Question

Problem 3 (30 pts) Billiard balls must weigh between 5.5 and 6 oz, and they must be 2.25 +0.005 in. in diameter. Treat the balls as particles and determine what conditions need to exist to have a moving ball A hit a stationary ball B so that A stops right after the impact. Assume A and B have the same diameter but not the same weight. Assume that the COR e = 1. Finally, given the ball specifications, is this condition realistic?

Answer 1

Solution:

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Given that,

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If the pr-impact velocity of A had a non-zero component velocity perpendicular to the line of impact (LOI), then this component of velocity would be conserved through the impact. This consideration implies that for A to stop after impact at the very least, its pre impact velocity must be entirely parallel to the LOI.

Write the pre impact velocities of balls A and B: Ū;=u And Ug = Apply the conservation of linear momentum in the direction of movement, mu+m,08 = m, 0.1, +m,ut Substitute the known values, mu; = m,+mu (1) Write the expressions for coefficient of restitution, 01-01 vi-D Substitute o for v e = De=03-0 U = ev, +01 (2)

Substitute the expression for u, in equation (1) mu = mv + mv mun = m,+m(ev; +01) (m - em, 0 =0 (m + m) (m, -emg ) (m, +mg) To check whether or not it is possible for A to stop, set 0 , =0 above expression, we get (m,-em, uz = 0 (m+mo) m = mye For the COR, e=1 m = mx Hence, the mass of balls A and B are equal in mass. Therefore we can say that it is not possible to have a moving ball A hit a stationary ball B so that A stops right after the impact.