Lecture 12Momentum ConservationInelastic collision in 1-D: ExampleSlide 4Exercise Momentum is a Vector (!) quantitySlide 6Slide 7A perfectly inelastic collision in 2-DSlide 11Elastic CollisionsBilliardsBilliards: Without external forces, conservation of momentum (and energy Ch. 10 & 11)Force and Impulse (A variable force applied for a given time)Slide 16Force and ImpulseAverage Force and ImpulseExercise 2 Force & ImpulseBoxing: Use Momentum and Impulse to estimate g “force”Slide 21Chapter 10: Energy (Forces over distance)Slide 23Physics 207: Lecture 12, Pg 1Lecture 12Goals:Goals:Assignment: Assignment: HW6 due Wednesday 3/3HW6 due Wednesday 3/3For Tuesday: Read all of chapter 10For Tuesday: Read all of chapter 10• Chapter 9: Momentum & ImpulseChapter 9: Momentum & Impulse Solve problems with 1D and 2D Collisions Solve problems having an impulse (Force vs. time)•Chapter 10Chapter 10 Understand the relationship between motion and energy Define Potential & Kinetic Energy Develop and exploit conservation of energy principlePhysics 207: Lecture 12, Pg 2Momentum ConservationMomentum conservation (recasts Newton’s 2nd Law when net external F = 0) is an important principle (usually when forces act over a short time)It is a vector expression so must consider Px, Py and Pz if Fx (external) = 0 then Px is constant if Fy (external) = 0 then Py is constant if Fz (external) = 0 then Pz is constantconstant that implies 0 PPdtddtddtmddtdmamEXTP)vvF(0 if and EXTFPPPPhysics 207: Lecture 12, 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 12, Pg 4Inelastic collision in 1-D: ExampleWhat is the momentum of the bullet with speed v ? Key question: Is x-momentum conserved ? vVbefore afterxaaaavm V)( 0 M v Mmm P BeforeP BeforeP AfterP After V)/1( v mMPhysics 207: Lecture 12, Pg 5Exercise 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 12, Pg 6Exercise Momentum is a Vector (!) quantityLet a 2 kg block start at rest on a 30° incline and slide vertically a distance 5.0 m and fall a distance 7.5 m into the 10 kg cartWhat is the final velocity of the cart?x-direction: No net force so Px is conserved.y-direction: Net force, interaction with the ground sodepending on the system (i.e., do you include the Earth?) Py is not conserved (system is block and cart only)5.0 m30°7.5 m10 kg2 kgPhysics 207: Lecture 12, Pg 7Exercise Momentum is a Vector (!) quantity Initial FinalPx: MVx + mvx = (M+m) V’x M 0 + mvx = (M+m) V’xV’x = m vx / (M + m) = 2 (8.7)/ 12 m/sV’x = 1.4 m/sx-direction: No net force so Px is conservedy-direction: vy of the cart + block will be zero and we can ignore vy of the block when it lands in the cart.5.0 m30°7.5 mNmg1) ai = g sin 30° = 5 m/s22) d = 5 m / sin 30° = ½ ai t210 m = 2.5 m/s2 t2 2s = t v = ai t = 10 m/s vx= v cos 30° = 8.7 m/s ijxy30°Physics 207: Lecture 12, Pg 10A perfectly inelastic collision in 2-DConsider a collision in 2-D (cars crashing at a slippery intersection...no friction).vv1vv2 VV before afterm1m2m1 + m2If no external force momentum is conserved.Momentum is a vector so px, py and pz Physics 207: Lecture 12, Pg 11A perfectly inelastic collision in 2-Dvv1vv2 VV before afterm1m2m1 + m2x-dir px : m1 v1 = (m1 + m2 ) V cos y-dir py : m2 v2 = (m1 + m2 ) V sin If no external force momentum is conserved.Momentum is a vector so px, py and pz are consevedPhysics 207: Lecture 12, 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 12, Pg 13BilliardsConsider the case where one ball is initially at rest. ppa ppb FFPPa beforeafterThe final direction of the red ball will depend on where the balls hit.vvcmPhysics 207: Lecture 12, 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 ppafter ppb FFPPafter beforeafterPhysics 207: Lecture 12, Pg 15Force and Impulse (A variable force applied for a given time) Gravity: At small displacements a “constant” force tSprings 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, JJ, of the force is a vector defined as the integral of the force during the time of the collision.Physics 207: Lecture 12, Pg 16Force and Impulse (A variable force applied for a given time) Fpttpdd td tpddtFJ)/(J a vector that reflects momentum transfertti tft Impulse JJ = area under this curve !(Transfer of momentum !)Impulse has units of Newton-secondsPhysics 207: Lecture 12, Pg 17Force 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 12, Pg 18Average Force and Impulset FtFtt t big, FFavav smallt small, FFavav bigFFav av FFav avPhysics 207: Lecture 12, Pg 19Exercise 2Force & 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 most momentum after the force acts ?F F lightheavyPhysics 207: Lecture 12, Pg 20Boxing: Use Momentum and Impulse to estimate g “force”Physics 207: Lecture
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