For a particle moving along x-axis, acceleration is given by $a = 2 V^{2} .$ If the speed of particle is $V_{0}$ at $x = 0$ then the speed of particle as a function of $x$ is given by $V = V_{0} e^{n x} .$ The value of $\sqrt{n}$ is _______

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Detailed Solution

$a=2 υ^{2}$
$\Rightarrow \frac{d υ}{d t} = 2 υ^{2}$
$\Rightarrow \frac{d υ}{d x} \times \frac{d x}{d t} = 2 υ^{2}$
$\Rightarrow υ \frac{d υ}{d x} = 2 υ^{2}$
$\Rightarrow \frac{d υ}{d x} = 2 υ$
$\Rightarrow \int_{υ_{0}}^{υ} \frac{d υ}{υ} = \int_{0}^{x} 2 d x$
$\log_{e} υ |_{υ_{0}}^{υ} = (2 x) |_{0}^{x}$
$\Rightarrow \log_{e} (\frac{υ}{υ_{0}}) = 2 x$
$\Rightarrow υ = υ_{0} e^{2 x} {∵ I ƒ \log_{e} x = α \Rightarrow x = e^{α}}$