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)

$$F_{net} = mg - kv$$

$$ma = mg - kv \implies a = g - \frac{k}{m}v$$

1. Behaviour of acceleration

At $$t = 0$$: The mass is just starting to fall, so its speed $$v = 0$$. This means the initial acceleration is at its maximum, i.e., $$a = g$$.

As $$t$$ increases: The body speeds up. As $$v$$ increases, the term $$\frac{k}{m}v$$ also increases, which causes the net acceleration $$a$$ to decrease.

As $$t \to \infty$$: Eventually, the drag force equals the gravitational force ($$kv = mg$$). At this point, the acceleration becomes zero, and the body reaches a constant speed called terminal velocity ($$v_T$$).

The acceleration curve must start at a high value and decay exponentially toward zero.

2. Behaviour of velocity

At $$t = 0$$: The body starts from rest ($$v = 0$$).

As $$t$$ increases: Since the acceleration is positive, the speed increases. However, because the acceleration is decreasing over time, the rate of increase of the speed also slows down.

As $$t \to \infty$$: The speed approaches the terminal velocity $$v_T = \frac{mg}{k}$$ asymptotically.

The speed curve must start at the origin and rise asymptotically toward a horizontal line.

Option C is the only graph where the acceleration ($$a$$) starts at a peak and decays to zero, while the speed ($$v$$) starts at zero and levels off at a terminal value.