એક પ્રવાહી એક સમક્ષિતિજ નળી કે જેનો આડછેદ બદલાતો હોય તેમાં જે સ્થાને \(P\) પાસ્કલ દબાણ હોય ત્યાં \(v\;ms^{-1}\) વેગથી વહે છે. બીજા સ્થાને જ્યાં દબાણ \(\frac{ P }{2}\) હોય ત્યાં તેનો વેગ \(V\;ms^{-1}\) છે. જો પ્રવાહીની ઘનતા \(\rho\, kg\, m ^{-3}\) અને પ્રવાહ ધારારેખી હોય તો \(V\) કેટલો હશે?

A \(\sqrt{\frac{ P }{2 \rho}+ v ^{2}}\)
B \(\sqrt{\frac{ P }{\rho}+ v ^{2}}\)
C \(\sqrt{\frac{2 P }{\rho}+ v ^{2}}\)
D \(\sqrt{\frac{ P }{\rho}+ v }\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\sqrt{\frac{ P }{\rho}+ v ^{2}}\)

Step-by-step Solution

Detailed explanation

Applying Bernoulli's Equation \(P _{1}+\frac{1}{2} \rho v _{1}^{2}+\rho gy _{1}= P _{2}+\frac{1}{2} \rho v _{2}^{2}+\rho gy _{2}\) \(P +\frac{1}{2} \rho v ^{2}=\frac{ P }{2}+\frac{1}{2} \rho V ^{2}\) \(\frac{2 P }{2 \rho}+\frac{1}{2} \frac{\rho v ^{2}}{\rho} \times 2= V ^{2}\)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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