Calculus: Analytic Geometry and Calculus, with Vectors

1.6 Introduction to velocity and acceleration 41

(^43) Let us suppose that a man who marries should select for his wifea woman
whose age is 10 years more than half his age. Construct a graph for use of
bachelors who are accustomed to picking information from graphs in the Wall
Street journal and everywhere else but are unaccustomed to making abstruse
mathematical calculations.
44 Let time t be measured in seconds so that, aswe can see by replacing x
by t in Figure 1.58, sin i increases from 0 to 1 and decreases back to 0 in7r (about
3) seconds. If you can acquire the ability to moveyour pencil point in the xy
plane in such a way that its coordinates (x,y) at time t arex = sin t and y =
Isin ti, you will get a V for victory.

1.6 Introduction to velocity and acceleration

matics and physics are accustomed to difficulties involved in correlating
studies of graphs, vectors, velocities, and accelerations in mathematics to
studies of diagrams, forces, velocities, and accelerations in physics.
There is a reason why it is not easy to achieve complete correlation. In
order to be able to solve just one of his easiest problems involving motion
of a body or particle, a physics student requires a little information about
several basic concepts. This section is introduced at the end of our first

chapter because it may be a desirable or even necessary part of some
educational programs. Students can be advised to read it to obtain

preliminary ideas about their external world but, so far as this course is
concerned, can be advised to postpone the learning of the mathematics in
it. Some and perhaps most teachers will proceed directly to the next

chapter and will devote a classroom hour to this section only if and when
their students face the prospect of studying falling bodies in their physics
courses before they encounter derivatives and integrals in their mathe-
matics courses. The next chapter, Chapter 2, treats vectors in space of
three as well as fewer dimensions. While physicists can regret that this
delays our full treatment of velocities and accelerations, they can also
rejoice in the fact that the delay permits production of a much more use-
ful treatment of the matter.f
As the preface states, the first third of this book contains all or nearly
all of the analytic geometry and calculus that students normally encounter
in their introductory full-year college and university courses in physics.
In a few weeks, formulas like

(1.611) s =

2

gtz + vot + so

(1.612) vdt=gt+vo

(1.613) a d

= da

d'ate _ g

t This is a very conservative statement. Vectors, like numbers, are important things
and there are many reasons why they should be encountered early and frequently when
geometry and calculus are studied.