Hubbard’s Leaky Bucket If you are given an empty bucket like the one in Fig. 8, having a h...

Hubbard’s Leaky Bucket If you are given an empty bucket like the one in Fig. 8, having a hole in the bottom from which all the water has leaked, can you tell how long ago the bucket was full? Of course not, and the answer can be related to nonunique solutions. The equation describing the height h of the water level comes from Torricelli’s Law:

where the positive constant k depends on the size and shape of the bucket and of the hole.

(a) Even before setting a time scale, we can see that Picard’s Theorem will not apply. Explain why.

(b) Formulate a general initial condition and solve the DE by separation of variables. Why is it impossible to tell how much time has elapsed? Draw several possible solution curves.

(c) Show that the total emptying time is given by   where h0 is the initial height of the water in the bucket.

Figure 8 Hubbard’s leaky bucket.