Hi everyone. I am hoping someone could help me work out the rate of change of angle of an object.

Basically, I have the arrow shown above moving at a velocity. But once it enters the 'field' of the circle shown above I want it to rotate towards the circle.

The part I find difficult is making sure that once it gains contact with the circle, the angle of the arrow is precisely equal to that of the tangent of the point of contact. (Hopefully that made sense)

I'm not sure whether the rate of change of angle would be constant or accelerating(if so at what rate?). Anyway, I'm hoping it would result in a motion that is shown by the red line.

Any help is appreciated!