Uniqueness is not just an abstraction designed to please the oretical mathematicians. For...

Uniqueness is not just an abstraction designed to please the oretical mathematicians. For example, consider a cylindrical drum filled with water. A circular drain is opened at the bottom of the drum and the water is allowed to pour out. Imagine that you come upon the scene and witness an empty drum. You have no idea how long the drum has been empty. Is it possible for you to determine when the drum was full?

(a) Using physical intuition only, sketch several possible graphs of the height of the water in the drum versus time. Be sure to mark the time that you appeared on the scene on your graph.

(b) It is reasonable to expect that the speed at which the water leaves through the drain depends upon the height of the water in the drum. Indeed, Torricelli’s law predicts that this speed is related to the height by the formula v2 = 2gh, where g is the acceleration due to gravity near the surface of the earth. Let A and a represent the area of a cross section of the drum and drain, respectively. Argue that A ∆h = av ∆t, and in the limit, A dh/dt = av. Show that dh/dt = .

(c) By introducing the dimensionless variables ω = αh and s = ßt and then choosing parameters

where h0 represents the height of a full tank, show that the equation dh/dt = becomes . Note that when w = 0, the tank is empty, and when w = 1, the tank is full.

(d) You come along at time s = s0 and note that the tank is empty. Show that the initial value problem, , where w(s0) = 0, has an infinite number of solutions. Why doesn’t this fact contradict the uniqueness theorem? Hint: The equation is separable and the graphs you drew in part (a) should provide the necessary hint on how to proceed.