A spherical solild of volume V is made of a material of density `rho_(1)`. It is falling through a liquid of density `rho_(2)(rho_(2) lt rho_(1))`. Assume that the liquid applies a viscous froce on the ball that is proportional ti the its speed v, i.e., `F_(viscous)=-kv^(2)(kgt0)`. The terminal speed of the ball is

Options

A. ` sqrt((Vg (rho _1 - rho _2 ))/(k)) `
B. ` (Vg rho _1)/(k) `
C. `sqrt((Vg rho _1 )/(k)) `
D. ` (Vg (rho_1 - rho _2 ))/(k)`

Answer

A

Explanation

The forces acting on the solid ball when it is falling through a liquid are mg downwards, thrust by Archimedes principle upwards. The viscous force rapidly increases with velocity, attaining maximum when the ball reaches the terminal velocity. Then the acceleration is zero `mg-Vrho_(2)g-kv^(2)=ma`
is volume where, v is the terminal velocity.
When the ball is moving with terminal velocity a = 0.
Therefore `Vrho_(1)g-Vrho_(2)g-kv^(2)=0" "Rightarrow" "v=sqrt((Vg(rho_(1)-rho_(2)))/(k))`