Today’s Topic: IMPULSE AND MOMENTUM CONSERVATION

Today’s topic:IMPULSE AND MOMENTUM CONSERVATION

Review of Last Week’s Lecture Elastic Potential Energy:x: displacementfrom equilibriumx 0: equilibrium position Work-Energy Theorem:Wtot Wg Wel Wnon cons K K 2 K1Wnon cons ( K 2 U g 2 U el2 ) ( K1 U g1 U el1 )Let U U g U el Potential EnergyE K U Wnon cons E2 E1Wnon cons 0 Total Mechanical Energy(friction, drag, work done by muscles, etc.)E2 E1Conservation of mechanical energyU (x) Force and Potential Energy:Fx dUdxdUdxdUdxx

Skate Park Animation

Slow motion video of last week’s spring launch: What does the spring do, other than shooting up(and falling down)? Does vibrational/rotational motion store energy? What kind of energy? Did you account for this energy in last weeksworkshop? Only about 2% of the total energy in vibration, muchless in rotation!

Rotational State PopulationsSpring launch may serve as model for molecules“desorbing” (i.e., detaching) from a surface:F. M. Zimmermann and W. Ho, Surface ScienceReports 22, 127-247 (1995).

A spring can vibrate in many “normal modes”:The higher the number of“nodes”, the greater thevibrational frequency.True not only for springs, but any solid!Do these modes continue to infinity (infinite # ofnodes)?No, wavelength is limited by spacing between atoms:Lattice Vibrationsor“Phonon Modes”

Use femtosecond laser spectroscopy to measure phononvibrations in LuMnO3 crystal:Reflectivitychange x 10-5Pump-Probe Delay (picoseconds)S. Lou, F. M. Zimmermann, R. A. Bartynski, N. Hur, andS. Cheong, Physical Review B 79, 214301 (2010).

MOMENTUM & IMPULSENEWTON’S2nd Law: F maWrite differently: dv d F m (mv )dt dt (Units: kg m/s N s)p mvDefine Momentum: dp F dtNet force Rate of change of momentumConsider this relationship further: dp F dp ( F )dtdtp2t2 dp F dt p2 p1p1t1Define Impulse: t2 J ( F )dt p2 p1t1Vector that equalschange in momentum

Had: Work-Energy Theorem , now have: Impulse – Momentum Theorem: J p2 p1Consider a variable force acting on an objectfrom time t1 to t2 (e.g., basketball dribble)t2F (t )J F (t ) dtt1Ft1t2Integral of actual force fromaverage force times intervaltJ F (t2 t1 )t1 to t 2 is equal to t t2 t1COMPARISON:MOMENTUM vs. KINETIC ENERGY: p is a vectorp v;KE is a scalarKE v 2 p related to time over which force acts !KErelated to distance over which force acts !

i-ClickerA 10-kg box, initially at rest, moves along a frictionlesshorizontal surface. A horizontal force to the right is appliedto the box. The magnitude of the force changes as afunction of time as shown.A. The impulse in the first 2 seconds is 2 kg·m/sB. The impulse from 5 seconds to 8 seconds is -6kg·m/sC. The impulse in the first 2 seconds is 1 kg·m/sD. The impulse from 2 seconds to 5 seconds is 0kg·m/sE. The impulse cannot be determines with theinformation given

i-ClickerE. I want 10 points subtracted from my grade J p2 p1 J 0 p1

i-ClickerA 2-kg object accelerates in response to anapplied force. During the 5-second interval thatthe force is applied, the object’s velocity changesfrom 3 m/s east to 7 m/s west. Which is trueabout the magnitude of the impulse?A. It equals 20 kg·m/sB. It equals 8 kg·m/sC. It equals 8/5 kg·m/sD. It equals 4 kg·m/sE. It cannot be found with the informationgiven.

i-ClickerIn Case A, a metal bullet penetrates a wooden block. InCase B, a rubber bullet with the same initial speed andmass bounces off of an identical wooden block.Will the speed of the wooden block after the collisionbe greater in Case A, greater in Case B, or the same inboth cases?A. The speed will be greater in Case A because themetal bullet exerts a larger force on the block.B. The speed will be greater in Case B because thebullet changes direction.C. The speed will be the same in both cases becausethe bullets have the same mass and initial speedand give the block the same momentum.D. Cannot be determined.

