PHYS 1443 – Section 003 Lecture #10ReminderLinear MomentumLinear Momentum and ForcesConservation of Linear Momentum in a Two Particle SystemMore on Conservation of Linear Momentum in a Two Particle SystemExample 9.1Example 9.2Impulse and Linear MomentumExample 9.3Example 9.4CollisionsExample 9.5Elastic and Inelastic CollisionsElastic and Perfectly Inelastic CollisionsTwo dimensional CollisionsExample 9.9Wednesday, Oct. 16, 2002 PHYS 1443-003, Fall 2002Dr. Jaehoon Yu1PHYS 1443 – Section 003Lecture #10Wednesday, Oct. 16, 2002Dr. Jaehoon Yu1. Linear momentum and Force2. Linear Momentum Conservation3. Impulse and Linear Momentum4. Collisions in One and Two DimensionToday’s homework is homework #11, due 12:00pm, next Wednesday!!Wednesday, Oct. 16, 2002 PHYS 1443-003, Fall 2002Dr. Jaehoon Yu2Reminder•I need to see the following students:–Matt Andrews–David Hunt–Dhumil PatelWednesday, Oct. 16, 2002 PHYS 1443-003, Fall 2002Dr. Jaehoon Yu3Linear 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 dtvdmvmdtddtpdFam Wednesday, Oct. 16, 2002 PHYS 1443-003, Fall 2002Dr. Jaehoon Yu4Linear 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 meteoriteTrajectory a satellite •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.•If a particle is isolated, the particle experiences no net force, therefore its momentum does not change and is conserved.Wednesday, Oct. 16, 2002 PHYS 1443-003, Fall 2002Dr. Jaehoon Yu5Conservation 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 3rd Law?Now how would the momenta of 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 p1 and #2 has p2 at some point of time.Using momentum-force relationship 121dtpdF And since net force of this system is 0constpp 12ThereforeFThe total linear momentum of the system is conserved!!!dtpdF212and2112FF dtpddtpd12 12ppdtd0Wednesday, Oct. 16, 2002 PHYS 1443-003, Fall 2002Dr. Jaehoon Yu6More 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.ffippppi1212This can be generalized into conservation of linear momentum in many particle systems.systemxfsystemxiPPsystemyfsystemyiPPsystemzfsystemziPPWednesday, Oct. 16, 2002 PHYS 1443-003, Fall 2002Dr. Jaehoon Yu7Example 9.1Estimate an astronaut’s resulting velocity after he throws his book to a direction in the space to move to a direction.fipp 0From momentum conservation, we can writevAvBAssuming the astronaut’s mass if 70kg, and the book’s mass is 1kg and using linear momentum conservationBABBAvmvmv701Now if the book gained a velocity of 20 m/s in +x-direction, the Astronaut’s velocity is smiivA/3.020701BBAAvmvm Wednesday, Oct. 16, 2002 PHYS 1443-003, Fall 2002Dr. Jaehoon Yu8Example 9.2A type of particle, neutral kaon (K0) decays (breaks up) into a pair of particles called pions (+ and -) that are oppositely charged but equal mass. Assuming K0 is initially produced at rest, prove that the two pions must have mumenta that are equal in magnitude and opposite in direction.0KThis reaction can be written as0KpSince K0 is produced at rest its momentum is 0. K0ppSince this system consists of a K0 in the initial state which results in two pions in the final state, the momentum must be conserved. So we can write0KpTherefore, the two pions from this kaon decay have the momanta with same magnitude but in opposite direction.pp0ppppWednesday, Oct. 16, 2002 PHYS 1443-003, Fall 2002Dr. Jaehoon Yu9Impulse and Linear Momentum By integrating the above equation in a time interval ti to 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. dtpdF Net force causes change of momentum Newton’s second lawSo what do you think an impulse is?fittpdpdtFIfittWhat 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 acts on a short time but much greater than any other forces present.dtFpd pppiffittdtFWednesday,
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