Six small raindrops each of radius 1.5 mm, come down with a terminal velocity of 6 cm `s^(-1)`. They coalesce to form a bigger drop. What is the terminal velocity of the bigger drop?

Explanation

Radius of each small drop r = 1.5 mm
= 0.15 cm
Terminal-velocity for small drops-`v_(t) = 6 cm s_(-1)`
Let the density of water = `rho`
and the density of air = `rho`.
Then `v_(t) = (2r^(2))/(9eta) (rho - rho.)`
6 cm`s^(-1) = (2xx(0.15)^(2))/(9eta) (rho - rho.)` .............(i)
When the six drops combine, let the radius of the bigger drop be R. Then volume .bf bigger drop = 6 `xx` (volume of a.small drop)
`4/3 pi R^(3) = 6xx 4/3 pi r^(3)`
`R^(3) = 6r^(3)`
R = `(6)^(1/3)r`
`(6)^(1/3) (0.15)` cm
Let the terminal velocity for this drop be `V_(r)`
then `V_(T) = (2xxR^(3))/(9eta) (rho - rho.)`
` V_(T) = (2xx6^(2/3) (0.15)^(2))/peta (rho - rho.)g `
Dividing equation (ii) by equation (i)
We get `V_(T)/6 = 6^(2/3) `
`V_(T) = 19.81` ms^(-1)