MHT CET · Physics · Mechanical Properties of Fluids

Two rain drops of same radius \(r\) falling with terminal velocity \(V\) merge and form bigger drop with radius \(R\), terminal velocity is

A \(\frac{V R^2}{r^2}\)
B \(\frac{V R}{r}\)
C \(\frac{V r^2}{R^2}\)
D \(\frac{2 V R}{r}\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(\frac{V R^2}{r^2}\)

Step-by-step Solution

Detailed explanation

Expression for terminal velocity can be obtained by balancing weight of the droplet with viscous retardation force and the buoyancy force.
Terminal velocity is given by \(v=\frac{2}{9}\left(\frac{r^2(\rho-\sigma)}{\eta}\right)\)
\(\therefore v \propto r^2\)
Now, let \(V^{\prime}\) be the terminal velocity of the merged droplet
\(\begin{aligned}
& \therefore \frac{V}{V^{\prime}}=\frac{r^2}{R^2} \\
& \Rightarrow V^{\prime}=\frac{V R^2}{r^2}
\end{aligned}\)

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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