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The velocity of a particle travelling along a straight line is $υ = υ_{0} - k x,$ where $υ_{0}$ and $k$ are constants .If $x = 0$ when $t = 0$ , velocity of the particle as a function of time is

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a

$\frac{υ_{0}}{k} (1 - e^{- k t})$

b

$υ = - υ_{\circ} e^{k t}$

c

$υ = υ_{0} e^{- k t}$

d

$- k υ_{0} e^{- k t}$

answer is C.

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Detailed Solution

Since $υ = \frac{d x}{d t}$
$\Rightarrow \int_{0}^{t} d t = \int_{0}^{x} \frac{d x}{υ_{0} - k x}$
$\Rightarrow t | \begin{matrix} t \\ 0 \end{matrix} = - \frac{1}{k} \ln (υ_{0} - k x) | \begin{matrix} x \\ 0 \end{matrix}$

$\Rightarrow t = \frac{1}{k} \ln (\frac{υ_{0}}{υ_{0} - k x})$
$\Rightarrow e^{k t} = \frac{υ_{0}}{υ_{0} - k x}$
$\Rightarrow x = \frac{υ_{0}}{k} (1 - e^{- k t})$
Now, $υ = \frac{d x}{d t} = \frac{d}{d t} [\frac{υ_{0}}{k} (1 - e^{- k t})]$
$\Rightarrow υ = υ_{0} e^{- k t}$

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The velocity of a particle travelling along a straight line is υ=υ0−kx, where υ0 and k are constants .If x=0 when t=0, velocity of the particle as a function of time is