I think this is a problem requiring calculus, but there may be a geometric solution. My skills in these areas are now too rusty.

Suppose you have a vessel (e.g. tug boat) travelling along the x-axis towing an object with a tow rope of length R. Lets set the tug at the origin and the object at (x=-R,y=0). The tug at (0,0) now makes an instantaneous turn to port and heads north east at an angle theta to the x-axis. It travels a distance D in a straight line to arrive at a point T= (D.Cos(theta),D.Sin(theta)). What path did the towed object travel and where is it now (i.e. what x,y)? We know that the towed object is always a distance R from the tug. It does not follow in the wake of the tug but instead follows a curve.

What is the equation that describes the path of the towed object (in terms of the start positions, D and theta)? This problem assumes that the towed object has no mass/momentum and always moves in the current direction of the tow rope.

What happens if the towed object starts not in the wake of the tug and therefore not on the x-axis, e.g. at a point (-R,Y), where Y can be positive or negative ?

Does anyone have an insight into this problem? Have I put this problem in the correct forum section?

Thanks