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

Question

The motion of a body falling from rest in a resistive medium, is described by the equation (d v/d t)=a-bv, where a and b are constants. The velocity (v) at any time t is (A) a(.1-b2 t.) (B) (a/b)(.1-e- b t.) (C) abe- t (D) ab2(.1-t.)

Tardigrade Admin · Accepted Answer

(d v/d t)=a-bv or ∫ limits 0v (d v/a - b v) = ∫ limits 0t d t or -(1/b)[l o g (a - b v)]0v=t or ((a - b v/a))=e- b t or a-bv=ae- b t v=(a/b)(.1-e- b t.)

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

$a (1 - b^{2 t})$

$\frac{a}{b} (1 - e^{- b t})$

$ab e^{- t}$

$a b^{2} (1 - t)$

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

Q. The motion of a body falling from rest in a resistive medium, is described by the equation dtdv​=a−bv, where a and b are constants. The velocity (v) at any time t is

a(1−b2t)

ba​(1−e−bt)

abe−t

ab2(1−t)