Page 1Physics 207 – Lecture 12Physics 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 Conservation Momentum conservation (recasts Newton’s 2ndLaw 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, Pyand Pz if Fx(external) = 0 then Pxis constant if Fy(external) = 0 then Pyis constant if Fz(external) = 0 then Pzis constantconstant that implies 0 == PPrrdtddtddtmddtdmamEXTP)vvFrrrrr====(0 if and =EXTFrPPPPage 2Physics 207 – Lecture 12Physics 207: Lecture 12, Pg 3Inelastic collision in 1-D: Example A 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 afterxaaaavrm V)( 0 M vMmm+=+P BeforeP BeforeP AfterP After V)/1( vmM+=Page 3Physics 207 – Lecture 12Physics 207: Lecture 12, Pg 5Exercise Momentum is a Vector (!) quantityA. YesB. NoC. Yes & NoD. Too little information given A block slides down a frictionless ramp and then falls and lands in a cart which then rolls horizontally without friction In 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 Pxis conserved. y-direction: Net force, interaction with the ground sodepending on the system (i.e., do you include the Earth?) Pyis not conserved (system is block and cart only)5.0 m30°7.5 m10 kg2 kgPage 4Physics 207 – Lecture 12Physics 207: Lecture 12, Pg 7Exercise Momentum is a Vector (!) quantityInitial FinalPx: MVx+ mvx= (M+m) V’xM 0 + mvx= (M+m) V’xV’x= m vx/ (M + m)= 2 (8.7)/ 12 m/sV’x= 1.4 m/s x-direction: No net force so Pxis conserved y-direction: vyof the cart + block will be zero and we can ignore vyof 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∆t22s = ∆tv = ai∆t= 10 m/svx= v cos 30°= 8.7 m/sijxy30°Physics 207: Lecture 12, Pg 8Home ExerciseInelastic Collision in 1-D with numbersice(no friction)Do not try this at home!Before: Before: A 4000 kg bus, twice the mass of the car, moving A 4000 kg bus, twice the mass of the car, moving at 30 at 30 m/sm/simpacts the car at rest. impacts the car at rest. What is the final speed after impact if they move together?What is the final speed after impact if they move together?Page 5Physics 207 – Lecture 12Physics 207: Lecture 12, Pg 9Home exercise Inelastic Collision in 1-Dvvff=?finallymv = 0iceM = 2mVV00(no friction)initially2Vo/3 = 20 m/s000V22V Vor V)( VmmmMmMMmM+=+=+=Physics 207: Lecture 12, Pg 10A perfectly inelastic collision in 2-D Consider a collision in 2-D (cars crashing at a slippery intersection...no friction).vv1vv2VVbefore afterm1m2m1+ m2 If no external force momentum is conserved. Momentum is a vector so px, pyand pzθPage 6Physics 207 – Lecture 12Physics 207: Lecture 12, Pg 11A perfectly inelastic collision in 2-Dvv1vv2VVbefore afterm1m2m1+ m2 x-dir px: m1v1= (m1+ m2) V cos θ y-dir py: m2v2= (m1+ m2) V sin θ If no external force momentum is conserved. Momentum is a vector so px, pyand pzare consevedθPhysics 207: Lecture 12, Pg 12Elastic Collisions Elastic means that the objects do not stick. There are many more possible outcomes but, if no external force, then momentum will always be conserved Start with a 1-D problem.Before AfterPage 7Physics 207 – Lecture 12Physics 207: Lecture 12, Pg 13Billiards Consider the case where one ball is initially at rest. ppaθθθθppbFFPPaφφφφbeforeafterThe 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 Momentum x-dir Px: m vbefore= m vaftercos θ + m Vaftercos φ y-dir Py: 0 = m vaftersin θ + m Vaftersin φppafterθθθθppbFFPPafterφφφφbeforeafterPage 8Physics 207 – Lecture 12Physics 207: Lecture 12, Pg 15Force and Impulse (A variable force applied for a given time) Gravity: At small displacements a “constant” force t Springs often provide a linear force (-k x) towards its equilibrium position (Chapter 10) Collisions often involve a varying force F(t): 0 maximum 0 We 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)F∫∫∫==≡pttpddtdtpddtFJrrrr)/( J a vector that reflects momentum transfertti tf∆t ImpulseJJ = area under this curve !(Transfer of momentum !)Impulse has units of Newton-secondsPage 9Physics 207 – Lecture 12Physics 207: Lecture 12, Pg 17Force and Impulse Two different collisions can have the same impulse since JJ depends only on the momentum transfer, NOT the nature of the collision.∆t FtFt∆t same area∆t big, FF small∆t small, FF bigPhysics 207: Lecture 12, Pg 18Average Force and Impulse∆t FtFt∆t ∆t big, FFavavsmall∆t small, FFavavbigFFavavFFavavPage 10Physics 207 – Lecture 12Physics 207: Lecture 12, Pg 19Exercise 2Force & ImpulseA. heavierB. lighterC. sameD. can’t tell Two boxes, one heavier than the other, are initially at rest on a horizontal frictionless surface. The same
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