Terra Fermat Solution

The following is a solution to Puzzle #11 – Terra Fermat. Take a look if you haven’t already.

 

katnissWe’ll first define some distances, with reference to the diagram to the right. Let b be the perpendicular displacement of Katniss from the pathway. Let a be the perpendicular distance of Katniss from the cornucopia. Let x be the distance along the pathway at which Katniss leaves the water.

The length of time Katniss spends swimming is

\displaystyle \frac{\sqrt{b^2+x^2}}{u}

from Pythagoras’ theorem.

The length of time Katniss spends running along the pathway is

\displaystyle \frac{a-x}{v}

So the total time t taken to reach the cornucopia is

\displaystyle t(x)=\frac{\sqrt{b^2+x^2}}{u}+\frac{a-x}{v}

To find the path of least time, we differentiate the expression for t to find the function’s global minimum. The derivative of t is

\displaystyle \frac{dt(x)}{dx}=\frac{1}{u}\frac{x}{\sqrt{b^2+x^2}}-\frac{1}{v}

using the chain rule on the first term. At extrema, the derivative of t goes to 0:

\displaystyle \frac{dt}{dx}=0

Rearrangement yields

\displaystyle x=\frac{b}{\sqrt{(\frac{v}{u})^2-1}}

We can recast this in terms of the angle of approach \theta. From the diagram,

\displaystyle \sin\theta=\frac{x}{\sqrt{b^2+x^2}}

Substituting in the minimising value of x gives

\displaystyle \sin\theta=\frac{u}{v}

 

Let’s check the limiting cases of this equation.

If Katniss can run much, much faster than she can swim, the ratio \frac{u}{v} goes to 0. This means

\theta\to 0

So her best bet is to make a beeline for the pathway and minimise the time she spends in the water.

If Katniss’ running and swimming speeds are equal, the ratio \frac{u}{v} goes to 1. This means

\displaystyle \theta\to\frac{\pi}{2}

In this case, the water and the pathway are indistinguishable, so Katniss may as well swim straight towards the cornucopia.

In general, the answer is intermediate. Heading directly toward the cornucopia minimises the physical distance travelled, but takes longer since swimming is slower than running. Heading directly towards the pathway maximises the time spent running, but makes the overall physical distance travelled unnecessarily long. The solution is a compromise.

 

What might \theta be approximately? Assuming Katniss is an Olympian, she can swim at about 2 metres per second and sprint at 10 metres per second. So

\displaystyle \theta=\arcsin\Big(\frac{1}{5}\Big)

\theta=11.5^\circ

to three significant figures. This is a shallow angle, so shallow it seems the problem is barely worth worrying about. But this is assuming Katniss can spring at full pelt along the pathway. In reality (!) the pathway is uneven and irregular, and Katniss’ footwear would be wet from her having had to swim. This would significantly reduce v relative to u, in turn increasing \theta.

 

An alternative approach: cheat and quote Snell’s law! Katniss behaves like a light ray incident at the critical angle to a boundary. This is because light rays also travel so as to minimise the time spent travelling from one point to another – this idea is called Fermat’s principle of least time. It handily explains why a light ray travel in straight lines through homogeneous media, why its angles of incidence and reflection are equal when encountering a specular surface, and its refraction at a boundary between two media.

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