How to calculate a wheel's linear velocity if there is slipping I am working on a car simulation game and I need to know how to calculate a vehicle's velocity if the gas pedal was floored and the car started accelerating from rest with some slippage in the wheels. So, is the traction force completely gone, or it decreases? And how long will the wheels keep slipping till it rotates again? Given the angular velocity of wheels, mass of car, and both the static and kinetic friction coefficients.
 A: The traction force isn't completely gone; rather, the coefficient of friction, $\mu$, is that of kinetic rather than static (i.e. rolling) friction.  Typically, the kinetic coefficient of friction is less than the static one.
$$\mu_k < \mu_s$$
The engine is delivering some power $P$ to the wheels.  Rotational power is equal to the torque about the wheel's axle axis, $\tau$, multiplied by the angular velocity, $\omega$.
$$P_r = \tau\omega$$
Torque is equal to the mass moment of inertia multiplied by the angular acceleration.
$$\tau = I\alpha$$
$$P_r = I\alpha\omega$$
As angular velocity increases, assuming constant power, the angular acceleration decreases.  That is, the torque decreases.  Torque is also equal to the cross product of the radius from the axis of rotation to the force being applied.  Here, this is the friction force, $F_f$.
$$\tau = r\times F_f$$
For this setup, we can reasonably assume the radius and friction force are perpendicular, so the product is just $\tau = rF_f$.  Depending on your material, there will be some force that the static (rolling) friction, $F_{f,s}$, can hold based on its coefficient of friction.
$$F_{f,s} = \mu_s N$$
Here, $N$ is the normal force, or the load applied to the wheel in question. 
 Above this force value $F_{f,s}$, there will be slipping.  When there is slipping, we will have to use the following equation:
$$F_{f,k} = \mu_k N$$
Hopefully this helps.
