CAN YOU EXPLA$$N W$$TH SUBJECT FOR EXAMPLE WHY FORMULA EXPLANAT$$ON PLS ample $($conservation of mass$)$

Two rivers merge to form a larger river as shown in Fig. P$5\mathrm{.}20.$ At a location downstream from the junction $1$ before the two streams completely merge$2,$ the nonuniform velocity profile is as shown and the depth is $1\mathrm{.}8$ m $.$ Determine the value of $V.$

Soln.

Since the mass is conserved

$m_1+m_2^{}=m_3^{}=m_{0\mathrm{.}8V}+m_V^{}$

$A_1{V}_1+A_2{V}_2=A_{0\mathrm{.}8V}0\mathrm{.}8V+A_{V}V$

$V=\frac{A_1{V}_1+A_2{V}_2}{A_{0\mathrm{.}8V}(0\mathrm{.}8)+A_V}=\frac{0\mathrm{.}6**15**0\mathrm{.}6+1\mathrm{.}5**24**1\mathrm{.}2}{1\mathrm{.}8**9**0\mathrm{.}8+1\mathrm{.}8**21**1}=1\mathrm{.}1\frac{m}{s}$