Sliding friction, position/velocity as a function of time?

[mcevans]

If this question is completely stupid, forgive me, I'm a few years out of university and math and physics are quickly leaving my brain... :)

Say a particle has a non zero velocity in one direction and a non zero acceleration in the opposite direction. If you also account for some sliding friction force opposing this movement, is it still possible to calculate its position and velocity as functions of time?

I've been googling around for a while and can't seem to come up with what I want. I'm trying to write a little (extremely simplified) continuous physics simulator, and after some timestep dt I need to be able to say "new_x_position = velocity * dt + f(dt)" where f is some mystery function- right now the only things I can come up with are piecewise functions.

edit: I should clarify- the source of my confusion is that sliding friction can never exceed whatever applied force is causing the particle's acceleration, and must always oppose movement, even if the particle's direction changes during that timestep

 

[mcevans]

I guess this question could be summarised as, "Is kinetic friction inherently piecewise?"