A block is projected up a rough inclined plane of angle 45^ with an initial speed u . It travels a distance S₁ before coming to rest. If the same block is projected with the same initial speed up a perfectly smooth inclined plane of the same angle, it travels a distance S₂ before coming to rest. If S₁ : S₂ = 2 : 3 , the coefficient of kinetic friction between the block and the rough inclined plane is:

Step-by-step solution

Let the angle of inclination be = 45^ . On the smooth inclined plane, the deceleration is a₂ = g . Using the third equation of motion ( v^2 = u^2 - 2as ), the stopping distance is: S₂ = u^2 2g On the rough inclined plane, the deceleration is due to both gravity and kinetic friction: a₁ = g( + _k ) The stopping distance is: S₁ = u^2 2g( + _k ) Given the ratio S₁ : S₂ = 2 : 3 , we have: S₁ S₂ = u^2 2g( + _k ) u^2 2g = + _k = 2 3 Substitute = 45^ , noting that 45^ = 45^ = 1 2 : 1 2 1 2 + _k 1 2 = 2 3 Dividing numerato