The kinematic entity of velocity is transformed in the Theory of Relativity by the Lorentz transformation of frames; in visionics, by delayed images. Simpler results in visonics.

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Cosmic Adventure 5.11
VELOCITY TRANSFORMATION IN
RELATIVITY & VISONICS

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VELOCITY IN CLASSICAL PHYSICS
Cosmic Adventure 5.11a

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Motion in a Straight Line
When an object moves at a constant
speed 𝑣 along a straight line, it will
cover a certain distance ∆𝑥 within a
certain period of time ∆𝑡.
Distance units in cm, m, km, etc.
∆𝑡
∆𝑥

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Definition of Speed
The ratio between ∆𝑥 and ∆𝑡 is
what we call speed 𝑣 – a scalar
quantity.
If this motion is carried out in a
certain direction, we call it
velocity and it is identified as a
vector – a speed with a
direction.
However since we are dealing
with velocity along a straight
axis, both definitions will work
without any difference.
𝑣 =
Δ𝑥
Δ𝑡
SpaceRatio
Time

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Units of Speed
Distance is measured in
meters, kilometers, feet, or
miles. Time is in seconds,
minutes or hours. For
example, the unit of velocity
can be written as km per
second, or miles pet hour, etc.

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Geometric Representation
of Speed
Since distance (space) and time
are two independent quantities,
they can be represented by the
two perpendicular coordinates of
a Cartesian coordinate system: y-
axis looks after distance and x-
axis looks after time. Velocity
become the slanting line or slope
across space and time (the so
called ‘space-time’).
Distance(space)
Time
Δ𝑥
Δ𝑡
𝑣 =
Δ𝑥
Δ𝑡

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Cosmic Adventure 5.11b
VELOCITY IN RELATIVITY

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Moving Frame
To find the velocity in
relativistic transformations,
we again employ the same
reference system which we
use to find the position and
time.
In this system, one observer
at O’ moves along the
common x-axis at a constant
velocity v with respect to
another observer at O.
𝑠
𝑥
𝑥′
0 P0’
𝑣
Moving frame

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Position and Time
The relativistic position 𝑥′
and time 𝑡′
are as what we
have found in previous
discussions:
𝑥′ =
𝑥 − 𝑣𝑡
1 −
𝑣2
𝑐2
𝑡′
=
𝑡 − 𝑣𝑥/𝑐2
1 −
𝑣2
𝑐2
𝑠
𝑥
𝑥′
0
P
0’
𝑣

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Relativistic Velocity
Since only uniform motion is involved, it is valid to consider small distance
∆𝑥′
and time ∆𝑡′
. Then the equations are slightly changed to:
𝑥′
=
𝑥 − 𝑣𝑡
1 −
𝑣2
𝑐2
∆𝑥′
=
∆𝑥 − 𝑣∆𝑡
1 −
𝑣2
𝑐2
𝑡′
=
𝑡 − 𝑣𝑥/𝑐2
1 −
𝑣2
𝑐2
∆𝑡′
=
∆𝑡 − 𝑣∆𝑥/𝑐2
1 −
𝑣2
𝑐2

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The Transformation of Velocity
We then simply divide the small
distance ∆𝑥’ with the small
duration of time ∆𝑡’ to obtain our
velocity 𝑢′:
𝑢′ =
∆𝑥
∆𝑡′
′
=
∆𝑥 − 𝑣∆𝑡
1 −
𝑣2
𝑐2
÷
∆𝑡 − 𝑣∆𝑥/𝑐2
1 −
𝑣2
𝑐2
𝑢′
=
∆𝑥 − 𝑣∆𝑡
∆𝑡 − 𝑣∆𝑥/𝑐2
=
∆𝑥/∆𝑡 − 𝑣∆𝑡/∆𝑡
1 − 𝑣∆𝑥/∆𝑡𝑐2
=
𝑢 − 𝑣
1 −
𝑣
𝑐2 𝑢

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Differential Approach
A more sophisticated way is to take the
‘differentials’ of the transformed Lorentz
coordinates:
𝑑𝑥′
=
𝑑𝑥 − 𝑣𝑑𝑡
1 −
𝑣2
𝑐2
𝑑𝑡′
=
𝑑𝑡 − 𝑣𝑑𝑥/𝑐2
1 −
𝑣2
𝑐2
But the results will the same:
𝑢′ =
𝑢 − 𝑣
1 −
𝑣
𝑐2 𝑢

