A smooth block is released at rest on a 45° incline and then slides a distance d. The time taken to slide is n times as much to slide on rough incline than on a smooth incline. The coefficient of friction is

See fig.

Let t be the time taken by the block to cover a distance d.

Then using the formula, $\mathrm{s}=\mathrm{ut}+(1/2){\mathrm{g}}^{\mathrm{\prime }}{\mathrm{t}}^2,$ we get 

$\mathrm{d}=\frac{1}{2}\left(\mathrm{g} \sin {45}^{\circ }\right){\mathrm{t}}^2$   …(1)

$\left(\because \mathrm{u}=0\text{ and }{\mathrm{g}}^{\mathrm{\prime }}=\mathrm{gsin}{45}^{\circ }\right)$

Let t' be the time taken for rough plane.

Then $\mathrm{d}=\frac{1}{2}\mathrm{g}\left(\sin {45}^{\circ }-{\mathrm{\mu }}_{\mathrm{k}}\cos {45}^{\circ }\right){\mathrm{t}}^{\mathrm{\prime }2}$ 

$\mathrm{or} \mathrm{d}=\frac{1}{2}\mathrm{g}\left(\sin {45}^{\circ }-{\mathrm{\mu }}_{\mathrm{k}}\cos {45}^{\circ }\right)(\mathrm{nt}{)}^2$    …(2)

Equating equations (1) and (2), we get

$\sin {45}^{\circ }=\left(\sin {45}^{\circ }-{\mathrm{\mu }}_{\mathrm{k}}\cos {45}^{\circ }\right){\mathrm{n}}^2$

$\mathrm{or} \frac{1}{\sqrt{2}}\times \frac{1}{{\mathrm{n}}^2}=\frac{1}{\sqrt{2}}-\frac{{\mathrm{\mu }}_{\mathrm{k}}}{\sqrt{2}}\text{ or }{\mathrm{\mu }}_{\mathrm{k}}=1-\frac{1}{{\mathrm{n}}^2}$.