Which of the following option correctly describes the variation of the speed $v$  and acceleration $'a'$  of a point mass falling vertically in a viscous medium that applies a force $F = -kv,$ where $'k'$  is a constant, on the body? (Graphs are schematic and not drawn to scale)

As shown schematically in the figure, two vessels contain water solutions (at temperature $T$ ) of potassium permanganate $\left( KMnO _4\right)$ of different concentrations $n_1$ and $n_2\left(n_1>n_2\right)$ molecules per unit volume with $\Delta n=\left(n_1-n_2\right) \ll n_1$. When they are connected by a tube of small length $\ell$ and cross-sectional area $S , KMnO _4$ starts to diffuse from the left to the right vessel through the tube. Consider the collection of molecules to behave as dilute ideal gases and the difference in their partial pressure in the two vessels causing the diffusion. The speed $v$ of the molecules is limited by the viscous force $-\beta v$ on each molecule, where $\beta$ is a constant. Neglecting all terms of the order $(\Delta n)^2$, which of the following is/are correct? ( $k_B$ is the Boltzmann constant)-

$(A)$ the force causing the molecules to move across the tube is $\Delta n k_B T S$

$(B)$ force balance implies $n_1 \beta v \ell=\Delta n k_B T$

$(C)$ total number of molecules going across the tube per sec is $\left(\frac{\Delta n}{\ell}\right)\left(\frac{k_B T}{\beta}\right) S$

$(D)$ rate of molecules getting transferred through the tube does not change with time

Why not rain drops do not posses greater velocity than some velocity ? Explain.

The terminal velocity of a small sphere of radius $a$ in a viscous liquid is proportional to

The terminal velocity $\left( v _{ t }\right)$ of the spherical rain drop depends on the radius ( $r$ ) of the spherical rain drop as

A ball of mass $m$ and radius $ r $ is gently released in a viscous liquid. The mass of the liquid displaced by it is $m' $ such that $m > m'$. The terminal velocity is proportional to