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

$Fv=k{v}^2$

For terminal velocity, force on the ball must be zero

$k{v}^2+v{\rho }_{2}g=v{\rho }_{1}gv=\sqrt{\frac{vg({\rho }_1-{\rho }_2)}{K}}$