How to calculate distance covered by the beam? (friction force problem)

Question

Question:
A uniform beam of weight W and length L is initially in position AB. As the cable is pulled over the pulley C, the beam first slides on the floor and is then raised, with its end A still sliding. If $\mu$ be the coefficient of friction between the beam and the floor, calculate the distance a that the beam will slide before it will begin to rise. Ans. $L(1-\mu)$

My try:
I drew the F.B.D. like this,

Now, as the beam slides, so,
Force for which the beam slides F = $P\sin 45^0-\mu N$
and from equation of equilibrium,
$$\Sigma F_y=0\\ \implies P\cos 45^0+N=W\\ \implies N=W-P\cos 45^0$$
$\begin{align}\\ \therefore &F = P\sin 45^0-\mu (W-P\cos 45^0)\\ &{\implies \begin{aligned}\\ \frac Wg\times a'=P\sin 45^0&-\mu (W-P\cos 45^0)\\ \end{aligned}\\}\\ &{\implies \begin{aligned}\\ a'&= \frac gW[P\sin 45^0-\mu (W-P\cos 45^0)]\\ \end{aligned}\\}\\ \end{align}\\$
But, I can't understand how I can relate the distance a with the acceleration a', as there is no final velocity given. Please anyone suggest any approach.