Physics 203 – College Physics I Department of Physics – The Citadel Physics 203 College Physics I Fall 2012 S. A. Yost Chapter 7 Part 2 Momentum and Collisions

Physics 203 – College Physics I Department of Physics – The Citadel Announcements Read Ch. 7, but you can skip sections 7.7 and 7.9. A problem set HW07A was due today. A problem set HW07B on sections 4 – 6 (collisions) is due Thursday. Next class: Rotational Motion, Ch. 10. Read sec. 1 – 5 for next time. Homework set HW08 on sections 1 – 6 will be due next Thursday. Sections 7 – 8 will be combined with chapter 9 in a later set. Skip section 9.

Physics 203 – College Physics I Department of Physics – The Citadel Exam 2 The average was 36%, high 67.5%.

Physics 203 – College Physics I Department of Physics – The Citadel Momentum and Impulse Momentum is defined to be p = m v The change in momentum is the impulse.  p = F avg t = J This is a consequence of Newton’s 2 nd Law. In an isolated system of objects, the total momentum of the objects is always conserved. This is a consequence of Newton’s 3 rd Law. → → → → →

Physics 203 – College Physics I Department of Physics – The Citadel 1D Collisions In a one dimensional collision, two masses m 1 and m 2 approach with velocities v 1 and v 2, and then collide. Their velocities v 1 ’ and v 2 ’ are measured after the collision. m 1 v 1 + m 2 v 2 = m 1 v 1 ’ + m 2 v 2 ’  p 1 +  p 2 = 0 m1m1 m2m2

Physics 203 – College Physics I Department of Physics – The Citadel Elasticity of a Collision Not all collisions are equally “bouncy”. This is quantified by elasticity. In a perfectly elastic collision, the relative speeds of the colliding objects is the same, though the direction may be different.

Physics 203 – College Physics I Department of Physics – The Citadel Inelastic Collision One possibility is that the red ball can come in with velocity v, stick to the blue ball, and both can travel off with final velocity v’. mm vv’

Physics 203 – College Physics I Department of Physics – The Citadel Inelastic Collision m1m1 m2m2 v’ This case is called an inelastic collision. In this case, the relative velocity of the masses is zero after the collision, since the balls stick together.

Physics 203 – College Physics I Department of Physics – The Citadel Example For example, a railroad car of mass M 1 = 1200 kg could travel at v = 2.5 m/s and strike a second railroad car of mass M 2 = 2800 kg. If the cars collide and hitch together, how fast do they move together? 1200 kg 2.5 m/s 2800 kg

Physics 203 – College Physics I Department of Physics – The Citadel Example Initial momentum: p i = M 1 v = 1200 kg x 2.5 m/s = 3000 kg m/s. Final momentum: p f = (M 1 + M 2 ) v’ v’ = 3000 kg m/s / 4000 kg = 0.75 m/s kg 2.5 m/s 2800 kg

Physics 203 – College Physics I Department of Physics – The Citadel Energy in a Collision How much energy is converted to internal energy (vibrations, heat, etc.) in this collision? E i = ½ M 1 v 2 = ½ (1200 kg)(2.5 m/s) 2 = 3750 J. E f = ½ (M 1 +M 2 )v’ 2 = ½ (4000 kg)(0.75 m/s) 2 = 1125 J. E i – E f = 2625 J.

Physics 203 – College Physics I Department of Physics – The Citadel Question Which of the inelastic collision(s) shown bring(s) the car at left to a halt? Choose D if all of them, E if none of them. ABCABC

Physics 203 – College Physics I Department of Physics – The Citadel Question In which collision does the car on the left experience the greatest force? Choose D if they are all equal, E if more information is needed. ABCABC

Physics 203 – College Physics I Department of Physics – The Citadel Question In collision C, which car experiences the greater force? A = left, B = right, C = equal. ABCABC

Physics 203 – College Physics I Department of Physics – The Citadel Question Which of the labeled cars has the greatest change in energy, comparing all of the cases together? Pick A, B, C, D, or E. (One of these is correct.) ABDABD C E

Physics 203 – College Physics I Department of Physics – The Citadel Elastic Collision An elastic collision conserves energy. In one dimension, a collision is elastic if the relative speeds are the same, v 2 ’ – v 1 ’ = v 1 – v 2. m1m1 m2m2 v1’v1’ v2’v2’ m1m1 m2m2 v1v1 v2v2 but notice the sign reversal

Physics 203 – College Physics I Department of Physics – The Citadel Bat and Ball Example A bat is swung at a ball pitched at 35 m/s, hitting the ball back on a line-drive at 56 m/s. What was the speed of the bat before and after the collision if the bat has mass M = 850 g? Assume an elastic collision. v = - 35 m/s v’ = 56 m/s m = 145 g M = 850 g

