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# W7.2 - Physics for Scientists Engineers 1 Spring Semester...

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February 18, 2008 Physics for Scientists&Engineers 1 1 Physics for Scientists & Physics for Scientists & Engineers 1 Engineers 1 Spring Semester 2008 Lecture 24

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February 18, 2008 Physics for Scientists&Engineers 1 2 Totally Elastic Collisions in 1d Totally Elastic Collisions in 1d Totally elastic collision : Total momentum is conserved (valid for all collisions) Total kinetic energy is conserved Momenta after collision p f ,1 2 2 m 1 + p f ,2 2 2 m 2 = p i ,1 2 2 m 1 + p i 2 2 m 2 p f ,1 + p f = p i ,1 + p i p f ,1 = m 1 ! m 2 m 1 + m 2 p i ,1 + 2 m 1 m 1 + m 2 p i p f = 2 m 2 m 1 + m 2 p i ,1 + m 2 ! m 1 m 1 + m 2 p i
February 18, 2008 Physics for Scientists&Engineers 1 3 Totally Elastic Collisions in 1d Totally Elastic Collisions in 1d Expression for final velocities from initial ones Relative velocity after collision, : v f ,1 ! v f ,2 v f ,1 ! v f = m 1 ! m 2 ! 2 m 1 m 1 + m 2 v i ,1 + 2 m 2 ! ( m 2 ! m 1 ) m 1 + m 2 v i = ! v i ,1 + v i = ! ( v i ,1 ! v i ) Relative velocity changes sign! v f ,1 = m 1 ! m 2 m 1 + m 2 v i ,1 + 2 m 2 m 1 + m 2 v i v f = 2 m 1 m 1 + m 2 v i ,1 + m 2 ! m 1 m 1 + m 2 v i p f ,1 = m 1 ! m 2 m 1 + m 2 p i ,1 + 2 m 1 m 1 + m 2 p i p f = 2 m 2 m 1 + m 2 p i ,1 + m 2 ! m 1 m 1 + m 2 p i compare!

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February 18, 2008 Physics for Scientists&Engineers 1 4 Special Case 1: Equal Masses, Special Case 1: Equal Masses, m 1 1 = = 2 In this case terms proportional to m 2 -m 1 vanish! 2m 2 /(m 1 +m 2 )=2 m 1 /(m 1 +m 2 )=1 The two objects of equal mass simply exchange their momenta! Same is true for the velocities (obvious, because masses are identical): p f ,1 = p i ,2 p f = p i ,1 v f ,1 = v i v f = v i ,1
February 18, 2008 Physics for Scientists&Engineers 1 5 Special Case 2: One object initially at rest Special Case 2: One object initially at rest Pick one of the two objects and set its initial momentum to 0 (does not matter which, say object 1) For this special case of p i, 1 = 0 we get from the general formulas: momenta velocities Now we can distinguish 3 cases, depending on the ratio of the two masses, m 2 = m 1 , m 2 > m 1 , m 2 < m 1 : In all 3 cases object 1 moves in the same direction that m2 had initially, but … p f ,1 = 2 m 1 m 1 + m 2 p i ,2 p f = m 2 ! m 1 m 1 + m 2 p i v f ,1 = 2 m 2 m 1 + m 2 v i v f = m 2 ! m 1 m 1 + m 2 v i

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February 18, 2008 Physics for Scientists&Engineers 1 6 Special Case 2: One object initially at rest Special Case 2: One object initially at rest p i, 1 = 0, m 2 = m 1 : After the collision object 2 is now at rest, and object 1 moves with the initial velocity of object 2 (already covered in previous special case of equal masses) p i, 1 = 0, m 2 > m 1 : Object 2 keeps moving in same direction as it did before collision, but with reduced speed. Speed of object 1 is bigger than initial speed of object 2.
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W7.2 - Physics for Scientists Engineers 1 Spring Semester...

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