On each of the vertices of a pentagon with sides of length 1, a bug is placed. Each bug walks directly toward the bug that started at the adjacent vertex counterclockwise from its vertex. Of course, the bug at the adjacent vertex is also moving so the bugs will spiral in toward the center of the pentagon. How far does each bug travel?
This is similar to Four Bugs
. There each bug's motion was not offset by any component of the bug being followed moving away from the one doing the following, so the distance traveled was just the initial distance between them.
In the present case, there is a component in the followed bug's motion that is away from the but chasing him. The angle at each corner of a regular pentagon (actually, this is an assumption that the pentagon is regular) is 108°, or 90° + 18°. The followed bug's motion component away from its chaser is sin(18°) so each bug effective speed relative to its quarry relates to its absolute speed is 1 - sin(18°) ~= .6909830056250526 so the distance it needs to travel is the reciprocal of this, or about 1.447213595499958, or exactly 1/(1-sin(18°)).
Posted by Charlie
on 2019-07-24 12:28:08