A given object takes n times as much time to slide down a 45° rough incline as it takes to slide down a perfectly smooth 45° incline. The coefficient of kinetic friction between the object and the incline is given by

On smooth inclined plane, the acceleration of the body,a=gsin⁡θIf s be the distance traveled, thens=12gsin⁡θt12   …(1)On rough inclined plane, the acceleration is given bya=gsin⁡θ−μgcos⁡θ∴s=12(gsin⁡θ−μgcos⁡θ)t22   ...(2)From eqs. (1) and (2),t22t12=sin⁡θsin⁡θ−μcos⁡θBut t2=nt1ort22/t12=n2∴n2=sin⁡θsin⁡θ−μcos⁡θSolving, we getμ=n2−1n2×sin⁡θcos⁡θ=n2−1n2×sin⁡45∘cos⁡45∘ =(n2−1)n2×(1/2)(1/2)=n2−1n2=(1−1n2).