Ch06 2 S1 - Michigan State University

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Chapter 6Impulse and MomentumContinued

6.1 The Impulse-Momentum TheoremChapter 6 is about the COLLISION of TWO masses.To understand the interaction, both masses must be considered.Newton's 3rd Law plays a very important part.Collisions involve two new concepts: Impulse and Momentum.Impulse concept leads to the Momentum definition.Also applied to two (or more) masses blown apart by an explosion.

6.1 The Impulse-Momentum TheoremDEFINITION OF IMPULSEThe impulse of a force is the product of the averageforce and the time interval during which the force acts:!F averageforce vectorImpulse is a vector quantity and has the same directionas the average force.

6.1 The Impulse-Momentum Theorem

6.1 The Impulse-Momentum TheoremDEFINITION OF LINEAR MOMENTUMThe linear momentum of an object is the productof the objectʼs mass times its velocity:Linear momentum is a vector quantity and has the samedirection as the velocity.

6.1 The Impulse-Momentum TheoremIMPULSE-MOMENTUM THEOREMWhen a net force acts on an object, the impulse ofthis force is equal to the change in the momentumof the objectimpulse FNet Δt mv f mv i pf pi ΔpChange in momentum ΔpNewton’s 2nd Law becomes: FNet Δt

6.2 The Principle of Conservation of Linear MomentumDuring collision of two masses in free fallForce on 1generated by 2 F12Impulses due to internalforces cancel out in sum( W2 F21Force on 2generated by 1 W1 W1 W2 Δt (p1 p 2 )f (p1 p 2 )iImpulses due onlyto external forces)Total momentum Total momentumin the final system in the initial systemNet EXTERNAL forces will change the Total Momentum of a system of masses

6.2 The Principle of Conservation of Linear MomentumDuring collision of two masses in free fallForce on 1generated by 2 F12 W2 F21Force on 2generated by 1 W1( W1 W2 Δt (p1 p 2 )f (p1 p 2 )i)With only INTERNAL forces affecting motion (e.g., if external forces are balanced) 0 (p1 p 2 )f (p1 p 2 )i (p1 p 2 )f (p1 p 2 )iFinal value oftotal momentumInitial value oftotal momentumIf only INTERNAL forces affect motion,total momentum VECTOR of a system does not change

6.2 The Principle of Conservation of Linear MomentumPRINCIPLE OF CONSERVATION OF LINEAR MOMENTUMThe total linear momentum of an isolated system is constant(conserved). An isolated system is one for which the sum ofthe average external forces acting on the system is zero.Most Important exampleIf the only external forces are gravitational forces that arebalanced by normal forces, the total momentum VECTORof a system is conserved in a collision.

6.2 The Principle of Conservation of Linear MomentumIf the only external forces are gravitational forces that arebalanced by normal forces, the total momentum VECTORof a system is conserved in a collisionFor Hockey Pucks on the ice, thegravitational force on each puck is balancedby the normal force of the ice.Hockey Pucks after collisionHockey Pucks before collision p1i1 p 2i2 (p1 p 2 )iMomentum Conservation: p1fBANG12Only internal forces act (p1 p 2 )f (p1 p 2 )i1 (p1 p 2 )f p 2f2

Clicker Question 6.2Two hockey pucks bang into each other on frictionless ice.Each puck has a mass of 0.5 kg, and are moving directly towardeach other each with a speed of 12 m/s. What is thetotal momentum of the system of two pucks?a) 6.0 N sb) 12 N sc) – 6.0 N sd) –12 N se)0.0 N s

Clicker Question 6.2Two hockey pucks bang into each other on frictionless ice.Each puck has a mass of 0.5 kg, and are moving directly towardeach other each with a speed of 12 m/s. What is thetotal momentum of the system of two pucks?a) 6.0 N sb) 12 N sc) – 6.0 N sd) –12 N se)0.0 N sClicker Question 6.3After the pucks collide, what is the total momentum of the system?a) 6.0 N sb) 12 N sc) – 6.0 N sd) –12 N se)0.0 N s

6.2 The Principle of Conservation of Linear MomentumConceptual Example Is the Total Momentum Conserved?Imagine two balls colliding on a billiardtable that is friction-free. Use the momentumconservation principle in answering thefollowing questions. (a) Is the total momentumof the two-ball system the same beforeand after the collision? (b) Answerpart (a) for a system that contains onlyone of the two collidingballs.

