Summary

DescriptionAddingUnitVelocities.png	English: Adding unit proper-velocities: In this case object A is traveling in various directions with respect to object B as shown by the red wAB vectors in the figure above, while object B is traveling east with respect to object C as shown by the green wBC vectors in the figure above. This illustrates how vector-addition can help us figure out the proper velocity of object A with respect to object C, as shown by the blue wAC vectors in the figure above. Note that when adding proper-velocities (i.e. map-distance r per unit traveler-time τ) at high speed e.g. to get wAC from wAB and wBC, the out-of-frame component (here A with respect to B) must be rescaled (orange extension) to become (wAB)C by a factor ≥ 1 before the vector-addition is performed. At high speeds, this magnitude-rescaling is needed in general (even in the uni-directional case).
Date	20 January 2012, 12:35:40
Source	Own work
Author	P. Fraundorf
Other versions	This physics image could be re-created using vector graphics as an SVG file. This has several advantages; see Commons:Media for cleanup for more information. If an SVG form of this image is available, please upload it and afterwards replace this template with {{vector version available|new image name}}. It is recommended to name the SVG file “AddingUnitVelocities.svg”—then the template Vector version available (or Vva) does not need the new image name parameter.

Additional information

As pointed out here^[1], local proper-velocities w ≡ dx/dτ add as 3-vectors, provided that we rescale the magnitude of the "out-of-frame" vector i.e. in this case (w_AB)_C:

${\displaystyle {\vec {w}}_{\text{AC}}=\left({\vec {w}}_{\text{AB}}\right)_{\text{C}}+{\vec {w}}_{\text{BC}}=\gamma _{\text{AC}}{\vec {v}}_{\text{AC}}}$.

Coordinate-velocity 3-vectors do not similarly add. Moreover, both proper-time and proper-acceleration are frame invariants. This makes 3-vector velocities and accelerations much more useful at high-speeds in the traveler-kinematic^[2] than in the Galilean-kinematic.

An application of relative proper-velocity vector-addition is illustrated in the figure at right.

Footnotes

1. ↑ P. Fraundorf (2011/2012) "Metric-first & entropy-first surprises", arXiv:1106.4698 [physics.gen-ph].
2. ↑ P. Fraundorf (2012) "A traveler-centered intro to kinematics", arxiv:1206.2877 [physics.pop-ph].

Licensing

I, the copyright holder of this work, hereby publish it under the following license:

This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.

You are free:

• to share – to copy, distribute and transmit the work
• to remix – to adapt the work

Under the following conditions:

• attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
• share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

https://creativecommons.org/licenses/by-sa/3.0CC BY-SA 3.0 Creative Commons Attribution-Share Alike 3.0 truetrue