Lecture 13Momentum is conserved if no external forceInelastic collision in 1-D: ExampleSlide 4Momentum is a vectorA perfectly inelastic collision in 2-DSlide 7Exercise Momentum is a Vector (!) quantityElastic CollisionsBilliardsBilliards: Without external forces, conservation of momentum (and energy Ch. 10 & 11)Explosions: A collision in reverseSlide 18Slide 19Impulse (A variable force applied for a given time)Force and Impulse (A variable force applied for a given time)Force and ImpulseAverage Force and ImpulseExercise Force & ImpulseBoxing: Use Momentum and Impulse to estimate g “force”Slide 26Ch. 10 : Kinetic & Potential energiesSlide 29Physics 207: Lecture 13, Pg 1Lecture 13Goals:Goals:Assignments: Assignments: HW5, due tomorrow HW5, due tomorrow For Wednesday, Read all of Chapter 10For Wednesday, Read all of Chapter 10•Chapter 9Chapter 9 Employ conservation of momentum in 1 D & 2D Examine forces over time (aka Impulse)•Chapter 10Chapter 10 Understand the relationship between motion and energyPhysics 207: Lecture 13, Pg 2Momentum is conserved if no external force afterbeforeppPhysics 207: Lecture 13, Pg 3Inelastic collision in 1-D: ExampleA block of mass M is initially at rest on a frictionless horizontal surface. A bullet of mass m is fired at the block with a muzzle velocity (speed) v. The bullet lodges in the block, and the block ends up with a final speed V. In terms of m, M, and V :What is the momentum of the bullet with speed v ?vVbefore afterxPhysics 207: Lecture 13, Pg 4Inelastic collision in 1-D: ExampleWhat is the momentum of the bullet with speed v ? Key question: Is there a net external x-dir force ? If not, then momentum in the x-direction is conserved!vVbefore afterxaaaavm 0 M v m V)/1( v mMP BeforeP After V)( Mm Physics 207: Lecture 13, Pg 5Momentum is a vectorzzyxxmpmpmpkjimpvvvˆpˆpˆpvyzyxPhysics 207: Lecture 13, Pg 6A perfectly inelastic collision in 2-DConsider a collision in 2-D (cars crashing at a slippery intersection...no friction).v1v2 beforem1m2If no external force momentum is conserved.Momentum is a vector so px, py and pz V afterm1 + m2Physics 207: Lecture 13, Pg 7A perfectly inelastic collision in 2-Dv1v2 V before afterm1m2m1 + m2x-dir px : m1 v1 = (m1 + m2 ) V cos y-dir py : m2 v2 = (m1 + m2 ) V sin p1p2pPhysics 207: Lecture 13, Pg 8Exercise Momentum is a Vector (!) quantityA. YesB. NoC. Yes & NoD. Too little information givenA block slides down a frictionless ramp and then falls and lands in a cart which then rolls horizontally without frictionIn regards to the block landing in the cart is momentum conserved?Physics 207: Lecture 13, Pg 12Elastic CollisionsElastic means that the objects do not stick.There are many more possible outcomes but, if no external force, then momentum will always be conservedStart with a 1-D problem.Before AfterPhysics 207: Lecture 13, Pg 13BilliardsConsider the case where one ball is initially at rest. pa pb FPa beforeafterThe final direction of the red ball will depend on where the balls hit.Physics 207: Lecture 13, Pg 14Billiards: Without external forces, conservation of momentum (and energy Ch. 10 & 11)Conservation of Momentumx-dir Px : m vbefore = m vafter cos + m Vafter cos y-dir Py : 0 = m vafter sin + m Vafter sin pafter ppb FPafter beforeafterPhysics 207: Lecture 13, Pg 17Explosions: A collision in reverseA two piece assembly is hanging vertically at rest at the end of a 2.0 m long massless string. The mass of the two pieces are 60 and 20 kg respectively. Suddenly you observe that the 20 kg is ejected horizontally at 30 m/s. The time of the “explosion” is short compared to the swing of the string.Does the tension in the string increase or decrease after the explosion?BeforeAfterPhysics 207: Lecture 13, Pg 18Explosions: A collision in reverseA two piece assembly is hanging vertically at rest at the end of a 2.0 m long massless string. The mass of the two pieces are 60 and 20 kg respectively. Suddenly you observe that the 10 kg is ejected horizontally at 30 m/s. Decipher the physics:1. The green ball recoils in the –x direction (3rd Law) and, because there is no net force in the x-direction the x-momentum is conserved. 2. The motion of the green ball is constrained to a circular path…there must be centripetal (i.e., radial acceleration)BeforeAfterPhysics 207: Lecture 13, Pg 19Explosions: A collision in reverseA two piece assembly is hanging vertically at rest at the end of a 20.0 m long massless string. The mass of the two pieces are 60 & 20 kg respectively. Suddenly you observe that the 20 kg mass is suddenly ejected horizontally at 30 m/s. Cons. of x-momentumPx before= Px after = 0 = - M V + m v V = m v / M = 20*30/ 60 = 10 m/sTbefore = Weight = (60+20) x 10 N = 800 NFy = m ay (radial) = M V2/r = T – MgT = Mg + MV2 /r = 600 N + 60x(10)2/20 N = 900 NBeforeAfterPhysics 207: Lecture 13, Pg 20Impulse (A variable force applied for a given time) Gravity: At small displacements a “constant” forceSprings often provide a linear force (-kx) towards its equilibrium position (Chapter 10)Collisions often involve a varying force F(t): 0 maximum 0We can plot force vs time for a typical collision. The impulse, J, of the force is a vector defined as the integral of the force during the time of the collision.Physics 207: Lecture 13, Pg 21Force and Impulse (A variable force applied for a given time) Fpttpdd td tpdd tFJ)/(J a vector that reflects momentum transfertti tft Impulse J = area under this curve !(Transfer of momentum !)Impulse has units of Newton-secondsPhysics 207: Lecture 13, Pg 22Force and ImpulseTwo different collisions can have the same impulse since JJ depends only on the momentum transfer, NOT the nature of the collision.t FtFtt same areat big, FF smallt small, FF bigPhysics 207: Lecture 13, Pg 23Average Force and ImpulseAverage Force and Impulset FtFtt t big, Fav smallt small, Fav bigFFav av FFav avPhysics 207: Lecture 13, Pg 24Exercise Force & ImpulseA. heavierB. lighterC. sameD. can’t tellTwo boxes, one heavier than the other, are initially at rest on a horizontal frictionless surface. The same constant force F acts on each one for exactly 1 second. Which box has the
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