Introduction

Take an ordinary empty tin can with no top, punch a small hole in the bottom and stuff a bit of sponge into the hole. Now fill the can with water and let'er drain! In this setup, the rate of draining is nearly proportional to the water height, and proportionality implies that the water height decreases to zero exponentially. The ubiquitous exponential describes a wide range of real world phenomena, and we've already met one example in the last chapter, the cooling cup of coffee. There is an analogy between heat and liquid: the cup of coffee cools like liquid (that is, heat) drains away to the environment. Newton's law of cooling tells us that it's the temperature difference between the object and the environment that drives the flow of heat. In the tin can, that corresponds to the water height: the higher the level, the faster the water leaks out. Physically, the height is proportional to the water pressure at the bottom of the can where the hole is.

The leaking tank can model many other phenomena, such as the flow of electricity, wind (as air moves from a point of high barometric pressure to low), and, as we'll see, even descending musical pitches. And when radioactive carbon-14 decays to ordinary carbon- 12, the diminishing amount of C-14 plays the role of the diminishing amount of water in a leaking can. We can also push an empty can bottom-first into a lake and hold it there.