# 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.

## 1 Answer

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.

• If static friction helps the tyre remain constant relative to the ground, so it rolls forward, why do we say kinetic friction make the wheel roll as well? And when does the wheel continue rolling again after slipping? – user3407319 May 4 '17 at 3:17
• Also, do we say kinetic friction helps in the car movement like the static does, or it opposses it? I mean, does the kinetic friction act as tractive force itself and help the car in moving or it acts as an opposite force to the car's movement? And can i use this formula here: force moving = force applied by engine - force kinetic friction? – user3407319 May 4 '17 at 13:25
• Kinetic friction gives the propulsive power while the wheels are spinning relative to the ground. There is no other force propelling the car besides the force at this contact point. This can help the car move it if we slam on the accelerator, or it can oppose it if we slam on the brakes and start skidding. It depends on the relative motion. For the case of accelerating, at some point, based on the equations above, the wheels will stop spinning relative to the ground and they can be considered rolling. – gdbb89 May 6 '17 at 13:33
• That is correct. Both are friction forces, but their magnitude differs depending on the coefficient of friction. Check out the Wikipedia article: en.wikipedia.org/wiki/Friction – gdbb89 May 6 '17 at 13:38
• Use the above equations to determine when the force induced by the engine -> axle -> wheel on pavement drops below $\mu_k N$. – gdbb89 May 6 '17 at 13:43