Calculus: Analytic Geometry and Calculus, with Vectors

180 Functions, limits, derivatives

workaday world, we need still another definition. When we say thata
moving body is, at time t, "going in the direction of a vector w" we mean
that its velocity v has the direction of w, that is, v = kw, where k is a
positive scalar. In case w is a unit vector, the scalar k is the speed of the
body. It is not so easy to tell what the body is "doing" when v = 0 or
v does not exist. The ancient Greek philosophers tried to make people
think about motion, and we never quite know how much they smiled
when they insisted that an arrow cannot move where it is and cannot
move where it isn't and, hence, cannot move at all. Thoughts about such
matters can bring the conviction that definitions are not superfluous.
A few simple observations should be made. In case the bumblebee
buzzes around in a plane which we take to be the xy plane, the above
story is unchanged but calculations are simplified by the fact that z(t) = 0
for each t. In case the bumblebee buzzes around in (or on) a line which
we take to be the x axis, we have y(t) = z(t) = 0 for each t. In modern
terminology, scalars cannot be velocities but can be scalar components
of velocities. In case a particle moves on a coordinate axis or on a line
parallel to a coordinate axis, its velocity and acceleration are still vectors
but their scalar components in the direction of the axis are scalars which
we shall call the scalar velocity and scalar acceleration of the particle.
For example, if a particle is moving on an x axis in such a way that its x
coordinate at time t is the scalar (or number)

x = /It'+Bt2+Ct+D+Esin cot,

then the scalar (or number) v (not v) defined by

v =

YT

= 3At2 + 2Bt + C + Ew cos wt

is its scalar velocity at time t and the scalar (or number) a (not a) defined

by

2

a = ate = 611t + 2B - Ewe sin wt

is its scalar acceleration at time t. The speed is ldx/dtl. When the posi-

tive x lies to the right of the origin, the particle is "moving to the right"

at those times for which dx/dt > 0 and is "moving to the left" when

dx/dt < 0. It is not so easy to tell what the particle is "doing" when

dx/dt = 0 or dx/dt does not exist.

Problems 3.79

1 A stone is thrown downward 10 feet per second from the deck of a bridge.

The distances it will have fallen t seconds later is assumed to be

s = 10t + 16t2.