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Question

If n identical water droplets falling under gravity with terminal velocity v coalesce to form a single drop which has the terminal velocity $$4$$v, find the number n.


Solution

No of identical water dropletts falling falling under gravity$$=n$$
Thermal velocity$$=V$$
And then coalesce to form a single drop
The thermal velocity $$==4V$$
Find n
The terminal velocity of an object is directly proportional to the square of the radius of the drop. The volume of the small drop is $$\cfrac{4}{3}\pi r^3$$. The volume of the bigger drop formed by collection of  $$N$$ drops is $$N\cfrac{4}{3}\pi R^3$$
(R be the radius of the bigger droplet and r be the radius of the smaller droplet)
Thus the radius of the bigger drop is $$R=N^{1/3}\times r(\cfrac{V_2}{V_1})\\ \quad=(\cfrac{R}{r})^2)(\cfrac{V_2}{V_1})\\ \quad=(\cfrac{r-N^{1/3}}{r})^2\times\cfrac{V_2}{V_1}\\=N^{2/3}4V\\=N^{2/3}V\\N=8$$

Physics

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