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
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 / (1-2(sinθ)²) = D
sinθ = D - 2D(sinθ)²
2D(sinθ)² + sinθ - D = 0 which is quadratic in sinθ
sinθ = (-1 ±√(1-4*2D*-D))/(2*2D) = (-1±√(1+8D²))/(4D)
θ = sin^-1 [(-1 ±√(1-4*2D*-D))/(2*2D)]
90 - 2θ = 90 - 2sin^-1 [(-1 ±√(1-4*2D*-D))/(2*2D)]
which is consistent with your table.
Posted by Jer
on 2013-04-24 12:49:34