# Lecture Notes 10/6

## Physics 114 with Ammar at University of Kansas *

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Lecture Notes October 6th Elastic Collisions/CM Bold letters indicate vectors Inelastic Collisions Kinitial ≠ KFinal Pinitial= Pfinal Elastic Collisions Kinitial = KFinal Pinitial= Pfinal If masses are equal, when m1 hits m2; m1 will stop and m2 will continue with its (m1’s) velocity. If masses aren’t equal, the object at rest will be set in motion with some v and the other will recoil (ball thrown against a wall) Elastic Collisions in 1 dimension Before After Va Vb V’a V’b ma mb ma mb Ki = KF => ½mava2 + ½mbvb2 = ½mav’a2 + ½mbv’b2 Pinitial= Pfinal => mava + mbvb = mav’a + mbv’b When given: ma, Va, mb, Vb - you are able to find V’a and V’b You find the following results : (Va - Vb) = -(Va’ - Vb’) The relative velocity of A & B with which they approach each other equals the relative velocity with which they recede from each other. Example – 2 equal Masses ma = mb = m Vb = 0 Equation 1 = Ki = Kf Equation 2 = Pi = Pf Equation 3 = (Va - Vb) = -(Va’ - Vb’) mava + mbvb = mav’a + mbv’b va + 0 = v’a + v’b Equation 4 = va = v’a + v’b Equation 5 = Va = -V’a + V’b Eq. 4 - Eq. 5 => 0 = 2V’a + 0 => Equation 6 = V’a = 0 (projectile stops) Plug 6 into 4 to get: Equation 7 = Va = Vb (target takes on velocity of projectile) Center of Mass Xcm = (m1x1 + m2x2 + m2x2) / m1 + m2 + m3 Ycm = (m1y1 + m2y2 + m3y3) / m1 + m2 + m3 General CM = Σmix2 ÷ Σmi - Center of mass for objects with a high degree of symmetry can usually be guessed. (sphere) - If no external forces act, the center of mass cannot move => CM is fixed. Example A child pulls a sled towards him (on frictionless ice) until they meet. How far does the child move? mchild = 30 kg msled = 5 kg d = 10 m Xcm = 30(0) + 5(10) / 35× Xcm = 50/35 Xcm = 1.43 m 10 m = child = sled Example - Projectile Shoot off a rocket with mass m that explodes at the height of its flight. It is separated into 2 equal parts. CM d d d Example - Raft As you walk from one end to the other, the boat must move to the left in order to keep its CM in the same location.

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