It's a little difficult to figure out what you're trying to do here, so let me start out by rephrasing the question as I interpreted it.
An object has a known initial position and is initially facing a specified direction. It can only move in the direction it is currently facing, but this direction is variable; the object rotates. The rotation is described by an angular velocity which is a linear function of time. The magnitude of the velocity is also a linear function of time. You want to know where the object is after it has flown about for a set period, but you are unable to simulate its motion in timesteps small enough to simply keep track of where it is, how fast it is moving and in what direction at all times.
Since both absolute as angular velocity are linear functions of time, the differential equations describing this motion can be solved analytically, but their solutions are ugly nonetheless, involving a bunch of Fresnel functions:
Where (x(t), y(t)) is the object's position as a function of time, b is the angular acceleration, v is the velocity (in x- and y-components), a is the acceleration, also in components, ω is the angular velocity, φ is the angle of rotation and C(x) and S(x) are the Fresnel integrals.
As the result is probably equally disturbing as unhelpful all by itself, I have omitted calculations that lead to it. Unfortunately, no simple approximations are available either. I fear discrete simulation is the only reasonable way of tackling this problem.