M 1 v 2 1 i 1 2 m 2 v 2 2 i 1 2 m 1 v 2 1 f 1 2 m 2 v

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m 1 v 2 1 ,i + 1 2 m 2 v 2 2 ,i = 1 2 m 1 v 2 1 ,f + 1 2 m 2 v 2 2 ,f Trick: rewrite equations as m 1 ( v 2 1 ,i - v 2 1 ,f ) = m 2 ( v 2 2 ,f - v 2 2 ,i ) We have: m 1 ( v 1 ,i - v 1 ,f ) = m 2 ( v 2 ,f - v 2 ,i ) m 1 ( v 1 ,i - v 1 ,f )( v 1 ,i + v 1 ,f ) = m 2 ( v 2 ,f - v 2 ,i )( v 2 ,f + v 2 ,i ) or: 1st in 2nd: v 1 ,i + v 1 ,f = v 2 ,f + v 2 ,i v 2 ,i - v 1 ,i = - ( v 2 ,f - v 1 ,f ) This is easy to remember as: the relative velocity of 1 and 2 gets reversed! We can use this equation, together with P f =P i : now the equations are linear!
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Phys 2A - Mechanics m 1 v 1 ,i + m 2 v 2 ,i = m 1 v 1 ,f + m 2 v 2 ,f We have: v 2 ,i - v 1 ,i = - ( v 2 ,f - v 1 ,f ) Solve: Some interesting special cases cum examples: 1. v 2,i =0 v 2 ,f = 2 m 1 v 1 ,i + ( m 2 - m 1 ) v 2 ,i m 1 + m 2 v 1 ,f = ( m 1 - m 2 ) v 1 ,i + 2 m 2 v 2 ,i m 1 + m 2 v 1 ,f = m 1 - m 2 m 1 + m 2 v 1 ,i v 2 ,f = 2 m 1 m 1 + m 2 v 1 ,i 2. m 1 =m 2 v 1 ,f = v 2 ,i v 2 ,f = v 1 ,i (good old switcheroo) DEMO 3. m 1 <<m 2 v 1 ,f = 2 v 2 ,i - v 1 ,i v 2 ,f = v 2 ,i
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Phys 2A - Mechanics Alternate way of understanding m 2 >>m 1 : the “brick wall” collision Bouncing off a stationary brick wall: v 1 ,f = - v 1 ,i The primes for “in frame of reference where brick wall is stationary.” Now, go to frame where brick wall moves with velocity v 2 ( v 1 ,f - v 2 ) = - ( v 1 ,i - v 2 ) v 1 ,f = 2 v 2 ,i - v 1 ,i or as before! v 1 ,i = v 1 ,i - v 2 Relative motion: v 1 ,f = v 1 ,f - v 2 and Hence
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Phys 2A - Mechanics Demo: For tennis ball, use v 1 ,f = 2 v 2 ,i - v 1 ,i y Here basketball has just bounced of floor: Drop balls from height h. Speed at ground v 0 = 2 gh v 2 ,i = v 0 and just before collision with basket ball, v 1 ,i = - v 0 v 1 ,f = 3 v 0 So and max-height after collision 3 v 0 = 2 gh f h f = 3 2 h
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Phys 2A - Mechanics Work
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Phys 2A - Mechanics 1-dimension Particle (or body in particle approximation) moves along trajectory x=x(t) from x i = x(t i ) to x f = x(t f ).
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