To me this a very interesting problem, and have no knowledge whatsoever from experience on similar problems, so likely of no use. Also I see no picture nor even understand what you have posted.
My limited undertanding:
first, the ball B will hit the center target. Thrower is not in motion nor is the center of the MGR. Nor is the ang velocity of the MGR important, or our A child, for that matter except insofar as we have a total ang velocity. So the perspective from child a is really a superposition of his motion, the MGR's and the balls. The first two add in simple way.
so if we could generate a superposition of spatial coordinates as a function of time for both ball and child, then we could accurately describe motion as seen by the child?
Then if childs place in space C(x,y)= R*(cos(wt) + sin(wt)) and the ball along the x coordinate as 4.5-10*t, then there should be a way to portray the combined motions. But this is where I get stuck, not knowing initial x,y of child how to proceed? Obviously at 12o'clock looks much different to observer than at 6. But in general,
Vx=d(C(x,y))/dt+10,
Vy=d(C(x,y)/dt (these are partials obviously) so knowing both x' and x", y' and y" we can map out perceived trajectory. I very much doubt this cumbersome notation is what is asked for, just wanting to understand, and I still don't get it, for instance
If I were to translate these to words, I would think along the x-axis the ball either accelerates briefly, then still (depending on R and w vs 10) and then slows or vice versa, while on the Y depending on which starting point I chose, say 12noon, the overall motion would be like that of a heat seeking missile and assuming i could make the 1/4 turn before getting it between the eyeballs,would then veer to the left and slow down. There must be a formal way of mixing these frames of reference? Help.