Six new solutions to the "travel problem" are given below, but first a few comments about and to this group:
IMHO, physics teachers should present a mix of Philosophy, Methodology, Facts, & Inventive Capacity Stimulation, hereafter, P, M, F & ICS. In the few months I have lurked in this group (with a few posts) I have observed a low (P+ ICS)/(M+ F) ratio. I am not concerned about the relatively low number of "P posts" (post discussing fundamental assumptions about space, time and measurements; sensory limitations; honesty in reporting, ethics, etc.) because hopefully philosophy teachers are also addressing at least some of these issues, but the low "ICS post" level is more disturbing.
I may be unwisely repeating the circa 1890s physicist’s error by thinking that "directly observable physics" (Simple tools: rulers, clocks, strings, springs & sealing wax, etc. + careful observation by human eyes, aided by microscopes and telescopes, when needed.) is now a closed book. – Nothing much more to learn; thus, tomorrow’s physicist will mainly deal with "indirect physics." (Things he/she can’t directly sense and often are processes with non-human time scale.) This fact should, IMHO, stimulate more "ICS posts" than I have observed. ("Imagination" is also stimulated by art and creative writing courses, etc. so by speaking of "ICS" instead of "imagination," I am trying to focus more on characteristics that are more unique and essential to tomorrow’s physicist.)
Puzzles, with out-side-the-box solutions, can help increase ICS, which is harder to teach than (P+M+F), again IMHO. Thus, I was pleased when John posted the "travel puzzle" for which I, like many others, had thought "North Pole" was the unique solution. Once I freed myself from self-imposed presumptions, I found six new solutions and one pseudo-solution (called the "tropical solution" in prior posts) which only appears to be valid.
To increase your student’s ICS, I suggest that you tell them that there are at least 8 solutions (which are at least arguably "conceptually distinct") to the travel puzzle ("Go South one mile, then East one mile, then North one mile to return to your departure point.") and perhaps also give them the "tropical solution," asking if it is valid. (I have posted a detailed description of the "tropical solution" twice before, but no one seems interested enough to point out the flaw, so I don’t post again, unless asked.)
Six conceptually new Solutions:
1) "Roll-a-Coaster" solution – This solution appears to be only a trivial extension of the "circumnavigate South Pole 1/n times" solution, but as it has finite, non-zero departure area, instead of the "zero departure area" of the previously posted "South Pole" solution, it is "conceptually different." (The altitude of the Roll-a-Coaster peaks is arbitrary and this permits continuous latitude displacement of the starting point.) This solution naturally suggests another "South-Pole" solution:
2) "Trench / Tunnel" solution -- Circumnavigate Earth’s geographic axis (many times for maximum economy) inside the earth and /or Antarctic ice in an on-axis "small chamber" at end of a one-mile tunnel whose starting point is approximately one mile from the South Pole. This solution relaxes any mild presumption that travel must be on the Earth’s surface or in the air by airplane. If a circular tunnel with circumference of one mile replaces the "small chamber," this solution is also "conceptually different" because it permits more northern starting points, which are not permitted by other "South-Pole" solutions. This solution suggests still another, "conceptually distinct" set of related extensions to the obvious "North-Pole" solution:
3a) "Submarine / Helicopter" solution. -- Circumnavigate Earth’s geographic axis many times in Arctic Ocean water at least one mile deep. If this depth of ice-free water is not currently available under North-Pole ice, wait until the Arctic Ocean is again ice free. (The problem statement does not require the journey to occur in the current era, as many may presume.) Alternatively, start in a helicopter at least one mile above the ice cap significantly less than one mile from Earth’s rotational axis. If the starting point is very near the axis, then the helicopter begins the journey with a nearly vertical, one-mile descent. In any case, the descent and assent path could be along a straight line directed towards any on-axis point, which is south of the equatorial plane. Because of possible variations in this line and initial altitude, the starting point can be anywhere within a hollow cylindrical volume instead of the zero volume line of the common "North Pole" solution. This makes
this solution "conceptually different" from the obvious North-Pole solution with zero volume and zero area for the starting point.
3b) "Ice-shaft or diving-bell" solution – This solution is conceptually the same solution as (3a) but inside the arctic ice cap (When the ice age has returned, if that ice currently is not one mile thick.) or the journey begins inside a diving bell, secured to the floor of the Arctic Ocean, and requires a south-sloping mine shaft and "small chamber," somewhat similar to solution (2) but more to (3a) in the direction of the shaft.
4) "Hot-start" solutions -- This solution can, in principle, have the starting point anywhere on large (approximately 50,000 square miles!) surface of a cylinder whose axis is the Earth’s rotational axis. This is not feasible with current or foreseeable technology, except near the poles as already discussed, but is conceptually possible at almost any starting point, which is not much greater than one mile from Earth’s rotational axis and thus with this large departure area, it is "conceptually different" from other solutions.
