The terminal velocity of a tiny droplet is V. N number of such identical droplets combine together forming a bigger drop. Find the terminal velocity of the bigger drop.

Explanation

The maximum constant velocity acquired by a body while falling through a viscous fluid is called its terminal velocity.
Terminal velocity, `V = (2)/(9).r^(2) = ((rho - sigma)g)/(eta)`
`rArr (V)/(r^(2)) = (2)/(9)[((rho - sigma)g)/(eta)]`
`(V)/(r^(2)) = (2g)/(9 eta)(rho - sigma)` ...(1)
Similarly, the bigger drip,
`(V')/(R^(2)) = (2g)/(9 eta)(rho - sigma)` ...(2)
Dividing equ. (1) by (2),
`(V)/(V') = (r^(2))/(R^(2)) rArr V' = V((R )/(r ))^(2)` ...(3)
If N drops, then
Volume of one big drop = Volume of N droplets.
`(4)/(3)pi R^(3) = N((4)/(3)pi r^(3))` [`because` Volume of the sphere `V = (4)/(3) pi r^(3)`]
`R^(3) = N(r^(3))`
`R^(3) = N^(1//3)(r )`
`:.` Terminal velocity of bigger drop,
`= ((R )/(r ))^(2) xx V` from equation ...(1)
`= ((N^((1)/(3)))/((r )/(3)))^(2) xx V = (N^(1//3))^(2) xx V`
`= N^(2//3) xx V` from equation ...(2)