MHT CET · Physics · Mechanical Properties of Fluids
Eight identical small drops of water are falling down vertically through a medium, each with terminal velocity \(v\). If they combine to form a single drop, then its terminal velocity will be
- A \(3 v\)
- B \(6 v\)
- C \(5 v\)
- D \(4 v\)
Answer & Solution
Correct Answer
(D) \(4 v\)
Step-by-step Solution
Detailed explanation
Let \(R\), and \(r\) be the radii of the big and small drops respectively. The volume of the big drop \(=8 \times\) volume of one small drops
\(\begin{aligned} & \therefore \frac{4}{2} \pi R^3=8 \times \frac{4}{3} \pi r^3 \\ & \therefore R^3=(2 r)^3 \therefore R=2 r\end{aligned}\)
The terminal velocity of a drop \(v \propto(\text { radius })^2\)
\(\begin{aligned} & \therefore \frac{V}{v}=\frac{R^2}{r^2}=\frac{(2 r)^2}{r^3}=4 \\ & \therefore V=4 v\end{aligned}\)
\(\begin{aligned} & \therefore \frac{4}{2} \pi R^3=8 \times \frac{4}{3} \pi r^3 \\ & \therefore R^3=(2 r)^3 \therefore R=2 r\end{aligned}\)
The terminal velocity of a drop \(v \propto(\text { radius })^2\)
\(\begin{aligned} & \therefore \frac{V}{v}=\frac{R^2}{r^2}=\frac{(2 r)^2}{r^3}=4 \\ & \therefore V=4 v\end{aligned}\)
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