Question

The motion of a body falling from rest in a resistive medium, is described by the equation $\frac{d v}{d t}=a-bv,$ where $a$ and $b$ are constants. The velocity $\left(v\right)$ at any time $t$ is

• A. $a\left(\right.1-b^{2 t}\left.\right)$
• B. $\frac{a}{b}\left(\right.1-e^{- b t}\left.\right)$
• C. $abe^{- t}$
• D. $ab^{2}\left(\right.1-t\left.\right)$

Answer 1

Answer: (B)

$\frac{d v}{d t}=a-bv$
or $\int\limits _{0}^{v} \frac{d v}{a - b v} = \int\limits _{0}^{t} d t$
or $-\frac{1}{b}\left[l o g \left(a - b v\right)\right]_{0}^{v}=t$
or $\left(\frac{a - b v}{a}\right)=e^{- b t}$
or $a-bv=ae^{- b t}$
$v=\frac{a}{b}\left(\right.1-e^{- b t}\left.\right)$