A piece of ice slides from rest down a rough 33.0° incline in twice the time it takes to slide down a frictionless 33.0° in- cline of the same length. Find the coefficient of kinetic friction between the ice and the rough incline.

When the ice block slides over a frictionless inclined plane with angle of inclination θ, the net force along the length of the inclined plane is responsible for accelerating the block. With no friction, the only force acting along the length of the inclined plane is the component of weight mg sin θ .
Therefore,
mg sin θ = ma₁
a₁ = g sin θ
When the ice block slides over a rough inclined surface with angle of inclination θ , the net force along the length of the inclined plane is the resultant of the weight acting along the length (mg sin θ) and the kinetic force of friction (f_k) .
Therefore,

Here, μ_k is the coefficient of kinetic friction between the ice and the surface of inclined plane


Round off to two significant figures,
μ_k = 0.49
Therefore, the coefficient of kinetic friction is 0.49 .