Given a circular billiards table with center O,
radius r, and no pockets. A cue ball (with a
radius negligible compared to r) is placed at
point A such that 0<OA≤r. The ball is struck
and bounces off the cushion twice before
returning to point A.
How was the direction of the shot determined?
(In reply to
re: Playing with Geometers' Sketchpad  a method by Charlie)
I'd rater a construction of a solution but this is the best I can do so far.
I solved it in a similar way myself but I don't have the notebook with me so I'll derive the solution from what you have.
Just finishing what you started.
You want sinθ/ cos(2θ) = OA which I will call D.
sinθ / (12(sinθ)²) = D
sinθ = D  2D(sinθ)²
2D(sinθ)² + sinθ  D = 0 which is quadratic in sinθ
sinθ = (1 ±√(14*2D*D))/(2*2D) = (1±√(1+8D²))/(4D)
θ = sin^1 [(1 ±√(14*2D*D))/(2*2D)]
90  2θ = 90  2sin^1 [(1 ±√(14*2D*D))/(2*2D)]
which is consistent with your table.

Posted by Jer
on 20130424 12:49:34 