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Fox vs Duck
A duck is swimming at the centre of a circular lake. A fox is waiting at the shore, not able to swim, willing to eat the duck. It may move around the whole lake with a speed four times faster than the duck can swim. As soon as the duck reaches the surface, it can fly, but not within the lake.
Can the duck always reach the shore without being eaten by the fox?
Can the duck always reach the shore without being eaten by the fox?
Hint
The fox is chasing the duck brainlessly. If the duck's angular velocity is (somehow) more than the angular velocity of the fox, the duck may actually create some sort of phase lag. So, what will the duck's strategy be?
Solution
At a radius of slightly less than 
, the duck can swim in circles, forcing the fox to run around.
Once the duck is at a phase of pi from the fox, it starts swimming towards the shore.
Since the fox moves four times faster, the distance of the fox has to be larger then four times the distance of the duck.
As we can see,
, the duck can swim in circles, forcing the fox to run around.
Once the duck is at a phase of pi from the fox, it starts swimming towards the shore.
- The duck has to cover a distance of
- The fox has to cover a distance of
Since the fox moves four times faster, the distance of the fox has to be larger then four times the distance of the duck.
As we can see,
(1) 
(2) 
(3) 
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