This problem was deliberately open-ended. The figure above shows one specific example of how you might approach it. It shows a six foot tall fisherman at the left, standing up to her waist in water that is three feet deep. Her waist is at the surface of the water. A fish or target is on the bottom the 20 feet from the fisherman.
The figure above shows how we determine where the fish thinks the fisherman’s head is. We find the path followed by light traveling from the top of the fisherman’s head to the fish in the usual way. See the preceding scene. As in the preceding scene the fish’s brain thinks the fisherman’s head is on the apparent path — the line through the fish and the place where the light’s path intersects the surface of the water. This point is quite far above the actual location of the top of the fisherman’s head.
To determine where the fish thinks the fisherman’s waist is we have two choices. We can start at a point on the fisherman’s body just a fraction of an inch above the surface of the water and do the analysis pictured above. The apparent position of the fisherman’s waist is quite high above the surface of the water. The other option is to start at a point on the fisherman’s body just a fraction of an inch below the surface of the water. Light traveling from this point to the fish is traveling entirely in water and follows a straight line. The apparent position of any point on the fisherman’s body that is below the surface of the water is the same as its actual position. Thus, it looks to the fish as if the fisherman is cut in two with the upper half floating high above the water. The figures below show a kitchen counter experiment verifying this surprising result.
In the figures above the yardstick plays the role of the fisherman. Notice that our prediction is verified. We also see something new and perhaps unexpected, fuel for further questions. We see the part of the fisherman’s body that is underwater reflected in the water’s surface!!
Just Add Math 