Question

The Bhagirathi and the Alaknanda merge at Devprayag to form the Ganga with their speeds in the ratio 1:1.5. The cross-sectional areas of the Bhagirathi, the Alaknanda, and the Ganga are in the ratio 1:2:3. Assuming streamline flow, the ratio of the speed of Ganga to that of the Alaknanda is.
A. 7:9
B. 4:3
C. 8:9
D. 5:3

Answer 1

We have to use the equation of continuity in this question. compare and equate simply to solve the problems. We will simplify the given equation by simply comparing the given areas of the river. After that simply use them to find the required answer.

Complete step by step answer:
We know the equation of continuity
By using the equation of continuity
Area of Bhagirathi = A
Area of alaknanda = 2A
Area of ganga = 3A
${{V}_{B}}:{{V}_{AL}}:{{V}_{G}}\Rightarrow V:\dfrac{3}{2}{{V}_{{}}}:{{V}_{1}}$
By equation of continuity and solving we get

& AV+\dfrac{3}{2}A.2.V\Rightarrow 3A.{{V}_{1}} \\\ & {{V}_{ganga}}=\dfrac{4}{3}V \\\ & \dfrac{{{V}_{alaknanda}}}{{{V}_{ganga}}}=\dfrac{\dfrac{3}{2}V}{\dfrac{4}{3}V}=\dfrac{9}{8} \\\ \end{aligned}$$ ${{V}_{ganga}}:{{V}_{alaknanda}}$ = 8:9 The ratio of the speed of Ganga to the speed of the Alaknanda by using equation of continuity is 8:9 **The correct option is (C).** **Additional Information:** We know that the Equation of Continuity: ${{\rho }_{1}}{{A}_{1}}{{V}_{1}}={{\rho }_{2}}{{A}_{2}}{{V}_{2}}$ The continuity equation states that within the case of steady flow, the quantity of fluid flowing past one point must be an equivalent because the amount of fluid flowing past another point, or the mass flow is constant. It is essentially a statement of the law of conservation of mass. The explicit formula of continuity is the following: ${{\rho }_{1}}{{A}_{1}}{{V}_{1}}={{\rho }_{2}}{{A}_{2}}{{V}_{2}}$ Where ρ is density, A is cross-sectional area and v is the flow velocity of the fluid. The subscripts 1 and 2 indicate two different regions in the same pipe. When fluids move through a full pipe, the quantity of fluid that enters the pipe must equal the quantity of fluid that leaves the pipe, albeit the diameter of the pipe changes. This is a restatement of the law of conservation of mass for fluids. **Note:** The continuity equation in fluid dynamics describes that in any steady-state process, the rate at which mass leaves the system is equal to the rate at which mass enters a system. u = flow velocity vector field.