Question

The motion of a body falling from rest in a resisting medium is described by the equation $\frac{dv}{dt}=A-Bv$ where $A$ and $B$ are constants. Then

• A. initial acceleration of the body is $A$
• B. the velocity at which acceleration becomes zero is $A/B$
• C. the velocity at any time $t$ is $\frac{A}{B}\left(1-e^{Bt}\right)$
• D. All of the above are wrong

Answer 1

Answer: (A), (B), (C)

Given acceleration, $\frac{d}{v}=A-Bv$
(a) When $t=0,v=0$, therefore initial acceleration,
${\left(\frac{dv}{dt}\right)}_{t=0}=A$
(b) When acceleration is zero, then $\frac{dv}{dt}=0$.
Hence, $A-Bv=0$
or $v=A/B$
(c) $\frac{dv}{A-Bv}=dt$
Integrating it within the limits of motion, i.e., as time changes from 0 to $t$, velocity changes 0 to $v$, we have
$-{\left[\frac{{\log }_e(A-Bv)}{B}\right]}_0^v=[t{]}_0^t$
$\Rightarrow {\log }_e(A-Bv)-{\log }_{e}A=-Bt$
or $\frac{A-Bv}{A}=e^{-Bt}$
or $v=\frac{A}{B}\left(1-e^{-Bt}\right)$