CA 5.11 Velocity Transform in Relativity & Visonics
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Transcript of CA 5.11 Velocity Transform in Relativity & Visonics
© ABCC Australia 2015 new-physics.com
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
Space Ratio
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’).
Dis
tanc
e (s
pace
) 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 P 0’
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
© ABCC Australia 2015 new-physics.com
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 −𝑣𝑐2 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 −𝑣𝑐2 𝑢
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© ABCC Australia 2015 new-physics.com
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 B Image
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 B Image
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 B 𝑥1 = 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 B Apparent 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∆𝑡1𝑣/𝑐
= 1 +𝑣
𝑐𝑣∆𝑡1 = 1 +
𝑣
𝑐𝑥1
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 +𝑣𝑐 𝑥1
1 +𝑣𝑐 ∆𝑡1
=𝑥1
∆𝑡1
= 𝑣
𝑢1 = 𝑢2 = 𝑣
That is, the observed velocity is
the same as the actual velocity!
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© ABCC Australia 2015 new-physics.com
ACCELERATION
To be continued in Cosmic Adventure 5.12