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Cosmic Adventure 5.11c
VELOCITY TRANSFORM IN VISONICS

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Previous Equations
The position and timing of
an object at constant motion
has been discussed in the
session on moving objects.
So the relevant equations
are those that have been
formulated.
Cosmic Adventure 5.4

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𝑣
A
Real clock A Real clock B
Situation 1
At time ∆t = 0, both clocks are at the starting position A. Clock A is at rest
while clock be is moving at velocity 𝑣.
Distance 𝑥 = 0
Time ∆t = 0

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𝑣
A B
Real clock A Real clock B
Situation 2
After time = ∆𝑡1, clock B has travelled to B, covering a distance 𝑥1 = 𝑣∆𝑡1.
Both clocks now register the same time, that is, ∆𝑡1.
𝑥1 = 𝑣∆𝑡1

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𝑣
A B
Real clock A Real clock BImage
𝑐
Time of image B
Reading ∆𝑡1
Situation 3 Image Emission
At this moment of time = ∆𝑡1, clock B sends an image (registering time
∆𝑡1) towards clock A, while keeps on traveling away from B.
Clock B goes on

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𝑣
A B C
Real clock A Real clock BImage
𝑐
Situation 4
This image takes time ∆𝑡2 to reach A at speed c. At the same time clock B
has reached C with BC= ∆𝑥1= 𝑣∆𝑡2. The time is then ∆𝑡3 = ∆𝑡1 + ∆𝑡2
𝑥1 = 𝑣∆𝑡1 = 𝑐∆𝑡2 ∆𝑥1= 𝑣∆𝑡2

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𝑣
A B𝑥1 = 𝑣∆𝑡1 = 𝑐∆𝑡2
Image
C
𝑐
Real clock A Real clock B
Actual time
∆𝑡3= ∆𝑡1 + ∆𝑡2
Apparent time
= ∆𝑡1
Situation 5
Actual time
∆𝑡3= ∆𝑡1 + ∆𝑡2
∆𝑥 = 𝑣∆𝑡2
= 𝑣∆𝑡1 × 𝑣/𝑐

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𝑣
A BApparent position
𝑥1 = 𝑣∆𝑡1 = 𝑐∆𝑡2
Image
C
𝑐
Real clock A Real clock B
Actual time
∆𝑡3= ∆𝑡1 + ∆𝑡2
Apparent time
= ∆𝑡1
Final Situation 6
Actual time
∆𝑡3= ∆𝑡1 + ∆𝑡2
∆𝑥 = 𝑣∆𝑡2
= 𝑣∆𝑡1 × 𝑣/𝑐
Actual position
𝑥3 = 𝑣∆𝑡1 + 𝑣∆𝑡2

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Observation 1 – Positions & Time
The apparent position of clock B is:
𝑥1 = 𝑣∆𝑡1 = 𝑐∆𝑡2
The clock reading of image B is ∆𝑡1.
So the apparent time is:
∆𝑡1
The actual position of clock B is :
𝑥3 = 𝑥1 + ∆𝑥 = 𝑣∆𝑡1 + 𝑣∆𝑡1 𝑣/𝑐
= 1 +
𝑣
𝑐
𝑣∆𝑡1 = 1 +
𝑣
𝑐
𝑥1
The actual time of B is (same as A):
∆𝑡3 = ∆𝑡1 + ∆𝑡2 = 1 +
𝑣
𝑐
∆𝑡1

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Observation 2 - Velocities
The apparent velocity is:
𝑢1 =
𝑥1
∆𝑡1
= 𝑣
The actual velocity is:
𝑢2 =
𝑥3
∆𝑡3
=
1 +
𝑣
𝑐
𝑥1
1 +
𝑣
𝑐
∆𝑡1
=
𝑥1
∆𝑡1
= 𝑣
𝑢1 = 𝑢2 = 𝑣
That is, the
observed velocity is
the same as the
actual velocity!

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ACCELERATION
To be continued in Cosmic Adventure 5.12