homework and exercises - Plank stopping time

Problem: A plank of length $L$ and mass $m$ lies on a frictionless floor, if the plank has an initial velocity $v$, what is the stopping time of the block if it meets a floor with friction constant $\mu$.

My attempt:

I took a little piece of block with mass $\Delta m$, we know that $\Delta m = m\frac{\Delta \ell}{L}$.

Now if I apply the Newton laws in this piece, I get the following equation:

$$N_{piece}= \Delta mg $$

Thus my friction is $f_r=\mu\Delta mg$

If I apply Newton laws to the whole plank, I get:

$$-f_r=ma$$ $$\mu\Delta mg=ma$$ $$\mu mg\frac{\Delta \ell}{L}=ma$$

$$a=\mu g\frac{\Delta \ell}{L}$$

I have this equation, but I don't know how to relate it with time.