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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/ib/paper-1/17d/i/solution

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Past exam of the mathematics course of the University of Cambridge / 2007 / ib / Paper 1 / 17D / i / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-14

The free jet is at atmospheric pressure. Bernoulli's equation from the nozzle to height

z

gives

U (z)^{2} /2 + g z = U_{0}^{2} /2

. Conservation of volume flux gives

A (z) U (z) = A_{0} U_{0}

. Hence

U (z) = U_{0} 1 - \frac{2 g z}{U _{0}^{2}}, A (z) = A_{0} (1 - \frac{2 g z}{U _{0}^{2}})^{- 1/2} .

(1)

For the prescribed well-below-apex regime,

U (z) = U_{0} [1 - g z / U_{0}^{2} + O ((g z / U_{0}^{2})^{2})]

and

A (z) = A_{0} [1 + g z / U_{0}^{2} + O ((g z / U_{0}^{2})^{2})]

. The slowing jet therefore thickens. These expressions use the slender-jet, nearly uniform cross-sectional speed approximation underlying the question; the formal vanishing-speed apex is outside that approximation.

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