In a straight race from center to shore, Secret Service guy can run the max half-circumference faster than the rabbit can swim the radius.
So to beat the man, the rabbit needs to do more work than that. Kinda thinking a spiral pattern might work but I'm too lazy to do that math.
Let's try something simpler. If the rabbit can get himself sufficiently far in one direction, while forcing Secret Service man to be 180 degrees opposite, from there he clearly can win the race. If the pond has radius r, and the rabbit is x away from the center, then a straight swim to the edge is distance (r-x), in which time the man can run 4(r-x) and has to cover pi*r distance. So the rabbit has to be within pi*r/4 of the shore, or almost 1/4 of the way to the shore or closer, to win. The remaining question is can the rabbit get to that point, and force Secret Agent Man to the diametrically opposite side?
Well, rabbit swims 1/4 as fast as Man can run. So if rabbit is anywhere closer to the center than r/4, he can begin to swim in a circle and cover the angle faster than the man can run, and therefore swim in that circle until he forces man to be 180 degrees opposite. Then he can bolt for the shore, and to Rabbit Freedom.
So rabbit can swim to some point in between r/4 from the center and (1 - pi/4)r from the center. Regardless where the man is, Rabbi can then begin swimming his loop until the man is 180 opposite, then dash.
So to beat the man, the rabbit needs to do more work than that. Kinda thinking a spiral pattern might work but I'm too lazy to do that math.
Let's try something simpler. If the rabbit can get himself sufficiently far in one direction, while forcing Secret Service man to be 180 degrees opposite, from there he clearly can win the race. If the pond has radius r, and the rabbit is x away from the center, then a straight swim to the edge is distance (r-x), in which time the man can run 4(r-x) and has to cover pi*r distance. So the rabbit has to be within pi*r/4 of the shore, or almost 1/4 of the way to the shore or closer, to win. The remaining question is can the rabbit get to that point, and force Secret Agent Man to the diametrically opposite side?
Well, rabbit swims 1/4 as fast as Man can run. So if rabbit is anywhere closer to the center than r/4, he can begin to swim in a circle and cover the angle faster than the man can run, and therefore swim in that circle until he forces man to be 180 degrees opposite. Then he can bolt for the shore, and to Rabbit Freedom.
So rabbit can swim to some point in between r/4 from the center and (1 - pi/4)r from the center. Regardless where the man is, Rabbi can then begin swimming his loop until the man is 180 opposite, then dash.