I can't imagine a single dimension racing game could be very interesting. Anyway, I don't know anything about cars but you can make it as complicated as you'd like.
Basically you need a function that reaches a peak since cars will not indefinitely accelerate.
Some examples would be
sin(x), x = 0 to pi
ax^2 + bx = 0 for a < 0 and b > 0
x * e^(-(x - n)), x = 0 to n
Bessel Function of the first kind for n = 1 and x = 0 to ~3.85
You can be as complicated as you'd like with the function. Then to find the velocity and position at any given point take the anti-derivative with respect to time, once to find the velocity, and again to find the position. You don't necessarily need to do that last part you can model it anyway you like through trial and error which will likely need to happen anyway. If you'd like to go this route I'd suggest picking an easy function like the quadratic equation, the second example, as it's easy to work with and you'll find enough information on the interwebs to get a generic answer for both without having to do the calculus if you've not taken an introductory course in it.
Edit: I've thought about this a bit more and I think the easiest thing to do would be to use some type of piecewise function. This will give you a bit more control as to what's going on.
x*e^(x), x <= max acceleration
then when x = max acceleration
x, x = max to 0.
Since this will update rather frequently, however you have that set up, then the result can just be added constantly to a velocity variable and the position updated accordingly. No need for calculus. And again this will give you a lot more control as you can more easily decide how quickly to accelerate and how quickly it will stop accelerating.