6.2 The Principle of Conservation of Linear MomentumPRINCIPLE OF CONSERVATION OF LINEAR MOMENTUMThe total linear momentum of an isolated system is constant(conserved). An isolated system is one for which the sum ofthe average external forces acting on the system is zero.In the top picture the net external force on thesystem is zero.In the bottom picture the net external force on thesystem is not zero.

6.2 The Principle of Conservation of Linear MomentumSkaters on the ice. Push off is an “explosion”. System of two massesNet External Force on system of two skaters is zero.Total momentum is conserved pTotal ,i m1v1i m2 v 2i initial momentum sum 0Total momentum is conserved pTotal ,f pTotal ,i pTotal ,f 0Before pTotal ,f m1v1f m2 v 2f 0 final momentum sum Momentum vector of mass 2 m2 v 2f m1v1f is opposite toMomentum vector of mass 1

6.2 The Principle of Conservation of Linear MomentumExample: Ice SkatersStarting from rest, two skaterspush off against each other onice where friction is negligible.One is a 54-kg woman andone is a 88-kg man. The womanmoves away with a speed of 2.5 m/s. Find the recoil velocityof the man. pTotal ,i 0v2f p pMomentum Conservation: Total ,fTotal ,im1v1f m2 v2f 054 kg ) ( 2.5m s )m1v1f(v2f 1.5m sm288 kgv1f

6.2 The Principle of Conservation of Linear MomentumApplying the Principle of Conservation of Linear Momentum1. Decide which objects are included in the system.2. Relative to the system, identify the internal and external forces.3. Verify that the system is isolated.4. Set the final momentum of the system equal to its initial momentum.Remember that momentum is a vector.

6.2 Collisions with the EarthThe total linear momentum is conserved when two objectscollide, provided they constitute an isolated system.In this collision with the earth, the ball alone is notan isolated system. The ball’s y-component of momentumchanges in the collision from p to p.Ball’s momentum isNOT conserved p ball ,i p p ball ,f p p ball ,i p p ball ,f pIn the collision with the earth, the ball and the earthconstitute an isolated system. After collision, what isthe y-component of momentum for the earth? initial: pTotal ,i p ball ,i p earth,i p 0 final: pTotal ,f p ball ,f p earth,f p p earth,fMomentum Conservation p 2 ppTotal ,i pTotal ,fEarth,f p p p earth,f Due to the large mass of the earth, this momentump earth,f 2 presults in an imperceptible change in the earth’s velocity.

6.2 Relationship between momentum and kinetic energymagnitude of momentum p mvp 2 m2 v 2 ;p 2 1 m2 v 2 1 2 2 2 mv2mmkinetic energyp2K 2mCompare the kinetic energies of a car, mC 2.00 103 kg, vC 30.0m/s,And the earth, mE 6.00 1024 kg , with the same momentum as the car.pE pC mC vC (2.00 103 kg)(30.0m/s) 6.00 104 kg m/spC2(6.00 104 kg m/s)25KC 9.00 10J32mC4.00 10 kgpE2(6.00 104 kg m/s)2 16KE 6.00 10J zero242mE6.00 10 kgThe Earth can absorb a significant momentum, but it absorbs zero kinetic energy.

6.2 Collisions in One DimensionOn ice, a puck hits a wall. The speed of puck hitting the wall andthe speed coming off the wall are measured to be the same.Clicker Question 6.4Which statement below is true about this collision?a) momentum of the puck is conservedb) the system consists of the puck before and after the collisionc)kinetic energy is not conserved in the collisiond)total energy is not conserved in the collisione)momentum is conserved in a system containing the earth and puck.

