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p09fl2 - Lecture 2 Two-body problem(4 Sep 09 A Relative...

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Lecture 2: Two-body problem (4 Sep 09) A. Relative motion of two bodies 1. Use notation R ~ R [bold face for vectors] so that dot for time deriva- tive is ˙ R . 2. Formulate Newton’s laws using two bodies with internal forces – second and third laws: d p 1 /dt = F 21 ; F 21 = - F 12 and central forces F 21 = [ - d Φ /dr ]( r 1 - r 2 ) /r 12 = [ - d Φ /dr r 12 . 3. Center-of-mass and relative coordinates defined for masses m 1 and m 2 , M = m 1 + m 2 : (L & L, Sec. 13) R = m 1 r 1 + m 2 r 2 M ; r = r 1 - r 2 with inverse relations r 1 = R + m 2 M r ; r 2 = R - m 1 M r 4. The corresponding relations for center-of-mass momentum and relative momentum are (anticipating quantum mechanics) P = p 1 + p 2 ; p = m 2 p 1 - m 1 p 2 M p 1 = m 1 M P + p ; p 2 = m 2 M P - p 5. For the mechanical momenta p j = m j ˙ r j , the relative momentum is, with reduced mass μ : p = μ ˙ r ; μ = m 1 m 2 m 1 + m 2 ; 1 μ = 1 m 1 + 1 m 2 6. Conservation of total (center-of-mass) momentum (only internal forces) d dt P = F 21 + F 12 = 0 1
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using Newton #3 explicit solution for R ( t ). 0 = d dt P = M d 2 dt 2 R R = R 0 + V t where V is the constant center-of-mass velocity.
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