n drops of water, each of radius 2mm, fall through air at a terminal velocity of $8 c m s^{- 1}$ .If they coalesce to form a single drop, then the terminal velocity of the combined drop is $32 c m s^{- 1}$ . Find n.

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answer is 8.

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Detailed Solution

$\frac{4}{3} π R^{3} = n \times \frac{4}{3} r^{3}$ or $R^{3} = n r^{3}$ or $R = n^{1 / 3} r$

Or $R = 2 n^{1 / 3} m m$

$v_{0} α r^{2}, v_{0}^{1} α R^{2}$

Now, $\frac{v_{0}^{1}}{v_{0}} = \frac{R^{2}}{r^{2}} = \frac{4 n^{2 / 3}}{4}$

or $\frac{32}{8} = n^{2 / 3}$ or $n^{2 / 3} = 4$

or $n = 4^{3 / 2} = \sqrt{64}$

or n=8

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