' n ' identical small drops of water are falling down vertically through a medium each with a terminal velocity ' v '. If they combine to form a single drop, then its terminal velo

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' n ' identical small drops of water are falling down vertically through a medium each with a terminal velocity ' v '. If they combine to form a single drop, then its terminal velocity is xv . The value of x is

TestHub · Accepted Answer

Terminal velocity of a small drop of radius $r$ is given by $v = \frac{2}{9} \frac{r ^{2} ( ρ - σ ) g}{η}$

Thus, $v \propto r^{2}$

When $n$ drops combine to form a single drop of radius $R$ , the total volume remains conserved.

$\frac{4}{3} π R^{3} = n \times \frac{4}{3} π r^{3}$

$R = n^{1/3} r$

The terminal velocity of the new drop is $V \propto R^{2}$

$V = (\frac{R}{r})^{2} v$

$V = (n^{1/3})^{2} v = n^{2/3} v$

Comparing with $V = xv$ , we get $x = n^{2/3}$

Answer: $n^{2/3}$

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