In a game of billiards, a player has decided to do a \"bank shot\" in order to \

Question

In a game of billiards, a player has decided to do a "bank shot" in order to "sink" the eight ball in the corner pocket. He imparts a velocity v = vi to the cue ball, which impacts the eight ball. After the impact, the eight ball moves at an angle alpha to the y axis. The eight ball then impacts the upper cushion. After impacting the cushion, the eight ball follows the path shown and goes into the corner pocket. The cue ball and the eight ball have equal mass, m, and coefficient of restitution, e. The impact between the eight ball and the cushion is perfectly elastic. Assume the surface is friction-less and the balls move without rolling. Determine the eight ball's speed, v_s", after reflection from the cushion in terms of v, alpha, and e. If the length l = 3/2h, what is alpha?

Explanation / Answer

let cue ball moves with v2 after collision making angle of Q with y axis and eight ball has velocity v1 after collision ; applying conservation of momentum ; v=v1sin(alpha)+v2sinQ ; e=[v1sin(alpha)-v2sinQ]/v ; ; v1=v*(e+1)/2sin(alpha) ; now ball collided with cushion elastically....since it is elastic collision,ball will return with same speed=v1 ; time taken to travel l-htan(alpha) distance should be equal to time taken to travel h distance vertically; v1cos(alpha)/h =v1sin(alpha)/(l-htan(alpha)); 2htan(alpha)=l , for l=3/2 h ,,alpha=36.86 degres ;; ANS=V_s= v1=v*(e+1)/2sin(alpha) ; for l=3/2 h ,alpha=36.86 degrees

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