Question

Snell's Law at its heart is a minimization problem. Light takes the path which takes the least time. The classic comparison is a lifeguard at a beach who needs to pick the correct path to get to a struggling swimmer in the least time. Assume the lifeguard can run at 2 meters/second on land, but can only swim at 1 meter/second.

Have the lifeguard start at (0,5). The shoreline runs along the x-axis. The swimmer who needs help will have coordinates that depend upon your birthday.

The X position will be 10 + the day of the month of your birthday (so if you are born on the 6th, x=16). The Y position will be -10 minus the month of your birth (so February would be -12).

For your homework, calculate the time needed for three paths. The first is a straight line from the starting position to the swimmer. The second minimizes the time spent in the water (where the lifeguard is slowest), the person runs to the X position of the swimmer on the shore, and then swims along Y directly. Finally, use Snell's Law or calculus to find the path of least time.

Show all work clearly. Sketches/diagrams will be very helpful.

Answer 1