Circular motion and centrifugal force

Assuming a race car drives around in a circle of radius r, center (0,0), linear velocity v, and ignoring centrifugal forces and friction I can calculate the position at any time, t. Angular velocity

$$\omega = \frac{v}{r}$$

degrees turned through (assume at time t = 0 car is at position (r,0)):

$$\theta = \omega t$$

So the positional (x,y) coordinates, p, at time t, are

$$p(t) =(r\cos(\theta) , r\sin(\theta ))$$

Please correct me if I am wrong up to this point. Now I want to include slipping, so that the car will be thrust outwards by the centrifugal (?) force, with friction force countering it. Supposing the centrifugal force is stronger, then there will be a resultant force in the direction tangential to the linear velocity at time t.

Suppose this force is $F$, then I want to find the modified position of the car at time t, as it won't be travelling in a circle but, I think, will be travelling in an increasing spiral (this is theoretical, I haven't taken into account spinning around its axis or crashing into a barrier).

So this is where I am going to use very naive maths/physics to try and find the modified position.

Assume the car has mass $m$.

Then the acceleration outward from the centre of the circle will be

$$a = \frac{F}{m}$$

Now I'm going to assume acceleration is constant to make my life simpler, and I can fudge a value for outward velocity u,

$$u = \int_{0}^{t} a dt = at$$

and more dodgy physics, I can calculate the position vector by multiplying by t, assuming constant velocity outwards,

$$\text{position} = (u t \cos(\theta),u t \sin(\theta))$$

Remembering at time t, the angle is $\theta$

I can see that the new position, let's say $q(t)$, is given by

$$q(t) = p(t)+(u t\cos(\theta), u t\sin(\theta)) =p(t)+ (\frac{Ft}{m} t\cos(\theta),\frac{Ft}{m}t\sin(\theta))$$

One problem I have noted, just looking at this is that $q(t) \tilde{}t^{2}$ since to get the position from acceleration I multiply by t twice, which seems too much, the car will spiral too fast.

I am very skeptical this is correct. Please correct me where I have gone wrong.

Edit:

Also I just pulled the value $F$ from thin air. Could you explain how I could calculate it based on the cars velocity, mass, radius of circle, coefficient of friction of surface or any other variables?