I recently watched The Hunger Games: Catching Fire. By means of a convenient plot device, Katniss ends up back in the arena. She is immediately presented with a test of her physical intuition.
In the centre of the arena is a circular lake. The lake is divided into twelve circular sectors of equal size by rocky pathways. The pathways converge on an island bearing the cornucopia, housing a bounty of weaponry.
The twenty-four tributes start in one of the circular sectors on a small pedestal, some distance from the cornucopia.
Suppose Katniss can swim at speed and run at speed . For simplicity we’ll assume both and are constant. Assume also that .
Question: At what angle should Katniss approach the nearest rocky pathway such that she reaches the cornucopia in the least possible time?
Hint: It is easiest to define the angle as being that which Katniss’ path subtends with the normal to the pathway.