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# CA 5.11 Velocity Transform in Relativity & Visonics

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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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### CA 5.11 Velocity Transform in Relativity & Visonics

1. 1. © ABCC Australia 2015 new-physics.com Cosmic Adventure 5.11 VELOCITY TRANSFORMATION IN RELATIVITY & VISONICS
2. 2. © ABCC Australia 2015 new-physics.com VELOCITY IN CLASSICAL PHYSICS Cosmic Adventure 5.11a
3. 3. © ABCC Australia 2015 new-physics.com 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
4. 4. © ABCC Australia 2015 new-physics.com 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
5. 5. © ABCC Australia 2015 new-physics.com 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.
6. 6. © ABCC Australia 2015 new-physics.com 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
7. 7. © ABCC Australia 2015 new-physics.com Cosmic Adventure 5.11b VELOCITY IN RELATIVITY
8. 8. © ABCC Australia 2015 new-physics.com 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
9. 9. © ABCC Australia 2015 new-physics.com 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
10. 10. © 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
11. 11. © ABCC Australia 2015 new-physics.com 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
12. 12. © ABCC Australia 2015 new-physics.com 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
13. 13. © ABCC Australia 2015 new-physics.com
14. 14. © ABCC Australia 2015 new-physics.com Cosmic Adventure 5.11c VELOCITY TRANSFORM IN VISONICS
15. 15. © ABCC Australia 2015 new-physics.com 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
16. 16. © ABCC Australia 2015 new-physics.com 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
17. 17. © ABCC Australia 2015 new-physics.com 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
18. 18. © ABCC Australia 2015 new-physics.com 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
19. 19. © ABCC Australia 2015 new-physics.com 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
20. 20. © ABCC Australia 2015 new-physics.com 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
21. 21. © ABCC Australia 2015 new-physics.com 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
22. 22. © ABCC Australia 2015 new-physics.com 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
23. 23. © ABCC Australia 2015 new-physics.com 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!
24. 24. © ABCC Australia 2015 new-physics.com
25. 25. © ABCC Australia 2015 new-physics.com ACCELERATION To be continued in Cosmic Adventure 5.12