Answer to Question #207942 in Math for Vardhan

Question

A 40cm diameter pipe conveying water, branches into two pipes of diameters 20cm and 10cm respectively if the average velocity in 40cm diameter pipe is 2m, find velocity in 20cm diameter pipe, if velocity in 10cm diameter pipe is 1.5 m/s.

Answer 1

Diameter of pipe 1 $d_1=40\ cm=0.4\ m$

C/S Area of pipe 1 $A_1 =\dfrac{\pi(d_1)^2}{4}=0.04\pi\ m^2$

Diameter of pipe 2 $d_2=20\ cm=0.2\ m$

C/S Area of pipe 2 $A_2 =\dfrac{\pi(d_1)^2}{4}=0.01\pi\ m^2$

Diameter of pipe 3 $d_3=10\ cm=0.1\ m$

C/S Area of pipe 3 $A_3 =\dfrac{\pi(d_3)^2}{4}=0.0025\pi\ m^2$

Discharge through pipe 1 

$$Q_1=A_1v_1=0.04\pi\ m^2 (2\ m/s)$$

$$=0.08\pi\ m^3/s$$

From continuity $Q_1=Q_2+Q_3$

Discharge through pipe 3 

$$Q_3=A_3v_3=0.0025\pi\ m^2 (1.5\ m/s)$$

$$=0.00375\pi\ m^3/s$$

Discharge through pipe 2

$$Q_2=Q_1-Q_3=0.08\pi\ m^3/s-0.00375\pi\ m^3/s$$

$$=0.07625\pi\ m^3/s$$

Velocity at pipe 2

$$v_2=\dfrac{Q_2}{A_2}=\dfrac{0.07625\pi\ m^3/s}{0.01\pi\ m^2 }=7.625\ m/s$$

$v_2=7.625\ m/s$