NOTE: The "type-three" solutions and the "hot start" solution all relax the strong self-imposed presumption that travel must start from Earth’s dry surface, and not inside earth, polar ice or Arctic Ocean water or from the Arctic Ocean sea floor. Also used in all these solutions is the fact that any motion, not directly towards the center of the Earth, that is never westbound, nor ever in either of two of the specified directions is travel in the third specified direction, even if most of the displacement during the required mile is radial. (If you do not accept this interpretation of the travel instructions, you also deny that airplanes can land while flying south because they too are partially "moving radially." I.e. approaching more closely the center of the Earth while "landing towards the south.") Many existing mines with elevators in vertical shafts go deeper than the practical "type-three" solutions.
5) "Wait-in-Space" solution -- This most distinct solution has two strong presumptions that must be overcome. (A) The first presumption is that "South," "East" & "North" are directions defined in the Earth’s usual geographic coordinates and not in celestial coordinates in which these directions are also well defined. I don’t think there is any additional solution in the celestial coordinate system commonly used for star maps because it is essentially the same as Earth’s geographic system, but the celestial coordinate system with the ecliptic, rather than the equator, dividing the sky into Northern and Southern Hemispheres holds another solution, which is "very expensive" if done with current technology, but that may not be important, because this solution is not possible for millions of years. I.e. one must also abandon the very strong presumption (B) that travel is in the current era instead of much later when Tidal Friction (my fourth hint, TF) has made the year almost exactly 365
days long. (Previously posted hint were: Space Ship, Celestial Coordinates & Year Long.) Relaxation of presumption "B," might also make a dry-land "tunnel solution" with a departure point near the North Pole feasible, but I don’t know enough about "plate tectonics" to claim this. Perhaps, long before this, the next ice age will lower ocean levels so that a "dry" point, below the ice, near the North Pole significantly less than one mile from the rotational axis exists. (I.e. like the "diving bell" solution but no diving bell required.)
The "wait-in-space" solution requires that one launch the rocket / space ship almost exactly at local midnight or noon (to avoid it being destroyed by Earth’s orbital motion, either immediately or approximately one year later by returning Earth). Resting the sun’s gravity for a year "motionless in space" is what makes this solution "very expensive," unless millions of years from now, when this solution exists, "dark energy" or some other "negative gravity system" has been developed.
Launch is made in the year that tidal friction has made the rotation rate of Earth correct to place the longitude of the launch platform at the terminal point of the journey. By directional definitions of the coordinates system, the Eastbound leg of the journey will be entirely in the ecliptic, but it can have whatever curvature is necessary to insure that the terminal point of the three-leg journey has the same distance from the center of the Earth as the launch platform. (Not necessary to postulate a circular orbit for Earth.)
Although the North and South legs of the journey are confined to planes that are perpendicular to the ecliptic, these legs of the journey may be different curves, which produce a net displacement from the ecliptic. This fact is used to compensate for the fact that if one has timed the launch to place the longitude of the launch platform correctly at the termination of the journey, then the rotation of the Earth and equal N&S displacement legs would make one miss latitude of the launch platform. That is, a valid solution requires three adjustable parameters so that the journey terminates at the launch platform’s (1) longitude, (2) latitude, and (3) distance for the center of the Earth. These three parameters are: (1) When launch is made, (2) a differential displacement produced by diffeerently curved North and Southbound legs, and (3) the curvature of the Eastbound leg, but these "adjustables" may not be exactly "orthogonal" parameters as my discussion of them might lead you to
initially think. I don’t think there is any additional freedom in the rate of travel, other than it may be technically easier to be slowly traveling for the whole year than to hang "motionless in space and wait for Earth to return. (I don’t know much about "dark-energy/mater engines.")
; < )
6) "Irresponsible or Mohammed" solutions – (So named because Mohammed wanted the mountain to come to him, at least in what may be a grossly-prejudice western culture.) There is a whole set of these irresponsible, impractical solutions, but I mention only one. During a slow journey, one blast pieces of the moon into space with nuclear explosives to change Earth’s orbit and bring the launch platform back to your space ship as the Earth / moon mutual rotation is changed.
END of Solutions. Now returning to my concerns:
In addition to being concerned about how physics students are taught, I am very concerned about keeping an adequate stream of them coming to the classroom, when other field are less taxing and often more financially rewarding. Fortunately, physics is no longer essentially a "males-only" domain, but recent changes in US emigration policies have deprived US universities of many bright Asian students who previous came to study and then remained. I have tried to recruit some bright students, who are not currently interested in any of the harder sciences, by writing a book designed to be interesting to them, as it is scary. - A small perturbation of Earth’s orbit induces a rapid-onset ice age beginning in 2008.
More information about how this could be physically possible, yet currently undetected threat, is available at www.DarkVisitor.com, where it also tells how to read book for free and lists all of the physics that is woven into the story, without conventional teaching. As a "hook" to catch the student not currently interested in science, the first four chapters mainly give the history of the principle characters and how their family fortunes were built; however, the text is really a physic book in disguise.
I mentioned that book can be read for free, here and at the website, mainly to deflect charges that I am "spamming." If you think I am, despite "free reading instructions" and the fact the book is published under a pseudonym to remove any charge that it is "fame motivated," then please visit the site before publicly making either charge.
Thank you. – Dr. W. R. Powell, a retired physics professor and worker associated with The Johns Hopkins Univ. and its Applied Physic Laboratory for essentially 30 years, before retiring.
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