A block starts moving up an inclined plane of inclination $30^{\circ}$ with an initial velocity of $v_{0}$. It comes back to its initial position with velocity $\frac{v_{0}}{2} .$ The value of the coefficient of kinetic friction between the block and the inclined plane is close to $\frac{ I }{1000},$ The nearest integer to $I$ is......
JEE MAIN 2020, Difficult
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$a_{1}=g(\sin \theta+\mu \cos \theta)$
$a_{2}=g(\sin \theta+\mu \cos \theta)$
$\therefore \frac{1}{2} v _{0} \frac{ v _{0}}{ a _{1}}=\frac{1}{2}\left(\frac{ v _{0}}{2}\right)\left(\frac{ v _{0}}{2 a _{1}}\right)$
$\Rightarrow 3 \sin \theta=5 \mu \cos \theta$
$\therefore \mu=\sqrt{3} / 5$
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