1443-501 Spring 2002Lecture #10Dr. Jaehoon Yu1. Term exam results2. Linear Momentum 3. Momentum Conservation4. Impulse and Momentum5. What are Collisions?6. Elastic and Inelastic collisions7. Two dimensional collisionsHomework: http://hw.utexas.edu/studentInstructions.html – Do Homework #2.Term Exam Answer Key: A new link on my web page http://www-hep.uta.edu/~yuFeb. 25, 20021443-501 Spring 2002Dr. J. Yu, Lecture #102This mean value is what I am going to use to scale!!Some of you have done rather well.But most of you have done not as well as I was hoping for.What I expected was about twice as much as what your mean value is.Please do homework for extra credit and for yourselves!!Might introduce pop-quiz after mid-term.Feb. 25, 20021443-501 Spring 2002Dr. J. Yu, Lecture #103Linear MomentumThe principle of energy conservation can be used to solve problems that are harder to solve just 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 interval()dtvdmvmdtddtpd==Fam ==Feb. 25, 20021443-501 Spring 2002Dr. J. Yu, Lecture #104Linear Momentum and ForcesWhat can we learn from this Force-momentum relationship?Something else we can do with this relationship. What do you think it is?()vmdtddtpdF ==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 meteorite Trajectory a satellite • The rate of the change of particle’s momentum is the same as thenet force exerted on it.• When net force is 0, the particle’s linear momentum is constant.• If a particle is isolated, the particle experiences no net force, therefore its momentum does not change and is conserved.Feb. 25, 20021443-501 Spring 2002Dr. J. Yu, Lecture #105Conservation of Linear Momentum in a Two Particle SystemConsider a system with two particles that does not have any external forces exerted 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 the 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 relationshipdtpdFdtpdF212121 and ==And since net force of this system is 0constpp =+12Therefore()012122112=+=+=+=∑ppdtddtpddtpdFFFThe total linear momentum of the system is conserved!!!Feb. 25, 20021443-501 Spring 2002Dr. J. Yu, Lecture #106More on Conservation of Linear Momentum in a Two Particle SystemWhat does this mean?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 interactionMathematically this statement can be written as Whenever two or more particles in an isolated system interact, the total momentum of the system remains constant.constppp =+=∑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.ffippppi1212+=+This can be generalized into conservation of linear momentum in many particle systems.∑∑∑∑∑∑===systemzfsystemzisystemyfsystemyisystemxfsystemxiPPPPPP ; ;Feb. 25, 20021443-501 Spring 2002Dr. J. Yu, Lecture #107Example 9.1Estimate an astronaut’s resulting velocity after he throws his book to a direction in the space to move to a direction.BBAAfivmvmpp +=== 0From momentum conservation, we can writevAvBAssuming the astronaut’s mass if 70kg, and the book’s mass is 1kg and using linear momentum conservationBABBAvmvmv701−=−=Now if the book gained a velocity of 20 m/s in +x-direction, the Astronaut’s velocity is()( )smiivA/3.020701−=−=Feb. 25, 20021443-501 Spring 2002Dr. J. Yu, Lecture #108Example 9.2A type of particle, neutral kaon (K0) decays (breaks up) into a pair of particles called pions (π+and π-) that are oppositely charged but equal mass. Assuming K0is initially produced at rest, prove that the two pions must have mumenta that are equal in magnitude and opposite in direction.−++→ ππ0KThis reaction can be written as−++=ππpppK0Since K0 is 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 write−+−+−==+=ππππpppppK00Therefore, the two pions from this kaon decay have the momanta with same magnitude but in opposite direction.Feb. 25, 20021443-501 Spring 2002Dr. J. Yu, Lecture #109Impulse 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-ti is equal to the change of the momentum of the particle caused by that force. Impulse is the degree of which an external force changes momentum.The above statement is called the impulse-momentum theorem and is equivalent to Newton’s second law. dtFpddtpdF == ;Net force causes change of momentum èNewton’s second lawSo what do you think an impulse is?∫∫=∆=−=fifittifttdtFppppdpdtFIfitt∆=≡∫What are the dimension and unit of Impulse? What is the direction of an impulse vector? Defining a time-averaged force ∫∆≡fittdtFtF1Impulse can be rewritten tFI ∆≡If force is constant tFI ∆≡It is generally approximated that the impulse force exerted actson a short time but much greater than any other forces present.Feb. 25, 20021443-501 Spring 2002Dr. J. Yu, Lecture #1010Example 9.3A golf ball of mass 50g is struck by a club. The force exerted on the ball by the club varies from 0, at the instant before contact, up to some maximum value at which the ball is deformed and then back to 0 when the ball leaves the club. Assuming the ball travels 200m. Estimate the magnitude of the impulse caused by
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