A small steel ball is dropped into a long cylinder containing glycerine. Which one of the following is the…

Consider a small steel ball as shown below:

As shown in diagram, buoyancy force $F_B=F$.

Applying Newton's second law, we get

$mg-F_B-F_v=ma$

$\Rightarrow \left(\rho \frac{4}{3}\pi r^3\right)g-\left({\rho }_L\frac{4}{3}\pi r^3\right)g-6\pi \eta rv=m\frac{dv}{dt}$

Let $\frac{4}{3m}\pi R^{3}g\left(\rho -{\rho }_L\right)=K_1$ and $\frac{6\pi \eta r}{m}=K_2$, then we get

$\Rightarrow \frac{dv}{dt}=K_1-K_{2}v$

Integrating both sides, we get

$\Rightarrow {\int }_0^v\frac{dv}{K_1-K_{2}v}={\int }_0^{t}dt$

$\Rightarrow -\frac{1}{K_2}\ln {{\left[K_1-K_{2}v\right]}_0}^v=t$

$\Rightarrow \ln \left(\frac{K_1-K_{2}v}{K_1}\right)=-K_{2}t$

$\Rightarrow K_1-K_{2}v=K_1{e}^{-K_{2}t}$

$\Rightarrow v=\frac{K_1}{K_2}\left[1-e^{-K_{2}t}\right]$

Clearly, option B matches with the given variation of velocity with time.