Question

Water flows at speed v in a pipe of radius r. Neglecting viscosity, at what speed does the water flow through a constriction in which the radius of the pipe is r/3

Answer 1

Answer:

9v

Explanation:

Given that the water flows at speed v in a pipe of radius r.

Let $v_2$ is the speed of water flow where the radius of the pipe is r/3.

By using the continuity equation, the mass flow rate of water is the same at all the cross-sections of the pipe, wh have

$\rho_1A_1_v_1 =\rho_2A_2_v_2$

Where $\rho_1, A_1$ and $v_1$ are the density, area of the cross-section and speed at cross-section 1 and  $\rho_2, A_2$ and $v_2$ are the density, area of the cross-section and speed at cross-section 2.

As the density of the water remains constant. so $\rho_1=\rho_2$.

$\Rightarrow A_1_v_1 =A_2_v_2 \\\\\Rightarrow (\pi r^2)v=(\pi (r/3)^2v_2 \\\\\Rightarrow r^2v=\frac {r^2}{9}\times v_2 \\\\\Rightarrow v= \frac {v_2}{9} \\\\\Rightarrow v_2=9v.$

Hence, the speed of the water flow where the radius is r/3 is $9v$.