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(A) When a particle falls through a fluid,it is acted upon by three forces: gravitational force $(mg)$ acting downwards,buoyant force $(F_B)$ acting u

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(A) When a particle falls through a fluid,it is acted upon by three forces: gravitational force $(mg)$ acting downwards,buoyant force $(F_B)$ acting upwards,and a resistive force $(F_R)$ acting upwards. The net force on the particle is $F_{net} = mg - F_B - F_R$. Given that the resistive force is proportional to the square of its speed,we have $F_R = kv^2$,where $k$ is a constant. Initially,at $t = 0$,the speed $v = 0$,so the resistive force $F_R = 0$. The acceleration is maximum,and the speed starts increasing. As the speed $v$ increases,the resistive force $F_R = kv^2$ also increases. Consequently,the net force $F_{net} = mg - F_B - kv^2$ decreases,which means the acceleration $(a = F_{net}/m)$ decreases. Eventually,the resistive force increases to a point where the net force becomes zero $(mg - F_B - kv^2 = 0)$. At this point,the acceleration becomes zero,and the particle attains a constant terminal velocity. The graph of speed $v$ versus time $t$ should show an initial increase in speed with a decreasing slope,eventually approaching a constant value (asymptote). Graph $(a)$ correctly represents this behavior.

$A$ particle released from rest is falling through a thick fluid under gravity. The fluid exerts a resistive force on the particle proportional to the square of its speed. Which one of the following graphs best depicts the variation of its speed $v$ with time $t$ ?

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