1.6 Introduction to velocity and acceleration 43meters, we conclude that, so far as our problem is concerned, the mag-
nitude of the gravitational force may be considered to be a constant, that
is, the same at all places on our meter sticks. We can know that air
resistance retards the motion of moving bodies but, when heavy bodies
fall only a few meters, this produces consequences so small thatour
measurements are unaffected. Thus, so far as our measurementscan
tell, we are investigating the motion of a body which moveson a line
through the center of the earth with only a constant gravitational force
acting upon it.
In what follows, vectors are denoted by boldface letters as they usually
are in printed scientific works.t Study of physics books or the next
chapter reveals the meaning of the statement that the gravitational force
F which the earth exerts upon our falling body is mgu, wherein is a
positive number (the mass of the body), g is a positive number (the scalar
acceleration of gravity), and u is a unit vector which lies on the line along
which our body falls and is directed toward the center of the earth. The
velocity V and the acceleration a of our falling body are vectors, but they
are representable in the form v = vu and a = au, where v and a are real
numbers that are not vectors and are sometimes called scalars to empha-
size the fact that they are not vectors. Thus v is not a velocity, but it is
the scalar component of a velocity. We call v a scalar velocity. Similarly,
a is a scalar acceleration.
Fortified by at least a hazy understanding of the significance of our
problem, we use experimental data of a table or of Figure 1.63 to learn
about the scalar velocity v and the scalar acceleration a of our body. Let
tl and t2 be two different times and let sl and s2 be the displacements of our
body at these times. As the formula(1.64) S2 - Sl
t2 - ti= average scalar velocityindicates, the quotient on the left is called the average scalar velocity of our
body over the time interval from the lesser to the greater of ti and t2.
In case tl < t2 and si < S2, the quotient in (1.64) has a very familiar form.
Except that the units may be different, the quotient is a positive number
of miles divided by a positive number of hours and hence is a number of
miles per hour that we normally call an average speed instead of an
t We pause to observe that boldface letters cannot be conveniently made with pencils,
pens, crayons, and typewriters, and that a vector F (boldface) is often denoted by F.
Readers are advised to look at F (boldface) and imagine that there is an arrow on top of it
so they will, in effect, see the F which they write when they want to emphasize the fact
that the symbol is (or represents) a vector. Thus the formula F = ma becomes F = ma
when it is transferred from printed material to handwritten hieroglyphics. Sometimes the
arrows are printed to remove the necessity for use of imaginations, but we can, in effect,
be paid for using our imaginations because printing the arrows increases costs of books.