In the previous chapter we studied kinematics. Kinematics is not really ``physics''; it is better viewed as a specialized form of mathematics-with-units of great use (as a tool) to physicists and those interested in solving related problems involving time-dependent differential systems.
Physics is the study of dynamics. Dynamics is the description, due to Isaac Newton, of the actual forces of nature that, we believe, underlie the causal structure of the Universe. We are about to embark upon a simple description of nature that introduces the concept of a force. A force is considered to be the causal agent that produces the effect of acceleration in any massive object.
Newton was far from being the first person to attempt to describe the underlying nature of causality. Many, many others, including my favorite `dumb philosopher', Aristotle, had attempted this. The major difference between Newton's attempt and previous ones is that Newton did not frame his as a philosophical postulate per se. Instead he formulated it as a mathematical postulate and proposed a set of laws.
In physics a law is the equivalent of a postulated axiom in mathematics. That is, a physical law is, like an axiom, an assumption about how nature operates that not provable by any means including experience. A physical law is thus not considered ``correct'' - rather we ascribe to it a ``degree of belief'' based on how well it describes nature. It must withstand the test of repeated experiments in order to survive.
If a set of laws survive all possible tests, we consider it likely that they constitute a part of the true laws of nature; if they survive for a while but then fail some test (often at some length or time scale) then we may continue to call them laws (applicable within the appropriate milieu) but recognize that they are only approximately true and that they are superceded by some more fundamental laws that are closer (at least) to being the ``true laws of nature''.
Newton's Laws, as it happens, are in this latter category. They are ``exact'' (for all practical purposes) for massive, large objects and long times such as those we encounter in the everyday world of human experience (as described by SI scale units). They fail badly (as a basis for prediction) for microscopic phenomena involving short distances, small times and masses, for very strong forces, and for the laboratory description of phenomena occurring at relativistic velocities. Nevertheless, even here they survive in a distorted but still recognizable form, and the constructs they introduce to help us study dynamics still survive.
Interestingly, Newton's laws lead us to second order differential equations, and even quantum mechanics appears to be based on differential equations of second order or less. Third order and higher systems of differential equations seem to have potential problems with causality (where effects always follow, or are at worst simultaneous with, the causes in time); it is part of the genius of Newton's description that it precisely and sufficiently allows for a full description of causal phenomena, even where the details of that causality turn out to be incorrect.
Incidentally, one of the other interesting features of Newton's Laws is that Newton invented calculus to enable him to solve the problems they described. Now you know why calculus is so essential to physics: physics was the original motivation behind the invention of calculus itself. Calculus was also (more or less simultaneously) invented in the more useful and recognizable form that we still use today by other mathematical-philosophers such as Leibnitz, and further developed by many, many people such as Gauss, Poincare, Poisson, Laplace and others. In the overwhelming majority of cases, especially in the early days, solving one or more problems in the physics that was still being invented was the motivation behind the most significant developments in calculus and differential equation theory. This trend continues today, with physics providing an underlying structure and motivation for the development of the most advanced mathematics.