Unit 5: Conservation of MomentumScheduleConservation of Energy and Momentum in CollisionsSlide 4The Classic Inelastic Collision: A Ballistic Pendulum.Slide 6Elastic Collisions in One DimensionThe SetupSlide 9Slide 10Special Case 1: Equal MassesSlide 12Special Case 2: Unequal Masses Target at RestSlide 14Slide 15Slide 16Slide 17An Example: Proton ScatteringSlide 19To Finish Up01/14/1901/14/19 Physics 253Physics 25311Unit 5: Conservation of Unit 5: Conservation of MomentumMomentumForce and Momentum and Force and Momentum and Conservation of Momentum (9-Conservation of Momentum (9-1,9-2, 9-3)1,9-2, 9-3)•Collisions (9-4, 9-5, 9-6)Collisions (9-4, 9-5, 9-6)•Collisions, Center of Mass (9-7,9-Collisions, Center of Mass (9-7,9-8)8)•Catch-up, Rocket Propulsion, and Catch-up, Rocket Propulsion, and Quiz Three Review (9-9,9-10)Quiz Three Review (9-9,9-10)01/14/1901/14/19 Physics 253Physics 25322ScheduleSchedule•This Friday, March 30This Friday, March 30thth–No ClassNo Class•Wednesday, April 4Wednesday, April 4thth–Finish Conservation of MomentumFinish Conservation of Momentum–Review of Conservation of Energy and Review of Conservation of Energy and MomentumMomentum•Friday, April 6Friday, April 6thth–Test Conservation of Energy & MomentumTest Conservation of Energy & Momentum–Crib sheet: single 3”x5” note card.Crib sheet: single 3”x5” note card.01/14/1901/14/19 Physics 253Physics 25333Conservation of Energy and Conservation of Energy and Momentum in CollisionsMomentum in Collisions•In two-body collisions difficult to In two-body collisions difficult to describe the force describe the force Newton’s Newton’s 22ndnd Law hard to use. Law hard to use.•But if external forces are absent But if external forces are absent or negligible, conservation of or negligible, conservation of momentum connects the initial momentum connects the initial and final states of the system.and final states of the system.•If the two objects suffer no If the two objects suffer no deformation during the collision deformation during the collision then kinetic energy is also then kinetic energy is also conserved and equal before and conserved and equal before and after the collision. This is called after the collision. This is called an an elastic collision.elastic collision.•Actually some of the kinetic Actually some of the kinetic energy may be lost to energy may be lost to deformation heat. This is called deformation heat. This is called an an inelastic collision.inelastic collision.2'222'1122221121212121constant KE :Collision Elastic vmvmvmvm energylost decreases KE :Collision Inelastic'1'121 KKKK01/14/1901/14/19 Physics 253Physics 25344KE ConstantElasticKE DecreasesInelastic01/14/1901/14/19 Physics 253Physics 25355The Classic Inelastic Collision: The Classic Inelastic Collision: A Ballistic Pendulum.A Ballistic Pendulum.•A nifty device to A nifty device to measure the speed measure the speed of a projectile by of a projectile by converting kinetic converting kinetic energy to potential energy to potential energyenergy•A beautiful A beautiful application of both application of both momentum and momentum and energy conservationenergy conservation01/14/1901/14/19 Physics 253Physics 25366'')()( :forceexternal no is thereSince after.just andblock thehitsbullet thebeforeonconservati momentumConsider vmMmvvMmmvghvghMmvMmghMmvMmUKUK2)()(21)(00)(21:arc spedulum' theof topat the then collideyelasticallblock andbullet after thejuston conservatienergy Consider '2'2'2211ghmMmvghvvmMmv2)(:former theintolatter thengSubstituti.2 :onconservatienergy From)( :onconservati momentum Fromresults twohave We''01/14/1901/14/19 Physics 253Physics 25377Elastic Collisions in One DimensionElastic Collisions in One Dimension•When momentum and kinetic energy are When momentum and kinetic energy are both conserved, we have two powerful both conserved, we have two powerful handles for describing a collision.handles for describing a collision.•The best practical examples would be The best practical examples would be billiard balls or air-hockey pucks billiard balls or air-hockey pucks –The normal force and gravity are equal so the The normal force and gravity are equal so the main external forces sum to zero.main external forces sum to zero.–Friction is essentially zero.Friction is essentially zero.•We ignore rotational motion because at We ignore rotational motion because at the instant of collision unimportant.the instant of collision unimportant.01/14/1901/14/19 Physics 253Physics 25388The SetupThe Setupleft. theto 0 negative andright the toare 0 velocitiespositive soright thex tochoose We. and denoted are velocities final The . and be and mass of objects two theof s velocitieinitial Let the'2'12121)(v)(vvvvvmmAlthough the figure has objects with positive velocity, the discussion is generaland describes motion in either direction01/14/1901/14/19 Physics 253Physics 25399)v)(vv(vm)v)(vv(vm)-v(vm)v(vmvmvmvmvmvvmvvmvmvmvmvmvmvmvmvm''2'22'22'11'11122222212112'222'112222112'22'112'222'11222211'22'112211 :mass toaccording termscollectingby and asrewritten becan equation energy The ).()( asrewritten becan equation momentum The equations. general twoon thesefocusjust slet' nowFor !determinedentirely are s velocitiefinal themasses theand velocity incoming theknow weif instanceFor unknowns. for two solvecan wemeans equations Two .situations ofnumberany for used becan and powerfulextremely are These21212121 :equationson conservati twohave We01/14/1901/14/19 Physics 253Physics 2531010 this.understand us help willcases special Some direction!opposite in the andcollision after the and before same theisobjects twoebetween th speed relative theIn word, collision.on-headany for trueis which )(see wengreorganizi and mass! canceling first, by theequation second thedividecan We and )()( slide previous theFrom'2'1'1'2212'2'112'22'22'11'1112'22'11vvvvvv)v(v)v(v)v)(vv(vm)v)(vv(vmvvmvvm01/14/1901/14/19 Physics 253Physics 2531111Special Case 1: Special Case 1: Equal MassesEqual Masses•A billiard ball of mass A billiard ball of mass m moving with speed m moving with speed vv11 collides head-on collides head-on and