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Re: rapidity



Thanks to those who responded to my "rapidity" questionnaire. For those who didn't see my original post, I asked several questions aimed at discovering whether people who teach special relativity are aware of and teach the concept of rapidity.

There were 20 respondents who have taught SR at the university/college level. Of those 5 are familiar with the concept of rapidity and 2 have used it in their teaching. There were 12 respondents who have taught SR at the high school level. Of those, 2 are familiar with the concept of rapidity and 1 has used it in teaching.

I was motivated to ask by the review of an article that I and a colleague wrote. In the article we introduce the concept of rapidity and give examples of how it can be used in introductory teaching of SR. We also argue for why it is a good idea to do so. The reviewer rejected the paper claiming the "The use of rapidity in discussing velocity addition is in widespread use at the level of introductory undergraduate courses." I do not believe this to be true, and am reassured by the responses to the questionnaire.

For those of you still wondering "What is rapidity?" I include here part of the introduction to the paper.

Imagine that a collision between two identical softballs is caused by throwing one at 15 m/s toward the other, which is stationary (Fig. 1). Ask a student of mechanics what the velocities of these balls are in the center of momentum frame, and you are likely to get the correct answer fairly quickly. In the center of momentum frame each ball is moving toward the other at 7.5 m/s (Fig. 2). The student's concept that this velocity should be halfway between the two original speeds is based on his or her intuition about Galilean velocity transformations.
Now repeat the problem for a proton moving at 0.92c which is traveling toward a stationary proton (Fig. 3). That the answer is not equal and opposite velocities of 0.46c conflicts with intuition, which makes thinking about relativistic problems difficult.
Having gone through this exercise, the student may be glad to learn that there is a quantity related to velocity which transforms in a Galilean-like way, thus restoring some ability to solve relativistic problems intuitively. Take the inverse hyperbolic tangent (a function commonly found on scientific calculators) of the ratio of each proton's speed to that of light.
tanh-1(0.92) = 1.6 (1)
tanh-1(0.00) = 0.0 (2)
These numbers transform in a Galilean-like way. In the center of momentum frame, both protons have an tanh-1(u/c) equal to 0.8, the value halfway between the protons' values in the original frame. The velocities in the new frame can be found by taking the hyperbolic tangent.
(3)
Because of its utility in such problems, rapidity, which is commonly given the symbol y, is defined by
y = tanh-1(ux/c) (4)
where ux is the velocity along the x-axis. In the above example, the x-axis was chosen along the line of flight of the moving proton.


**********************************************
* Rob Davies, Ph.D.
* Lecturer and Laboratory Manager
* University of Denver
* Department of Physics and Astronomy
* Denver, CO. USA
* rodavies@du.edu
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