Question

A spherical solid ball of volume $V$ is made of a material of density ${\rho }_1$. It is falling through a liquid of density ${\rho }_1({\rho }_2<{\rho }_1)$. Assume that the liquid applies a viscous force on the ball that is proportional to the square of its speed $v$, i.e., $F_{viscous}=-k{v}^2(k>0)$. The terminal speed of the ball is

• A. $\sqrt{\frac{Vg({\rho }_1-{\rho }_2)}{k}}$
• B. $\frac{Vg{\rho }_1}{k}$
• C. $\sqrt{\frac{Vg{\rho }_1}{k}}$
• D. $\frac{Vg({\rho }_1-{\rho }_2)}{k}$

Answer 1

The correct option is A $\sqrt{\frac{Vg({\rho }_1-{\rho }_2)}{k}}$

The condition for terminal speed $\left(v_t\right)$ is
Weight = Buoyant force + Viscous force

$\therefore V{\rho }_{1}g=V{\rho }_{2}g+k{v}_t^2$

$\therefore v_t=\sqrt{\frac{Vg({\rho }_1-{\rho }_2)}{k}}$