Circular Billiards Table (Posted on 2013-04-23)

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?

	Submitted by Bractals
Rating: 3.0000 (1 votes)
Solution:	(Hide)
Let B and C be the first and second bounce points. The path will be ΔABC. We wish to find ∠OAB. Let p = |OA|. We will assume that the angle of incidence equals the angle of reflection at a bounce. Therefore, ∠ABO = ∠OBC and ∠BCO = ∠OCA. Since ΔBOC is isosceles, ∠OBC = ∠BCO. Let φ = ∠ABO = ∠OBC = ∠BCO = ∠OCA. Using the law of sines on triangles OAB and OAC, r p p ----------- = ------------ = --------- sin(∠OAB) sin(∠OBA) sin(φ) p r = ----------- = ------------ sin(∠OCA) sin(∠OAC) Let θ = ∠OAB = ∠OAC. Summing the interior angles of ΔABC gives θ = 90° - 2φ Applying the law of sines again to ΔOAB gives r p -------- = ------- sin(θ) sin(φ) or r*sin(φ) = p*sin(θ) = p*sin(90° - 2φ) = p*cos(2φ) = p*[1 - 2*sin2(φ)] or 2*p*sin2(φ) + r*sin(&phi) - p = 0. Therefore, sin(φ) = [√(r2 + 8*p2)-r]/(4*p). Thus, ∠OAB = 90° - 2*arcsin{ [√(r2 + 8*p2)-r]/(4*p) }. QED Note: If p=r, then ΔABC is equilateral;

	Subject	Author	Date
	re(2): Playing with Geometers' Sketchpad -- a method	Jer	2013-04-24 12:49:34
	re: Playing with Geometers' Sketchpad -- a method	Charlie	2013-04-24 00:40:10
	Playing with Geometers' Sketchpad	Charlie	2013-04-23 23:21:21
	re(2): Obvious??	brianjn	2013-04-23 23:04:34
	re: Obvious??	Kenny M	2013-04-23 22:36:36
	Obvious??	brianjn	2013-04-23 21:17:58