Question

The motion of a body falling from rest in a resisting medium is described by the equation $\frac{d v}{d t}=A-B v$ 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 $\frac{A}{B}\left(1-e^{B t}\right)$
• D. All of the above are wrong

Answer 1

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

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