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))/(9 eta) (rho - rho') g`
6 cm `s^(-1) = (2 xx (0.15)^(2))/(9 eta) (rho - rho')g" "...(i)`
When the six drops combine, let the radius of the bigger drop be R.
Then volume of bigger drop = 6 `xx` (volume of a small drop)
`rArr (4)/(3)pi R^(3) = 6 xx (4)/(3)pi r^(3)`
`rArr R^(3) = 6r^(3)`
`rArr R = (6)^((1)/(3))r`
= `(6)^(1//3)(0.15)`cm
Let the terminal velocity for this drop be `V_(T)`
Then `V_(T) = (2 xx R^(2))/(9 eta) (rho - rho')g`
`rArr V_(T) = (2 xx 6^(2//3)(0.15)^(2))/(9eta) (rho - rho')g" "...(ii)`
Dividing equation (ii) by equation (i),
We get `(V_(T))/(6) = (6)^(2//3)`
`V_(T) = 6^(5//3) m s^(-1)`
`V_(T) = 19.81 m s^(-1)`