You hold a 3 kg metal block against a wall by applying a horizontal force of 40 N, as shown in the figure in representations. The coefficient of friction for the metal-wall pair of materials is 0.6 for both static friction and sliding friction. Does the block slip down the wall?

You need to identify whether the momentum in the y direction is negative (if it is, that would mean it was slipping down the wall).

Start by computing the change in momentum for both the x direction and the y direction.

$ x: \Delta p_x = (F_{head} - F_N) \Delta t = 0 $

$ y: \Delta p_y = (F_N - mg) \Delta t, \,\,assuming\, it\, slides $

Combining these two equations, we have

$ \Delta p_y = (F_{head} - mg) \Delta t = (0.6(40N) - (3 kg)(9.8 N/kg)) \Delta t $

$ \Delta p_y = (-5.4 N) \Delta t $

Since there is a nonzero impulse in the -y direction, the block will slip downward with increasing speed.