Fox vs. Duck

The fox is chasing the duck brainlessly. If the duck's angular velocity while swimming circles is theoretically more than the angular velocity of the fox circling the lake, the duck may actually create some sort of phase lag. So, what will the duck's strategy be?

At a radius of slightly less than $\frac{r}{4}$, the duck can swim in circles, forcing the fox to run around.



Once the duck is at an angle of $\pi$ from the fox, it starts swimming towards the shore.

• The duck has to cover a distance of $\frac{3r}{4}$
• The fox has to cover a distance of $\frac{2\pi \cdot r}{2}$

Since the fox moves four times faster, the distance of the fox has to be larger than four times the distance of the duck.

As we can see, \begin{equation}\frac{3r}{4} * 4 < r* \pi \end{equation} \begin{equation} 3r  < 3.14r \end{equation} \begin{equation} 3  < 3.14 \end{equation} Therefore, the duck will survive!

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