## Mathematical Training of Lifeguards

### Light route

Some of the readers that have followed us until now might ask questions about the relationship between the original problem and the title of this section, that includes the word “light”. But this can be easily explained: the velocity of propagation of the light depends on the environment. And the path of a light ray that starts at point $$A$$ and goes through $$B$$ has a shape that minimizes6 the time it takes to complete that path (Fermat’s Rule). For instance, the propagation velocity of light is, in the air, much greater than the one in the water. Hence, the light ray changes direction on the transition surface between the two environments, with angles satisfying the previously mentioned rule. The ratio between the light propagation velocities, in the air and in the water, is called the refraction index of water with respect to air.

The next figure shows three photos, taken exactly from the same place, of a mug with a coin at the bottom, almost invisible in the first one and that progressively becomes visible, when the mug is filled with water.

Click on the image to see the animation

The reason for this phenomenon, is that the light rays, that come from the coin, change direction when they reach the surface of the water.

6 In the given examples, the total path is really minimized, but what we can generally say is that the path followed by light locally minimizes the time spent, that is, minimizes it in terms of all available paths that are close enough.