Laws of Motion Part 2

Concept:A particle falling through a resistive medium experiences a velocity-dependent drag force, leading to exponential approach to terminal velocity.Explanation:The net force on the particle is: $m \frac{dv}{dt} = mg - kv$Separate variables and integrate with initial condition $v(0) = 0$:$\int_0^v \frac{dv}{mg - kv} = \frac{1}{m} \int_0^t dt$This gives: $v(t) = \frac{mg}{k} \left(1 - e^{-\frac{k}{m}t}\right)$At $t = 0$, $v = 0$; as $t \to \infty$, $v \to \frac{mg}{k}$ (terminal velocity).The velocity increases exponentially and approaches a constant value asymptotically.Hence the graph starts at the origin and gradually flattens.Shortcut:With resistive force $F = -kv$, the velocity grows and saturates to $mg/k$.Only graph D shows this exponential rise to a constant asymptote from zero.Answer:Option D