction upward), we havea(t) = -g, with g = 9.8 m/s2. Thus Eq. (2) gives v(t) = v(to)-phy 241 MenŽndez 4Chapter 1g(t-t0), which we can plug into Eq. (3) to obtain the familiarexpression x(t1) = x(t0) + v(t0)(t1-t0) - 1/2g(t1-t0)2.Of course, so far we have only shown that there is a case inwhich the acceleration is completely determined by Nature.Our analysis does not demonstrate that this will be alwaysso. On the other hand, if motion in Nature can be under-stood from a simple and unified theory, the free fall case wejust discussed should be a special application of a moregeneral principle. We therefore assume that Nature deter-mines accelerations and that the solution to our problem isgiven by Eq. (2) and Eq. (3). Ultimately, our assumptionswill have to be verified by experiments.If acceleration is the central quantity that determines themotion of an object, we need a set of rules to obtain theacceleration. These rules are NewtonÕs laws.How Naturedeterminesacceleration:DynamicsThe key idea is