Rowing to the shore of a perfectly circular lake whose circumference is patrolled by a monster that can run $4x$ as fast as you can row

Question

You are on a rowboat in the middle of a large, perfectly circular lake. On the perimeter of the lake is a monster who wants to eat you, but fortunately, he can't swim. He can run (along the perimeter) exactly $4x$ as fast as you can row, and he will always run towards the closest bit of shore to you. If you can touch shore even for a second without the monster already being upon you, you can escape. Suggest a strategy that will allow you to escape, and prove that it works.

Well the solution to this puzzle is to make your angular speed greater than the monster by revolving around the smaller circle and get to the diametrically opposite point relative to him and then just run for the shore in radial direction. To make your angular speed greater than the monster you have to revolve around a circle whose radius will be less than R/4 but i am skipping those details here.

I am thinking of another solution where i will always run in a perpendicular direction to him, but not able to prove that this strategy will or won’t work. Need some guidance here. I think the curve that i will make will be logarithmic spiral but not able to come to equations.

Answer 1

The monster travels with velocity $[-4\cos(t),4\sin(t)]$, and you start traveling with velocity $[-\sin(t),-\cos(t)]$ in response, always moving perpendicularly. You will realize that your distance function is now $[\cos(t)-1,-\sin(t)]$. So, you're not escaping that lake unless the magnitude of your velocity, which I'm taking as 1, is greater than $\frac{R}{2}$.
What if you're in a very small lake, so small that $||v||>\frac{R}{2}$? Can you escape now? Let's say the monster is instead just a jogger, attempting to lap around the circular lake as many times as possible with some speed $J$, not caring about your position. The jogger starts at the north point of the lake, and you start moving perpendicular to him. I'm going to rewrite your equations of motion and the lake's boundary in the form $(x-h)^2+(y-k)^2=r^2$: $$x^2+y^2=R^2,(x+1)^2+y^2=1$$ Then, we can solve for the two points of intersection by subtracting them, getting $$2x+1=R^2-1 \Rightarrow x=-\frac{R^2}{2}$$ $$y=\pm R\sqrt{1-\frac{R^2}{4}}$$ Assuming you started by going south, that means you'll first hit the edge of the lake at $$\left(-\frac{R^2}{2},R\sqrt{1-\frac{R^2}{4}}\right)$$ The amount of time this takes, $t$, can be solved for by taking $$\cos(t)-1=-\frac{R^2}{2} \Rightarrow t_{boat}=\cos^{-1}\left(1-\frac{R^2}{2}\right)$$ This also shows you'll never hit shore if $R>2$, since $\cos^{-1}(x)$ isn't defined outside $-1\leq x\leq 1$.
Meanwhile, our jogger, whose position function is $\left[R\sin\left(Jt\right),R\cos\left(Jt\right)\right]$, will hit this spot at time $$R\sin(Jt)=-\frac{R^2}{2} \Rightarrow t_{jogger}=\frac{1}{J}\sin^{-1}\left(-\frac{R}{2}\right)$$ You'll notice if you graph these two that $t_{jogger} < t_{boat}$ if $J>0.5$. The monster would have to be half of your boating speed for you to feasibly escape with this method, and if it's four times your boating speed it could get there in one-eighth the time you would take to boat. You can't escape with this method, unfortunately.