Galileo started to present the classical coordinate systems and the equations of relative positions and timing as a basis for further discussion.

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POSITIONING IN CLASSICAL MECHANICS
Cosmic Adventure 4.7

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Bodily Orientation
When we wish to tell other
people about the location of
an object, we would say that
the object is on our left,
right or front to show the
relationship of its position to
our bodies.
In doing so, we are using our
bodies as a set of reference
frames to which the object is
referred.
Right
Over
Front
Left
Under
Back

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Orientation System
Anyone would know that the
terms left, right, front, back, up,
and down actually embraced all
the direction of space around us.
Left and right are in line with
each other and form a pair, so are
front and back, up and down.
They form three pairs of
complementary directions, each
pair orthogonal or perpendicular
to both of the others.
Right
Over
Front

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Extends to all space
These axes can be
extended to any
lengths thus covering
all space around us. We
can point to a close-by
house or a distant
galaxy in the same
manner without the
fear of incurring
ambiguity.

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Cartesian coordinate system
This set of rectangular coordinates,
or bodily coordinates has been used
intuitively and unknowingly by
people in ancient times, or even in
pre-historic times, the same way we
use it in our daily life today. But the
first person who consolidated the
idea into a definable system was the
great French philosopher René
Descartes (1596 –1650). So the
setup is often referred to as the
Cartesian coordinate system or
simply the Cartesian coordinates.

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3-Dimensional Setup
There are many other kinds
of coordinate systems, but
the three dimensional
Cartesian coordinate system
has always been and is still
the most popular system
used in physics.
𝑋𝑍
𝑌

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3-Dimensional Setup
The powerful feature of this
system is that it is able to
extend to infinity in space.
𝑋𝑍
𝑌

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But what we are dealing with now is
rectilinear motion, so a one
dimensional system is good enough
to do the job. Sometimes we add the
y-axis just to indicate that it is
actually a three dimensional
coordinate system.
𝑋
𝑌

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Two Dimensional Coordinate System
By dealing with only one dimensions, life is much easier. At most we can
add the y-axis to indicate that it is a coordinate system for clarity, so we
are using a one or two dimensional system as deemed appropriate from
now on.
𝑥0

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0
P
The simplest system
𝑥
So here is our classical reference system in
its simplest form. It is just our common
sense of locating an object. The object is
now at P with coordinate 𝑥. That is it is 𝑥
units of length away from the origin O.

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Cosmic Adventure 4.8
TIME IN CLASSICAL PHYSICS

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Jury: Besides space, can the
classical system also tell how
the observers recognize time?
Newton: That is for sure.
Classical physics is a complete
system that covers both space
and time. We do have various
devices to measure time. The
most popular ones are the
water clocks and the sand
clocks.

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Your gadgets are not
that suitable for our
purposes. They are
inaccurate and hard to
calibre. Here is a clock
which is a better device
to illustrate our
discussions. It has a
good mechanical system
to make it an excellent
piece for timing.

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The Observed Object
The best object to be observed in
the study of Relativity is the clock.
It offers an addition advantage
over any other objects.
Besides being able to be moved
from place to place, it can also
register the time at the
corresponding position. To make
things simpler, we use one hand
only, that is, we only use clocks
that register seconds. So we have
now a 12 second clock.
Seconds
12 second clock
Ordinary clock

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0
P
Galileo: That is very nice. It is surely
more effective. Thanks Angela!
Time in the classical coordinate
system behaves the same as its
position. Time at one corner on Earth
is the same as time at any other
corner.
𝑥

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Synchronized Clocks
But to compare time, we need
to first synchronize the
clocks.
The best way to synchronize
clocks is by setting their
hands to exactly the same
configurations at the same
location.
After this, we can put them in
different frames of reference
and use them to measure
events.

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𝑠
0
In the linear system as we have, the
clocks all run at the same rate, thus
telling the same time at any instant. So
here is the equation which represents
time in any location:
𝑡 = 𝑡′
𝑥

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𝑥 = 𝑠
𝑦
𝑧
𝑡
So the classical physics of positioning and
timing for a single system is prescribed
by these classical equations:

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𝑠
0
𝑥
𝑥’
0′
Two Systems But if there are two systems
referring to the same object, then:
𝑥′
= 𝑥 − 𝑠
𝑥 = 𝑥′
+ 𝑠
𝑃

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So these are the two sets of equations treasured by classical physics:
System x:
𝑥′ = 𝑥 − 𝑠
𝑦′ = 𝑦
𝑧′ = 𝑧
𝑡′ = 𝑡
System x’:
𝑥 = 𝑥′ + 𝑠
𝑦 = 𝑦′
𝑧 = 𝑧′
𝑡 = 𝑡′

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OBJECTS ON THE MOVE
To be continued on Cosmic Adventure 4.9