The duck moves out to a circle 1/4 the radius of the full pond. At this radius, the duck has a faster angular velocity of the wolf, so the duck can go in a circle until the wolf is on the far side of the pond from the duck. At the point, the duck goes straight to the edge, as the duck can traverse the distance of 3/4 before the wolf can traverse the distance π .

Just a note because we don't all use the same browsers, π is supposed to render as pi. In Safari it looks really weird.

Alright, what is the maximum wolf speed for which this works? In general, if the wolf has speed v, the duck can go out the a distance 1/v and still have a greater angular velocity. The duck must then travel a distance 1-1/v before the wolf can travel a distance π . So, the duck requires that:

1-1/v < π/ v v < π+1

Thus, this strategy works until the wolf has speed π+1.

Next question:

Suppose we increase the speed of the wolf to 4.3 times the ducks speed, can the duck do any better? What is the maximum wolf speed for this new strategy?

I'll concede this isn't really a new question, but really it feels like one. The better strategy is alot harder to find and requires some more complicated math to prove it works and is optimal. However, I still promise that all shapes involved are simple geometrical shapes, lines and arcs of circles basically.

## No comments:

Post a Comment