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 => ˝mava2 + ˝mbvb2 = ˝mav?a2 + ˝mbv?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 = ?mix2 ÷ ?mi - 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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