Smooth block is released at rest on a 45° incline and then slides a distance d. If the time taken of slide on rough incline is n times as large as that to slide than on a smooth incline. Show that coefficient of friction. µ = (1-1/n ²).

When there is no friction, the block slides down the inclined plane with
acceleration.
a = g sin θ
when there is friction, the downward acceleration of the block is
a´ = g (sin θ – μ cos θ)
As the block slides a distance d in each case so
d= 1/2 at² = 1/2 a’t’²

a/a’ = t’²/t ² = (nt)²/t² = n²
or gsinθ/g(sinθ – μ cosθ) = n²