Commuting by Plane
(Escape from FLAT isLAND)
Physics 203: Profs. Martens Yaverbaum, Bean, Lu, Kitayama, Song, Alexander
John Jay College of Criminal Justice, the CUNY
You find yourself trapped in the middle of a desolate island. Maybe you are a "Survivor", maybe you are Gilligan. Either way, you lack food, water and tasteful commercial sponsorship.
Your goal is to get off the island, i.e. to travel as far away from the center as possible.
You find a bunch of accurate directions, but, due to some inexplicable plot twist, they are all ripped up so that the proper sequence is unintelligible:
1) *** Make a point at the center of your graph paper. Even if it is not precisely the center, it must be at the intersection of two blue lines.
2) Decide on some reasonable scale for your graph paper--such as "1 blue box = 1 square yard" or "3 blue boxes = 1 square yard", etc.
3) Either by theory or trial, decide on an order (sequence) in which to follow the directions. You must follow every direction EXACTLY ONCE.
DETERMINE THE ORDER FOR THESE 2-DIMENSIONAL DIRECTIONS which
MAXIMIZES TOTAL 2-DIMENSIONAL DISPLACEMENT
4) *** Obeying the scale you chose, follow the first direction: Draw an arrow from the origin (the center point you drew in step ). The tail of the arrow will be at the origin. At the other end of the arrow, make sure to draw an arrow-head or "tip".
5) *** From the tip of your first arrow, draw an arrow corresponding to the next direction. The tail of this arrow is at the tip of the first. This arrow, too, will culminate in a tip.
6) *** Continue in this manner until you have used up all the directions.
7) Think for a moment about directions that involve square roots of 2. There is a reason that they do. The quantity is intended to make things easier, not harder. At the end of the day, all arrows, if drawn properly will begin and end at points where two blue lines meet.
8) When you are done following all directions:
a] *** Calculate the TOTAL DISTANCE of your chosen route.
b] *** Calculate the TOTAL DISPLACEMENT of your chosen route.
9) *** Now pick a different color and try AT LEAST ONE MORE
route--no matter how strongly you feel about your theory. Consider this a
primitive but central scientific experiment. For a theory to be truly
meaningful, it must embody a coherent body of analysis and data.
10) Now that you have a bunch of data and possibly a theory:
1) *** show how your total displacement for route #2 could have been calculated without any graph paper or measurements.
2) Assume that you walk all 3 of the following paths in direct succession. you start each path right where you completed the previous one:
10 yards due north (that is, 90 degrees north of an axis that points east),
20 yards due north-east (that is, 45 degrees north of an axis that points east) and then
yards at 30 degrees south-west (that is, 30 degrees south of an axis that points west).
*** If you walk all these three paths, what is your final displacement?
*** that is: how far and in what direction (angle) do you end up from where you started?
*** FIND OUT BY A METHOD OF COMPUTATION, NOT by drawing/measuring!!!