Dynamics of a Block on top of a slab with friction between them

[haruspex]

Now can I say that $W_{\text{friction}}= \Delta l \mu_d g m_b$?

Yes.

I'm not sure because the work is done on 2 surfaces that are moving.

Depending on the frame of reference you choose, you could consider that the block moves $x$ one way and the slab moves $\Delta l-x$ the other way. So the block does work $\mu_kxm_bg$ against friction while the slab does work $\mu_k(\Delta l-x)m_bg$ against friction. The total work done against friction is always $\mu_k\Delta lm_bg$.

But what would happen when the block falls from the slab.

You understand that you do not need to consider that, right?

Is our system still valid or as soon as they detach, they can no longer be considered as a system?

A system is whatever you define it to be.
Certainly it will behave differently after detachment.

 

[Steve4Physics]

But what would happen when the block falls from the slab. Is our system still valid or as soon as they detach, they can no longer be considered as a system?

The question does not ask about what happens when the block falls from the slab. It wants the velocities at the moment before this happens.

 

[Thermofox]

You understand that you do not need to consider that, right?

Yeah, I perfectly do. I was just a thought I had, nothing more.

Depending on the frame of reference you choose, you could consider that the block moves $x$ one way and the slab moves $\Delta l-x$ the other way. So the block does work $\mu_kxm_bg$ against friction while the slab does work $\mu_k(\Delta l-x)m_bg$ against friction. The total work done against friction is always $\mu_k\Delta lm_bg$.

Ok now I understand it, Thank you immensely!

 

[Thermofox]

The question does not ask about what happens when the block falls from the slab. It wants the velocities at the moment before this happens.

Yeah I do realize that, It was just an "if case" I had.