A block is pushed up a rough inclined plane of 45°. If the time of descent is twice the time of ascent, the coefficient of friction is

Suppose the block is projected up the inclined with velocity u.

deceleration: ${\mathrm{a}}_1=\mathrm{g}(\sin \theta +{\mu }_{\mathrm{k}}\cdot \cos \theta )$
$\Rightarrow 0=u^2-2a_{1}s \Rightarrow s=\frac{u^2}{2a_1}$ 

Also, $0=\mathrm{u}-a_1\cdot t_a\Rightarrow t_a=\frac{u}{a_1}$. 

During descent, acceleration of the block,

 $a_2=g(\sin \theta -{\mu }_k\cos \theta )$
$\Rightarrow s=0\times t_d+\frac{1}{2}\times a_2\times {t_d}^2$

$\Rightarrow t_d=\sqrt{\frac{2s}{a_2}}=\frac{u}{\sqrt{a_1{a}_2}}$
$\Rightarrow \frac{t_d}{t_a}=\sqrt{\frac{a_1}{a_2}}=2$

$\Rightarrow \frac{\sin \theta +{\mu }_k\cos \theta }{\sin \theta -{\mu }_k\cos \theta }=4$
$\Rightarrow {\mu }_k=\frac{3}{5}\tan \theta =\frac{3}{5}\tan {45}^0=0.6$