The Equation of Continuity — Question 8

Question 8

A falling jet narrows because its speed increases

A thin vertical water jet leaves a nozzle downward at speed $v_0=2.0\,\mathrm{m/s}$ and diameter $D_0=10\,\mathrm{mm}$. Neglect air resistance, surface tension and pressure variations within the free jet, so each parcel falls with acceleration $g=10\,\mathrm{m/s^2}$. The jet is steady and incompressible.

1. Use particle kinematics and continuity to find jet diameter after falling $0.60\,\mathrm{m}$.

2. At what downward distance is its diameter $D_0/2$? Explain why halving diameter requires more than doubling speed.

Question 8 - Solution

A falling jet narrows because its speed increases

(a) Derivation and calculation

With downward distance $y$,$$v^2=v_0^2+2gy=2.0^2+2(10)(.600)=16\Rightarrow v=4.00\,\mathrm{m/s}.$$$$Av=A_0v_0\Rightarrow\frac{D}{D_0}=\sqrt{\frac{v_0}{v}}.$$$$\boxed{D=10.0\sqrt{2/4}=7.07107\,\mathrm{mm}}.$$

(b) Result and interpretation

$$D=D_0/2\Rightarrow A=A_0/4\Rightarrow v=4v_0=8.00\,\mathrm{m/s}.$$$$\boxed{y=\frac{v^2-v_0^2}{2g}=\frac{64-4}{20}=3.00\,\mathrm m}.$$Halving diameter quarters area, requiring four times the speed. The prediction assumes an unbroken steady jet.

Physical check

At the nozzle, $y=0$ recovers $v=v_0$ and $D=D_0$.