MHT CET · Physics · Mechanical Properties of Fluids

A gardening pipe having an internal radius ' \(R\) ' is connected to a water sprinkler having ' \(n\) ' holes each of radius ' \(r\) '. The water in the pipe has a speed ' \(v\) '. The speed of water leaving the sprinkler is

A \(\left(\frac{\mathrm{R}^2}{\mathrm{r}^2}\right) \mathrm{nV}\)
B \(\frac{R^2 v}{n r^2}\)
C \(\left(\frac{\mathrm{nr}^2}{\mathrm{R}^2}\right) \mathrm{V}\)
D \(\left(\frac{n R^2}{r^2}\right) V\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\frac{R^2 v}{n r^2}\)

Step-by-step Solution

Detailed explanation

Using the equation of continuity, we have
\(\mathrm{A}_1 \mathrm{v}=\mathrm{A}_2 \mathrm{v}^{\prime}\)
where \(v\) is speed of water in the pipe and \(v^{\prime}\) is the speed of water leaving the pipe.
Area of the pipe \(=\pi \mathrm{R}^2\)
Area of each hole in the sprinkler \(=\pi r^2\)
\(\therefore \quad\) Total area of the holes in the sprinkler \(=n \pi r^2\)
\(\pi R^2 v=n \pi r^2 v^{\prime}\) ...from (i)
\(\therefore \quad \mathrm{v}^{\prime}=\frac{\mathrm{R}^2 \mathrm{v}}{\mathrm{nr}^2}\)

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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