Problem 29

  A person is sitting in a small boat in the ocean. A shark is swimming under water at a depth of 4.5 m. When the shark is beyond a certain distance (measured horizontally) from the boat, the shark cannot be seen. Assume that the person's eyes are very near the surface of the water and find that distance.

SOLUTION: In order for the shark to not be seen, light rays coming from the shark must be refracted at greater than 90^o. Using n = 1.33 for the index of refraction of water, and applying Snell's Law, we see that

$$1.33 \sin\theta_c = 1.00 \sin90.0^o$$

$$\theta_c = \arcsin\frac{1.00}{1.33} = 48.6^o$$

Use the definition of tangent to find the distance to the shark

$$\tan48.6^o = \frac{x}{4.5 m}$$

x = 5.1 m

If the shark is farther than 5.1 m, a light ray from the shark will be incident with an angle greater than the critical angle. As a result, the ray would be totally internally reflected back into the water.