PHYS 1443 – Section 501Lecture #16AnnouncementsLinear MomentumLinear Momentum and ForcesLinear Momentum ConservationConservation of Linear Momentum in a Two Particle SystemMore on Conservation of Linear Momentum in a Two Particle SystemExample for Linear Momentum ConservationExample for Linear Momentum ConservationImpulse and Linear MomentumExample for ImpulseExample for ImpluseCollisionsExample for CollisionsElastic and Inelastic CollisionsElastic and Perfectly Inelastic CollisionsPHYS 1443 – Section 501Lecture #16Wednesday, Mar. 24, 2004Dr. Andrew Brandt• Momentum• Elastic and Inelastic CollisionsWednesday, Mar. 24, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt1Announcements• HW#6 on Ch. 7+part of Ch. 8 is due Weds 3/24 at midnight (note HW#7 on ch 8 due 3/29)• Test 2 on ch. 6-10 will be Weds Apr. 7• I will be in Rio next week (sorry!) My class will go on!Probably no e-notes, and 1 quiz guaranteed!Wednesday, Mar. 24, 2004 2PHYS 1443-501, Spring 2004Dr. Andrew BrandtWednesday, Mar. 24, 2004 3PHYS 1443-501, Spring 2004Dr. Andrew BrandtLinear MomentumThe principle of energy conservation can be used to solve problems that are harder to solve using Newton’s laws. It is used to describe motion of an object or a system of objects.A new concept of linear momentum can also be used to solve physical problems, especially the problems involving collisions of objects.vmp =Linear momentum of an object whose mass is m and is moving at a velocity of v is defined as 1. Momentum is a vector quantity.2. The heavier the object the higher the momentum3. The higher the velocity the higher the momentum4. Its unit is kg.m/s What can you tell from this definition about momentum?What else can use see from the definition? Do you see force?The change of momentum in a given time intervaldtpdam=()vmdtd=dtvdm=F=Linear Momentum and Forces()vmdtddtpdF ==What can we learn from this Force-momentum relationship?• The rate of the change of particle’s momentum is the same as the net force exerted on it.• When net force is 0, the particle’s linear momentum is constant as a function of time.• If a particle is isolated, the particle experiences no net force, therefore its momentum does not change and is conserved.What else can this relationship be used for?The relationship can be used to study the case where the mass changes as a function of time.Can you think of a few cases like this?Motion of a rocket Motion of a meteoriteWednesday, Mar. 24, 2004 4PHYS 1443-501, Spring 2004Dr. Andrew BrandtLinear Momentum ConservationWednesday, Mar. 24, 2004 5PHYS 1443-501, Spring 2004Dr. Andrew Brandt221121vmvmppii+=+'22'1121vmvmppff+=+Conservation of Linear Momentum in a Two Particle SystemConsider a system with two particles that does not have any external forces exerting on it. What is the impact of Newton’s 3rdLaw?Now how would the momentaof these particles look like?If particle#1 exerts force on particle #2, there must be another force that particle #2 exerts on #1 as the reaction force. Both the forces are internal forces and the net force in the SYSTEM is still 0. Let say that the particle #1 has momentum p1and #2 has p2at some point of time.Using momentum-force relationship 121dtpdF =dtpdF212=andWednesday, Mar. 24, 2004 6PHYS 1443-501, Spring 2004Dr. Andrew BrandtAnd since net force of this system is 0constpp =+12Therefore∑FThe total linear momentum of the system is conserved!!!2112FF +=dtpddtpd12+=()12ppdtd+=0=More on Conservation of Linear Momentum in a Two Particle Systemconstppp =+=∑12From the previous slide we’ve learned that the total momentum of the system is conserved if no external forces are exerted on the system.As in the case of energy conservation, this means that the total vector sum of all momenta in the system is the same before and after any interactionWhat does this mean?Mathematically this statement can be written as ffippppi1212+=+∑∑=systemxfsystemxiPP∑∑=systemyfsystemyiPP∑∑=systemzfsystemziPPThis can be generalized into conservation of linear momentum in many particle systems.Whenever two or more particles in an isolated system interact, the total momentum of the system remains constant.Wednesday, Mar. 24, 2004 7PHYS 1443-501, Spring 2004Dr. Andrew BrandtExample for Linear Momentum ConservationEstimate an astronaut’s resulting velocity after he throws a book.Wednesday, Mar. 24, 2004 8PHYS 1443-501, Spring 2004Dr. Andrew BrandtipFrom momentum conservation, we can writevAvB0=fp=BBAAvmvm +=Assuming the astronaut’s mass if 70kg, and the book’s mass is 1kg and using linear momentum conservation=−ABBmvmBv701−=AvNow if the book gained a velocity of 20 m/s in +x-direction, the Astronaut’s velocity is()=− i20701()smi / 3.0−=AvExample for Linear Momentum ConservationA type of particle, a neutral kaon (K0), decays (breaks up) into a pair of particles called pions (π+and π-) that are oppositely charged but have equal mass. Assuming the K0is initially produced at rest, prove that the two pions must have momenta that are equal in magnitude and opposite in direction.−++→ππ0KThis reaction can be written asWednesday, Mar. 24, 2004 9PHYS 1443-501, Spring 2004Dr. Andrew Brandt0KpSince K0is produced at rest its momentum is 0. K0π+π−pπ+pπ−Since this system consists of a K0in the initial state which results in two pions in the final state, the momentum must be conserved. So we can write0Kp−++=ππpp−++=ππpp0=−+−=ππppTherefore, the two pions from this kaon decay have momenta with same magnitude but opposite in direction.Impulse and Linear Momentum By integrating the above equation in a time interval tito tf, one can obtain impulse I.Impulse of the force F acting on a particle over the time interval ∆t=tf-tiis equal to the change of the momentum of the particle caused by that force. Impulse is the degree to which an external force changes momentum.The above statement is called the impulse-momentum theorem and is equivalent to Newton’s second law. dtpdF =Net force causes change of momentum ÎNewton’s second lawSo what do you think an impulse is?∫fittpdpdtFIfitt∆=≡∫What are the units of Impulse? Defining a time-averaged force ∫∆≡fittdtFtF1Impulse can be rewritten tFI ∆≡If force is constant tFI ∆≡dtFpd =pppif∆=−=∫=fittdtFWednesday, Mar. 24, 2004 10PHYS 1443-501, Spring 2004Dr. Andrew BrandtIt is generally assumed that the impulse force acts for a short time but is much greater than
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