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Dynamics_Part42

# Dynamics_Part42 - The initial velocity of the cue ball is V...

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The initial velocity of the cue ball is V = V j = V cos α n + V sin α t Tangential-Velocity Invariance. The velocity of the balls is unchanged in the t direction. Hence, for the cue ball, we have V I t = V sin α Normal-Momentum Conservation. Letting V I 8 denote the speed of the eight ball in the n direction after the impact, there follows mV cos α + m · 0 = mV I n + mV I 8 = V I n + V I 8 = V cos α Impact Relation. The final principle we use is the impact relation, viz., V I 8 V I n = e ( V cos α 0) = V I 8 V I n = eV cos α Completing the Solution. Subtracting the impact-relation equation from the n -momentum equation yields V I n = 1 2 (1 e ) V cos α Therefore, the velocity of the cue ball after the impact is V I = 1 2 (1 e ) V cos α n + V sin α t To transform to xy coordinates, we substitute for n and t from above and proceed as follows. V I = 1 2 (1 e ) V cos α ( sin α i + cos α j ) + V sin α (cos α i + sin α j ) = } 1 2 (1 e ) V sin α cos α + V sin α cos α ] i + } 1 2 (1 e ) V cos 2 α + V sin 2 α ] j = } 1 2 (1 + e ) V sin α cos α ] i + } 1 2 (1 e ) V cos 2 α + V sin 2 α
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