The Duck and Fox Puzzle
A duck is swimming in the center of a perfectly circular pond of radius 4 m, and a fox is on the land at the very edge of the pond. The fox is hungry and wants to eat the duck, but it cannot swim. The duck wants to get out of the pond but needs to reach the land in order to fly away. If the duck swims at 1 meter per second, and the fox can run on land at 4 meters per second, how can the duck escape?
There are many possible solutions to this problem; here is one example.
If the duck uses polar coordinates r, θ, she knows that to be as far away from the fox as possible, she will ideally have the opposite phase angle that the fox has:
θduck=θfox + π.
She also knows that the fox would prefer that their phase angles are the same, so to achieve her ideal situation, she needs to be able to adjust her phase angle faster than the fox can adjust his. In other words, she has to be able to swim with angular speed faster than the fox can run. Either animal's angular speed is the ratio of its linear speed to the radius of the circle that it's navigating:
ω = vr.
So, as long as the duck is swimming within 1 m of the center of the pond, she will be able to position herself directly opposite the center from the fox.
At that point, and knowing that swimming in a wider circle won't help, the obvious thing to try is to swim as fast as possible directly towards the bank that is 3 m away. It will take her 3 s to get there, and the fox will take π s (3.14 s),just slightly longer, so the duck will be able to escape!
Can you discover a better solution?
Click and drag the duck to the edge of the pond so that she can fly away without the fox catching her.
Speed of Fox
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