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

For a body moving with constant acceleration, the kinematics equation is
s=ut+1/2at²
If the initial speed is zero, then the time taken to reach a distance s is
t=√(2s/a)
i.e., t∝√(1/a)
In the case of a smooth inclined plane, a₁=g sin θ
In the case of rough inclined plane, a₂=g(sin θ - μcos θ)
Time taken to travel down the smooth inclined plane is t₁=t
Time taken to travel down the smooth inclined plane is t₂=nt
t₁/t₂=√(a₂/a₁)
⇒1/n² = (sin θ - μcos θ)/sin θ
⇒μ = tan θ (1 - 1/n²)