Q. The motion of a body falling from rest in a resistive medium is described by the equation $\frac{dv}{dt} = a - bv,$ where $a$ and $b$ are constants. The velocity at any time $t$ is
A) $a(1 - b^{2t})$
B) $\frac{a}{b}(1 - e^{- bt})$
C) $abe^{- t}$
D) $ab^{2}(1 - t)$
Explanation
(b) $\frac{dv}{dt} = a - bv$
or $\int_{0}^{v}{\frac{dv}{a - bv} = \int_{0}^{t}{dt}}$
or $- \frac{1}{b}\left\lbrack \log(a - bv) \right\rbrack_{0}^{v} = t$
or $\left( \frac{a - bv}{a} \right) = e^{- bt}$
or $a - \frac{b}{a}v = e^{- bt}$
$v = \frac{a}{b}(1 - e^{- bt})$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.