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A particle of mass m falls from rest through a resistive medium having resistive force, F = -kv, where v is the velocity of the particle and k is a constant. Which of the following graphs represents velocity (v) versus time (t)?
Now the forces acting on the particle are $$mg\ and\ F=-kv$$
so $$mg-kv=F_{net}$$
$$ma=mg-kv$$
$$m\times\ \frac{dv}{dt}=mg-kv$$
$$\frac{dv}{mg-kv}=\frac{dt}{m}$$
Integrating on both sides
$$\int\ \frac{dv}{mg-kv}=\int\ \frac{dt}{m}$$
$$-\frac{1}{k}\ln\ \left(\frac{\left(mg-kv\right)}{mg}\right)=\frac{t}{m}$$
$$v=\frac{mg}{k}\left(1-e^{-\frac{kt}{m}}\right)$$
Therefore option D is right
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