A spherical solid ball of volume V is made of a material of density `rho_1`. It is falling through a liquid of density `rho_2(rho_2ltrho_1)`. Assume that the liquid applies a viscous force on the ball that is proportional to the square of its speed v, i.e., `F_(viscous)=-kv^2(kgt0)`. The terminal speed of the ball is

Options

A. `(Vg rho_1)/(k)`
B. `sqrt((V g rho_1)/(k))`
C. `(Vg(rho_1-rho_2))/(k)`
D. `sqrt((Vg(rho_1-rho_2))/(k))`

Answer

D

Explanation

Weight of the ball = Viscous force + `F_b`
`m_b g=F_(v)+F_b`
`implies V rho_1 g=k v^2+V rho_2 g`
`implies k v^2=V(rho_1-rho_2)g`
`implies v = sqrt((V g(rho_1-rho_2))/(k))`.