19.1 A cue ball hits an eight ball dead center. Assume both have the same mass a

Question

19.1 A cue ball hits an eight ball dead center. Assume both have the same mass and all motion is along a straight line. The eight ball is initially at rest. In each of the following cases, (1) use conservation of momentum to determine the final velocity of each ball and (2) calculate the ratio of final to initial total kinetic energy. (3) Classify each case as elastic, inelastic or super-elastic. Assume the initial speed of the cue ball A) After the collision, the two balls move with identical final speeds in the same direction as the cue ball's is Vo initial velocity. B) After the collision, the cue ball reverses direction but maintains the same speed.

Explanation / Answer

Initial velocity of cue ball = Vo
Initial momentum of cue ball + eight ball = MVo + 0 = MVo (in the direction of velocity cue ball)
Initial kinetic energy = MVo2/2

A) Velocity of both balls is same. Let it be V.
Momentum of cue ball + eight ball after collision = 2M V
By momentum conservation 2MV = MVo
V = Vo/2 = velocity of cue as well as eight ball
Final kinetic energy = (2M)(Vo/2)2 / 2= MVo2/4
ratio of final to initial KE = 1/2
ratio is less than 1 , hence it is inelastic collision

B) Final velocity of cue ball = -Vo
Let final velocity of eight ball = V
momentum afetr collision = MV - MVo
By momentum conservation MVo = MV - Mvo
V = 2Vo
Final Kinetic energy = MVo2/2 + M(2Vo)2/2 = 5MVo2/2
Ratio = 5:1
Increase in KE, means collision is superelastic

C) Final velcoity of cue ball = 0
Let final velocity of eight ball = V
so final momnetum = MV
Hence MV = Mvo   or V = Vo
Final KE = 0 + MVo2/2
ratio of KE = 1
collision is elastic

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