In a game of billiards, the two ball and three ball are moving with velocities v

Question

In a game of billiards, the two ball and three ball are moving with velocities v2 = v2i and v3 = v3j respectively. The balls impact the nine ball, which is initially at rest. The two ball is at rest after the impact, while the three and nine balls move with velocities v'3 and v'9, respectively. All three balls have identical mass and the impacts are perfectly elastic. Assume the surface is frictionless and the balls move without rolling. 1. Determine v'3 and v'9 as functions of v2, v3 and the angles phi and theta 2. Assuming that phi = 15 and theta = 30, compute the ratio v2/v3 phi is the angle above the positive x-axis at which the 9-ball moves after the impact theta is the angle below the x-axis at which the 3-ball moves after the impact.

Explanation / Answer

a)By conservation of Momentum we have Assuming velocities as v'3 = Ai+Bj and v'9 = Ci+Dj Then v2 = A+C and v3 = B+D and also conservation of energy we have v2^2 + v3^2 = A^2 + B^2 + C^2 + D^2 So AC = -BD Solve this eqns to get the relations b)now v'3 = v'3*cos30 i - v'3*sin30 j and v'9 = v'9*cos15 i + v'9*sin15 j v2 = v'3*cos30 + v'9*cos15 and v3 = - v'3*sin30 + v'9*sin15 So v2/v3 = (v'3*cos30 + v'9*cos15)/(v'9*sin15 - v'3*sin30)

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