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?"

 

[astrokreng]

Hi there,

No need to apologize, it's always good to ask questions and refresh your knowledge!

To answer your question, yes it is still possible to calculate the position and velocity of a particle as functions of time even when sliding friction is involved. However, the calculations will be more complex and will require taking into account the forces acting on the particle.

First, let's define some variables:
- x(t) represents the position of the particle at time t
- v(t) represents the velocity of the particle at time t
- a(t) represents the acceleration of the particle at time t
- F(t) represents the net force acting on the particle at time t
- μ represents the coefficient of sliding friction between the particle and the surface it is sliding on

Now, using Newton's second law, we can write the following equation:
F(t) = ma(t)

We also know that the friction force (Ff) is given by the following equation:
Ff = μN
where N is the normal force acting on the particle (perpendicular to the surface).

So, the net force acting on the particle can be written as:
F(t) = ma(t) - μN

To calculate the position and velocity of the particle as functions of time, we can use the following equations:
x(t) = x0 + v0t + (1/2)at^2
v(t) = v0 + at
where x0 and v0 are the initial position and velocity of the particle at time t=0.

Now, to incorporate the sliding friction force, we need to modify the acceleration term in these equations. So, we can write:
x(t) = x0 + v0t + (1/2)(a(t) - μN/m)t^2
v(t) = v0 + (a(t) - μN/m)t

As you can see, the friction force will affect the acceleration of the particle, and therefore, its position and velocity as functions of time. To calculate the normal force (N), you will need to consider the forces acting on the particle in the direction perpendicular to the surface it is sliding on.

I hope this helps and clarifies any confusion you had. Good luck with your physics simulator!