So long as we deal only with motion along a straight
line, we are far from understanding the motions observed
in nature. We must consider motions along
curved paths, and our next step is to determine the laws
governing such motions. This is no easy task. In the
case of rectilinear motion our concepts of velocity,
change of velocity, and force proved most useful. But
we do not immediately see how we can apply them to
motion along a curved path. It is indeed possible to
imagine that the old concepts are unsuited to the description
of general motion, and that new ones must
be created. Should we try to follow our old path, or
seek a new one?
line, we are far from understanding the motions observed
in nature. We must consider motions along
curved paths, and our next step is to determine the laws
governing such motions. This is no easy task. In the
case of rectilinear motion our concepts of velocity,
change of velocity, and force proved most useful. But
we do not immediately see how we can apply them to
motion along a curved path. It is indeed possible to
imagine that the old concepts are unsuited to the description
of general motion, and that new ones must
be created. Should we try to follow our old path, or
seek a new one?
The generalization of a concept is a process very
often used in science. A method of generalization is
not uniquely determined, for there are usually numerous
ways of carrying it out. One requirement, however,
must be rigorously satisfied: any generalized concept
must reduce to the original one when the original conditions
are fulfilled.
We can best explain this by the example with which
we are now dealing. We can try to generalize the old
concepts of velocity, change of velocity, and force for
the case of motion along a curved path. Technically,
when speaking of curves, we include straight lines.
The straight line is a special and trivial example of a
curve. If, therefore, velocity, change in velocity, and
force are introduced for motion along a curved line,
then they are automatically introduced for motion
along a straight line. But this result must not contradict
those results previously obtained. If the curve becomes
a straight line, all the generalized concepts must
reduce to the familiar ones describing rectilinear motion.
But this restriction is not sufficient to determine
the generalization uniquely. It leaves open many possibilities.
The history of science shows that the simplest
generalizations sometimes prove successful and sometimes
not. We must first make a guess. In our case it is
a simple matter to guess the right method of generalization.
The new concepts prove very successful and help
us to understand the motion of a thrown stone as well
as that of the planets.
And now just what do the words velocity, change in
velocity, and force mean in the general case of motion
along a curved line? Let us begin with velocity. Along
the curve a very small body is moving from left to
right. Such a small body is often called a particle. The
dot on the curve in our drawing shows the position of
the particle at some instant of time. What is the velocity
corresponding to this time and position? Again Galileo's
clue hints at a way of introducing the velocity. We must,
once more, use our imagination and think about an
idealized experiment. The particle moves along the
curve, from left to right, under the influence of external
forces. Imagine that at a given time, and at
the point indicated by the dot, all these forces suddenly
cease to act. Then, the motion must, according to the
law of inertia, be uniform. In practice we can, of
course, never completely free a body from all external
influences. We can only surmise "what would happen
if. . . ?" and judge the pertinence of our guess by the
conclusions which can be drawn from it and by their
agreement with experiment.
The vector in the next drawing indicates the guessed
direction of the uniform motion if all external forces
were to vanish. It is the direction of the so-called
tangent. Looking at a moving particle through a
microscope one sees a very small part of the curve,
which appears as a small segment. The tangent is its
prolongation. Thus the vector drawn represents the
velocity at a given instant. The velocity vector lies on
the tangent. Its length represents the magnitude of the
velocity, or the speed as indicated, for instance, by the
speedometer of a car.
Our idealized experiment about destroying the motion
in order to find the velocity vector must not be
taken too seriously. It only helps us to understand what
we should call the velocity vector and enables us to
determine it for a given instant at a given point.
In the next drawing, the velocity vectors for three
different positions of a particle moving along a curve
are shown. In this case not only the direction but the
magnitude of the velocity, as indicated by the length
of the vector, varies during the motion.
To Be Contd...