A given object takes $n$ times the time to slide down $45^{\circ}$ rough inclined plane as it takes the time to slide down an identical perfectly smooth $45^{\circ}$ inclined plane. The coefficient of kinetic friction between the object and the surface of inclined plane is :
JEE MAIN 2024, Difficult
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Case-$1$ : No friction
$\mathrm{a}=\mathrm{g} \sin \theta$
$\ell=\frac{1}{2}(\mathrm{~g} \sin \theta) \mathrm{t}_1^2$
$\mathrm{t}_1=\sqrt{\frac{2 \ell}{\mathrm{g} \sin \theta}}$
Case-$2$ : With friction
$a=g \sin \theta-\mu g \cos \theta$
$\ell=\frac{1}{2}(g \sin \theta-\mu g \cos \theta) t_2^2$
$\sqrt{\frac{2 \ell}{g \sin \theta-\mu g \cos \theta}}=n \sqrt{\frac{2 \ell}{g \sin \theta}}$
$\mu=1-\frac{1}{n^2}$
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