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

The free jet is at atmospheric pressure. Bernoulli's equation from the nozzle to height $z$ gives $U(z{)}^2/2+gz=U_0^2/2$. Conservation of volume flux gives $A(z)U(z)=A_0{U}_0$. HenceFor the prescribed well-below-apex regime, $U(z)=U_0[1-gz/U_0^2+O((gz/U_0^2{)}^2)]$ and $A(z)=A_0[1+gz/U_0^2+O((gz/U_0^2{)}^2)]$. 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.