In this problem, I need to find the trajectory of a particle (as a function of time) which moves at a speed 's' but also turns at an increasing rate; angular acceleration α. The trajectory looks like a spiral which converges to a point. The particle has an initial position vector p and a velocity vector v. So without an angular velocity or 'turning effect' the particle should simply trace out the path "p(t) = ∫ v dt", however the particle has an angular acceleration which means that the velocity vector is rotated at the rate α, therefore vnew = R(Δθ) vold (R(θ) is the rotation matrix to rotate a 2d vector θ degrees). I'm looking at this problem in a 2d sense. An example of this would be a car that is travelling a constant speed but the driver starts turning the wheel at a constant rate so that the car turns sharper and sharper. I've tried tackling this problem by expressing the velocity vector and angular velocity as a complex numbers and integrating their product, as well as trying a method using the rotation matrix, but on both occasions I'm stuck with 'dt's' inside trigonometric functions while integrating.