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

We need to find the coefficient of kinetic friction for a $$45°$$ rough incline, given that the time to slide down is $$n$$ times that on a smooth incline.

Acceleration on a smooth 45° incline.

$$a_1 = g\sin 45° = \frac{g}{\sqrt{2}}$$

Acceleration on a rough 45° incline.

$$a_2 = g(\sin 45° - \mu_k \cos 45°) = \frac{g}{\sqrt{2}}(1 - \mu_k)$$

Next, relate the times using kinematic equations.

For the same distance $$s$$ starting from rest:

$$s = \frac{1}{2}a_1 t_1^2 = \frac{1}{2}a_2 t_2^2$$

Given $$t_2 = n \cdot t_1$$:

$$a_1 t_1^2 = a_2 (nt_1)^2 = a_2 n^2 t_1^2$$

$$a_1 = n^2 a_2$$

Next, solve for $$\mu_k$$.

$$\frac{g}{\sqrt{2}} = n^2 \cdot \frac{g}{\sqrt{2}}(1 - \mu_k)$$

$$1 = n^2(1 - \mu_k)$$

$$\mu_k = 1 - \frac{1}{n^2}$$

The coefficient of kinetic friction is $$1 - \frac{1}{n^2}$$.

The correct answer is Option 4: $$1 - \frac{1}{n^2}$$.