6.2 Elastic and Inelastic collisionsElastic collision -- One in which the total kineticenergy of the system after the collision is equal tothe total kinetic energy before the collision.Inelastic collision -- One in which the total kineticenergy of the system after the collision is not equalto the total kinetic energy before the collision; if theobjects stick together after colliding, the collision issaid to be completely inelastic.

6.3 Collisions in One DimensionElastic collisions (x components of the velocities)Equal masses:FinalInitialv1imm(m1 m2 m) masses cancel in both equationsMomentum conservation: v1i v1f v2fEnergy conservation:v1f 0mv2i 0v v v21i21f22fv2f v2imv1f2 v 22f 2v1f v2f v1f2 v 22f2v1f v2f 0Incoming mass stops, target mass gets initial momentumv1f 0;v2f v1i

6.3 Collisions in One DimensionElastic collisionsEqual masses:FinalInitialv1imm(m1 m2 m) masses cancel in both equationsMomentum conservation: v1i v1f v2fEnergy conservation:v v v21i21f22fv2f v2iv1f 0mv2i 0mv1f2 v 22f 2v1f v2f v1f2 v 22f2v1f v2f 0v1f 0;v2f v1iIncoming mass stops, target mass gets initial momentumFinalInitialv1iUnequal masses:v2i 0m2m1Momentum conservation:Energy conservation:12v1fv2fm1m2mm1v1i m1v1f m2 v2f v1i2 v1f2 2 m21 v1f v2f m1v1i2 12 m1v1f2 12 m2 v 22f v1i2 v1f2 m2m1v 22f( )vm2m1222f

6.3 Collisions in One DimensionElastic collisions with arbitrary masses (solve for final velocities)v1iUnequal masses:FinalInitialv2i 0m2m1v1fv2fmm2Momentum conservation:Energy conservation:(1) (2)m2m112m1v1i m1v1f m2 v2f m m v1i2 v1f2 2 2 v1f v2f 2 v 22fm1 m1 m1v1i2 12 m1v1f2 12 m2 v 22f v1i2 v1f2 v 222fm2m1v1f v2f v 22f 2v1f v2f v2f 2v1f m2m1m2m1( )vm2m12m2 2v 2fm1(2)22fv 22fv2fmm1 m2 vv1f 12 1 m21 v2f 2m1 2f 2m1 v2f v1im m 12 (3)Momentum conservation with (3) v v m2 v v m2 2m1 vm1 2fm1 m1 m2 1i1i1f1f(2m)v1f 1 m1 m2 2 v1i m m2 v1f 1 v1im m 12 (1)(4)

6.3 Collisions in One DimensionElastic collisions with arbitrary massesm1 m2v1f v1im1 m22m1and v2f v1im1 m2Equal masssolution is here too

6.3 Collisions in One DimensionElastic collisions with arbitrary massesm1 m2v1f v1im1 m2Little masshits big massm2 m1Let m1 02m1and v2f v1im1 m2Initialv1im1(x components of the velocities)m2Bounces backv1f v1im1v1f v1iFinalv2f 0m2and v2f 0

6.3 Collisions in One DimensionElastic collisions with arbitrary massesm1 m2v1f v1im1 m2Little masshits big massm2 m1Initialv1im1 m2Let m1 m2m1Bounces backLet m1 0Big masshits little mass2m1and v2f v1im1 m2v1iv1f v1iv1f v1iv1im1m2Small mass ejectedat twice the speedv2f 0Finalm2m1FinalInitialEqual masssolution is here toov1f v1iand v2f 0v1im1v2f 2v1im2and v2f 2v1i

Collisions involve two new concepts: Impulse and Momentum. Impulse concept leads to the Momentum definition. Also applied to two (or more) masses blown apart by an explosion. 6.1 The Impulse-Momentum Theorem . The total linear momentum of an isolated system is constant (conserved). An isolated system is one for which the sum of

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