Given the system in the figure:
The block slides on a horizontal surface and is acted upon by a force $P$ which varies in magnitude as shown. Knowing that the coefficients of friction between the block and the surface are $\mu_s = 0.6$ and $\mu_k = 0.25$ and that the block is initially at rest, I have to determine the velocity of the block at $t = 5 s$.
I'm using the principle of linear impulse and momentum to solve the problem. $$mv_1 + \sum \int fdt= mv_2$$ What I want to know is how to split the frictional force. Should I use $f_r=\mu_sN$ once $P$ drops below $\mu_sN$ or once it drops below $\mu_kN$. Also, what happens once $P$ is between $\mu_sN$ and $\mu_kN$, will the object remain in motion or will it stop?
What I mean is summarized in this equation: $$mv_1 + \int P dt -\mu_kN \times t_1 -\mu_sN \times (5-t_1) = mv_2$$ where $t_1$ is the time at which $P$ drops below $\mu_kN$.