Nonlinear Dynamics and Chaos
Nonlinear Dynamics and Chaos
2nd Edition
ISBN: 9780813349107
Author: Steven H. Strogatz
Publisher: PERSEUS D
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Chapter 2.5, Problem 6E
Interpretation Introduction

Interpretation:

For a water bucket with a hole in the bottom, it is to be shown that av(t) = Ah˙(t), where a is the area of the hole, A is the cross-sectional area of the bucket, v(t) is the velocity of the water passing through the hole, and h(t) is the height of the water remaining in the bucket at time t. The equation v2 = 2gh is to be derived. It is to be shown that h˙ = -Ch, where C = 2g(aA). Given h(0) = 0, it is to be shown that the solution of h(t) is non-unique for t < 0.

Concept Introduction:

By invoking the law of conservation of mass, the equation av(t) = Ah˙(t) can be proved.

By using the law of conservation of energy, the equation v2 = 2gh can be derived.

By using the equations av(t) = Ah˙(t) and v2 = 2gh, it can be shown that h˙ = -Ch, where C = 2g(aA).

It can be shown that the solution of h(t) is non-unique for t < 0 by integrating the equation h˙ = -Ch and using the initial condition h(0) = 0.

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