This document discusses relative motion and reference frames. It begins by explaining that relative motion occurs between two or more bodies, with each having its own coordinate system. When the observer and object are moving in a straight line, the coordinate systems can be aligned. It then presents two static reference systems, labeled with primes, that are separated by a distance s and both refer to the same object P. These can be combined into a single system. The document goes on to describe setting the two systems in motion, with the second system moving away from the first at a constant velocity v. It derives the relationship that the distance s between the systems after a time period Δt is s = vΔt. Finally, it provides the transformation equations

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OBJECTS ON THE MOVE
Cosmic Adventure 4.9

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Relative motion is the motion
between two or more bodies.
So, in dealing with motion, we have
two or more coordinate systems
each of which is attached to an
observer or an object.

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Alignment of Systems
When the observer and the object are in line as in rectilinear motion, any
two coordinate systems can be aligned to each other to form a simpler
observation system.
0’ P0’

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𝑠
0’
𝑥′
P
System 1 Primed (‘)
𝑥′′
0’ P
System 2 Primed (‘’)
Two Static Reference
Systems
We start off with two static
reference systems, all of
which are related to a third
object located at P.
They are separated by a
distance s. To identify them,
system 1 will bear a single
prime (‘) and 2 will bear a
double prime (‘)

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𝑠
𝑥′
𝑥′′
0’ P
When they refer to the same
object at P and are collinear,
they can be combined into
one reference system. But for
easier recognition, we prefer
to use two in graphics.
0’’
𝑠
0’
𝑥′
P
System 1 Primed (‘)
𝑥′′
0’ P
System 2 Primed (‘’)
❶
❷
Single system

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Two Reference Systems
with Synchronized Timing
Now we have set
primed two system
(O’’) in motion.
In the first place we
move frame two to
coincide with frame
one so that they both
count their time
starting time at t=0.
0’ → 0’’
𝑠 = 0
0’
𝑥′
P
0’’
𝑥′
′
P

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Time & Period
We understand that in
telling time we use 𝑡; but
in measuring time
duration we have to use
∆𝑡. Time is a location in
the time scale but
duration is a period – a
space between two times
𝑡1and 𝑡2. Then:
∆𝑡 = 𝑡2− 𝑡1
Time 𝑡
Duration
∆𝑡
𝑡2
𝑡1

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0’
𝑥′
P
𝑥′′
0’’ P
𝑠 = 𝑣∆𝑡
Relative Velocity v
If O’’ is on the move away
from the starting point at
constant velocity v, the
relative velocity between
them is v.
Then after a period of time
∆𝑡 the distance 𝑠 between
the two systems will be:
𝑠 = 𝑣∆𝑡

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𝑥′′
= 𝑥′ − 𝑣𝑡
𝑥′ = 𝑥′′
+ 𝑣𝑡
Then the relation between
the two systems will be:
0’
𝑥′
P
𝑥′′
0’’ P
𝑠 = 𝑣∆𝑡

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

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FROM GALILEAN TO EINSTEINIAN
Cosmic Adventure 4.10