Which one of the following graphs represents the velocity-time $(v - t)$ graph of a small spherical body falling in a viscous liquid?

When a small spherical body falls in a viscous liquid, it experiences three forces: its weight downwards, buoyant force upwards, and viscous drag force upwards.

The equation of motion is given by:
$mg - F_B - F_v = ma$

Using Stokes' law, $F_v = 6\pi \eta r v$, the equation becomes:
$mg - \dfrac{4}{3}\pi r^3 \sigma g - 6\pi \eta r v = m \dfrac{dv}{dt}$

This can be written in the form:
$\dfrac{dv}{dt} = A - Bv$
where $A$ and $B$ are positive constants.

Integrating with the initial condition $v = 0$ at $t = 0$, we get:
$v(t) = \dfrac{A}{B}(1 - e^{-Bt})$

This equation indicates that the velocity of the body initially increases with time, but the rate of increase (acceleration) continuously decreases. As $t \rightarrow \infty$, the velocity asymptotically approaches a constant maximum value known as the terminal velocity ($v_T = \dfrac{A}{B}$).

The first graph correctly represents this behavior, showing the velocity increasing and eventually leveling off to a constant value.

Answer: