1.1: NEWTON'S LAW OF COOLING Imagine some object like a cannon ball suspended in a swimming Summary: §1.1: NEWTON'S LAW OF COOLING Imagine some object like a cannon ball suspended in a swimming pool by a rope. The cannon ball is at some temperature; perhaps hotter thant the water, perhaps cooler. We can think of it as being a `point source' of heat flow in or out. We are interested in the tem- perature T (in degrees centigrade) as a function of time t (in min- utes). It will be simplest if we measure the temperature relative to the surrounding water, so T = 0 represents the water temperature. We would like to understand the function T(t). Another function we can consider is the derivative T (t) (in degrees per minute.) One way of approaching this problem is to throw away one of the variables, time t. We can imagine the STATE SPACE which is the plane with coordinates (T, T ). A point in the state space is a pair of num- bers representing a temperature, and a rate of change of tempera- ture. This represents a possible state the cannon ball could be in. It is hard to think about the rate of change without thinking about time. T T' T T' Collections: Mathematics