A spherical ball of radius $\text{'}r\text{'}$ and mass $\text{'}m\text{'}$ is dropped into a lake having liquid of density one-third of density of ball and viscosity $\eta .$ Choose incorrect statement for steady state having terminal speed $v_T.$

At steady state

$$\begin{gathered} F_{net}=0 \\ mg-F_b-F_D=0 \\ mg-{\rho }_{1}Vg=F_D \\ mg-\frac{{\rho }_s}{3}Vg=F_D \\ {F}_D=\frac{2}{3}mg \end{gathered}$$

Rate of Loss of gravitational P.E of ball  $=\frac{d}{dx}mgh=mg{v}_T$
Rate of Heat Loss = Power of Drag Force
$=F_{D}V=-\frac{2}{3}mgv$

$$\begin{gathered} F_{net}=mg-F_b-6\pi \eta rv \\ ma=mg-F_b-6\pi \eta rv \\ \frac{dv}{dt}=g-\frac{F_b}{m}-\frac{6\pi \eta r}{m}v \end{gathered}$$

So time constant  $\tau =\frac{m}{6\pi \eta r}$
at $t=\tau ,v\text{'}=\left(.63\right)v_T$