A stick AB of length L is placed vertically by the wall. At its lower end sits a
bug. The lower end B of the stick starts moving to the right with speed v, and at
the same moment the bug starts crawling along the stick with speed u relative
to the stick. What is the maximal height above the floor that the bug reaches
while it crawls along the stick? Top end A of the stick does not lose contact with
the wall.
(In reply to
solution by Charlie)
The
diagram (here) shows the path of the bug under four scenarios: u=v/2 (the rightmost path), u=v, u=v*sqrt(2) and u=2v (the leftmost path).
Only in the first case is the bug's path solely to the right, or in a clockwise path (there's got to be a small portion near the bottom right where it goes slightly leftward as it goes down). In all the other cases the bug makes a counterclockwise loop, starting to the right and then looping up and left, then down and left. Note the critical case at u=v*sqrt(2), where the bug's maximum height is reached precisely as he reaches the end of the stick (and therefore the wall), so his path at that moment is perpendicular to the wall.
Of course any path to the left of the yaxis (the wall) is not actually achieved as it's on the other side of the wall.
BTW, C in the diagram is the movable point B in the problem. Line AB in the diagram is just a template used to specify the length for the stick via the radius for the initial circle around C in Geometer's Sketchpad. The "calculation result" t1 = 1.00000 was the specification for u/v as the ratio of the radii of the circles representing the bug's and the stickend's distance from C.

Posted by Charlie
on 20190523 10:38:23 