Physics 203 – College Physics I Department of Physics – The Citadel Bat and Ball Example Let v 1, v 1 ’ be the speed of the bat before and after the collision. Momentum conservation : m 1 v 1 + m 2 v 2 = m 1 v 1 ’ + m 2 v 2 ’ Elastic collision: v 1 – v 2 = v 2 ’ – v 1 ’ v 2 ’ = 56 m/s m 1 = 850 g v 2 = - 35 m/s m 2 = 145 g

Physics 203 – College Physics I Department of Physics – The Citadel Bat and Ball Example m 1 v 1 + m 2 v 2 = m 1 v 1 ’ + m 2 v 2 ’ v 1 – v 2 = v 2 ’ – v 1 ’ 2m 1 v 1 + (m 2 – m 1 ) v = (m 1 + m 2 ) v’ 2m 1 v 1 = (m 1 + m 2 ) v’ + (m 1 – m 2 ) v 1.7 v 1 = 0.995(56 m/s) (–35 m/s) = 31.0 m/s v 1 = 18 m/s. v 1 ’ = v 2 ’ + v 2 – v 1 = 3 m/s. m 1 m 1

Physics 203 – College Physics I Department of Physics – The Citadel Force Due to a Stream of Mass This mini-UZI fires 16 rounds per second, with m = 8g each, into a wall at velocity v 0 = 360 m/s. What is the average force on the wall? Think about the impulse resulting in the change of momentum when the bullets are stopped by the wall.

Physics 203 – College Physics I Department of Physics – The Citadel Force Due to a Stream of Bullets The wall stops the bullets, applying a force to the bullets of F wb = dp/dt. In time t = 1/16 s, a mass m = 8 g hits the wall. The momentum lost is  p =  m v 0. The impulse is F wb =  p/t =  mv 0 /t =  46 N F wb

Physics 203 – College Physics I Department of Physics – The Citadel The stream of bullets could be used as a propulsion mechanism for the gun… this is exactly the mechanism of rocket propulsion. The wall plays no role in this; the gun pushes against the bullets. F = 46 N Force Due to a Stream of Bullets

Physics 203 – College Physics I Department of Physics – The Citadel Two people on roller blades throw a ball back and forth. After a couple of throws, they are (ignore friction) A. standing where they were initially. B. standing farther away from each other. C. standing closer together. D. moving away from each other. E. moving toward each other. Question

Physics 203 – College Physics I Department of Physics – The Citadel Recoil Suppose a box of mass m = 10 kg is thrown out of a small boat of mass M = 100 kg with a speed relative to the boat of 15 m/s. M m What is the speed of the boat after the box is thrown?

Physics 203 – College Physics I Department of Physics – The Citadel Recoil Momentum conservation implies MV + mv = 0. We were given the speed of the box relative to the boat: M m v V v 0 = 15 m/s = v – V

Physics 203 – College Physics I Department of Physics – The Citadel Recoil MV + mv = 0. v = v 0 + V. MV + m(v 0 + V) = 0. Solving for V gives  m v 0  (10 kg) (15 m/s) M + m 110 kg M m v V V = = =  1.4 m/s

Physics 203 – College Physics I Department of Physics – The Citadel Center of Mass The center of mass a set of objects is the average position of their mass. For two objects in 1D: x 1 m 1 + x 2 m 2 m 1 + m 2 m1m1 m2m2 x1x1 x2x2 CM x cm = x cm

Physics 203 – College Physics I Department of Physics – The Citadel Center of Mass In more dimensions you can use vectors to locate the CM: m 1 r 1 + m 2 r 2 + m 3 r 3 m 1 + m 2 + m 3 It moves with velocity m1m1 m2m2 r1r1 r3r3 CM m2m2 r2r2 r cm = m 1 v 1 + m 2 v 2 + m 3 v 3 m 1 + m 2 + m 3 v cm = P = M v cm

Physics 203 – College Physics I Department of Physics – The Citadel Motion of the Center of Mass If an external force F acts on an extended object or collection of objects of mass M, the acceleration of the CM is given by F = Ma cm. You can apply Newton’s 2 nd Law as if it were a particle located at the CM, as far as the collective motion is concerned. This says nothing about the relative motion, rotation, etc., about the CM. That comes up in chapter 8.

Physics 203 – College Physics I Department of Physics – The Citadel Motion of Extended Objects The motion of extended objects or collections of particles is such that the CM obeys Newton’s 2 nd Law.

Physics 203 – College Physics I Department of Physics – The Citadel Motion of the Center of Mass The CM of a wrench sliding on a frictionless table will move in a straight line because there is no external force. In this sense, the wrench may be though of as a particle located at the CM. cm motion

Physics 203 – College Physics I Department of Physics – The Citadel Motion of the Center of Mass For example, if a hammer is thrown, its CM follows a parabolic trajectory under the influence of gravity, as a point object would.

Physics 203 – College Physics I Department of Physics – The Citadel Motion of the Center of Mass For example, if a hammer is thrown, its CM follows a parabolic trajectory under the influence of gravity, as a point object would.