Question: A duck, that is being chased by a fox, saves itself by sitting at the centre of circular pond of radius r. The duck can fly from land but cannot fly from the water. Furthermore, the fox cannot swim. The fox is four times faster than the duck. Assuming that the duck and fox are perfectly smart, is it possible for the duck to ever reach the edge of the pond and fly away to its escape from the ground?
Answer: Since fox is four times faster than the duck, fox can travel half circumference before duck can reach the opposite point on circumference of pond. This can be shown mathematically.
Let v be the speed of duck and 4v be the speed of fox. Time taken by fox to travel half-circumference distance is πr ⁄ 4v and time taken by duck to reach circumference is r ⁄ v . Clearly πr ⁄ 4v < r ⁄ v.
This would make it seem like it is impossible for the duck to escape.
But the duck can swim in concentric circles, so that the fox has to continuously run along the circumference of the pond to stay on the same radius as the duck. If the duck swims near the edge of the pond, the fox could easily keep up since they would be covering approximately the same distance and the fox is four times faster. But what if the duck swam closer to the centre of the pond? The duck would have to cover a smaller circumference, and could use this strategy to put some distance between the fox and itself.
At a distance of r/4 from the centre of the pond, the circumference of the pond is exactly four times the circumference of the duck’s path. Thus, to stay on the same radius as the duck, the fox would barely keep up.
Say, the duck circles the pond at a distance r/4 – e, where e is an infinitesimal amount. So as the duck continues to swim along this radius, it would slowly gain some distance over the fox. Once the duck is able to gain 180 degrees over the fox, the duck would have to cover a distance of 3r/4 + e to reach the edge of the pond. In the meanwhile, the fox would have to cover half the circumference of the pond (i.e the 180 degrees). At that point,
πr ⁄ 4v > (3r/4 + e) / v, for small e
Thus duck can reach circumference of pond faster than fox can catch duck!