On the Shapes of Water Fountains and Times Tables - Abstract

Stephen Eberhart
Bridges: Mathematical Connections in Art, Music, and Science (2002)
Pages 314–314

Abstract

When I began teaching mathematics at Waldorf schools (newly recruited from being an orchestra musician), I relied heavily on the books of one of their past master teachers, Hermann von Baravalle, in one of which he suggests having students draw a sequence of parabolas from a common point by connecting dots up 3 units, up 1 unit, then down 1, 3, 5, 7, etc., and going left or right 1 unit each time for one pair of curves, 2 units for a second pair, and so on, calling the result a fountain. I tried this, and was immediately nailed by the typical back-row pupil to whom I have since come to be so grateful: "Fountains aren't flat on top!" A private session with the physics teacher and some resulting expressions for conservation of kinetic and potential energy later, we found that if individual water jets (emanating from a spherical nozzle under equal pressure, viewed in profile) are parabolic in Galileian ideal, then the locus of their vertices is a 2 x 1 sideways ellipse (passing through nozzle and peak) and their envelope of all jets is the parabola tangent to that ellipse at peak - nice!

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