= 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 $A(z)U(z)=A_0U_0$. Hence
$$
\boxed{U(z)=U_0\sqrt{1-\frac{2gz}{U_0^2}},\qquad
A(z)=A_0\left(1-\frac{2gz}{U_0^2}\right)^{-1/2}.}
$$
For 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.
Back to article page