CONSERVATION OF LINEAR MOMENTUMConsider two isolated objects that interact only by theirmutual force.(No netexternal force) FA on B FB on A FB on A FA on BAB(Newton’s 3rd Law) FA on B FB on A 0But dp AFB on A dtSo: dpBFA on B dt dp A dpB d ( p A pB ) 0dtdtdtFor isolated system (no external forces) total linearmomentum of the system is constant: P p A pB constantCONSERVATION OF LINEAR MOMENTUM

Conservation of momentum is valid for any number ofparticles interacting only with each other(No External Forces) P i piIs a vector quantity that is conservedEXAMPLE - physics of hockey:A Ranger and a Devils hockey player are fighting on theice. The Devils player (M 100 kg) throws a punch thatsends the Ranger (m 80 kg) off atvR 0.5 m/s.What is the speed of the Devils player,P pR pDPi PfvD ?vR fpRi 0; pDi 0 Pi 0 Pf 0 pR f pD f 0pR f (80 kg)( 0.5 m/s ); pD f (100 kg)vD f ( 40 kg m/s ) (100 kg)vD f 0 vD f(40 kg m/s ) 0.4 m/s(100 kg)vD f

i-ClickerTwo boxes are tied together by a string and are sitting atrest in the middle of a large frictionless surface. Betweenthe two boxes is a massless compressed spring. Thestring tying the two boxes together is cut suddenly andthe spring expands, pushing the boxes apart. The box onthe left has four times the mass of the box on the right.At the instant (after the string is cut) that the boxes losecontact with the spring, the speed of the box on the leftwill be A.)B.)C.)D.)Greater than the right boxLess than the right boxEqual to the right boxNot enough information provided

MOMENTUM CONSERVATION AND COLLISIONSCollision: Brief, strong interaction between objects. F FIfbetween objects, Neglect Fexti behaves as an isolated systemext PF i piF PI i piITotal momentum just after collision Total momentum just before collision Classify Collisions: Elastic Collision Total momentum andTotal kinetic energy conserved Inelastic Collision Momentum conservedKE is not (lost to internal energy) Completely Inelastic Collision Momentum conserved(Objects stick together)KE not. (KE Internal)Momentum conserved in any collisionKE conserved only in elastic collision

EXAMPLE: Completely Inelastic CollisionThe Ballistic Pendulum:v2v1hA bullet (ma , v1 ) is fired into clip of pendulumwhich swings to height h.What is v1 ?TWO PARTS ! Collision is completely inelastic Use PF PI to find state just after collision. PI ma v1 0PF (ma mb )v2 m mB v2v1 A mA Use conservation of mechanical energy:K 2 U g 2 K f U gf12(mA mB )v22 (mA mB ) gh v2 2 gh m mB 2 ghv1 A mA

ELASTIC COLLISION KE and P CONSERVED“Billiard Ball Collision”v AiBeforevBimAmB1-D Collisionalong x-axis(omit subscripts)?After1 PF PI mAv A2 mB vB2 mAv A1 mB vB111112 KEF KEI 2 mAv A 2 2 mB vB 2 2 mAv A1 2 mB vB1222 PAGE OF ALGEBRA 3(vB2 v A2 ) (vB1 v A1 )MAGNITUDE OF RELATIVE VELOCITYUNCHANGED AFTER COLLISION2

EXAMPLE: Pocket The Eight Ballv A1Before collision:mA movingmB at restmAmB1 mAv A1 0 mAv A2 mB vB23 v A1 0 vB2 v A2 vB2 v A22mAm A mBm A mBm A mB v vA1A13 IMPORTANT CASESImB mA v A2 0 ; vB2 v A1II mB mA v A2 v A1 ; vB2 2v A1III mA mB v A2 v A1 ; vB2 0v A1mAmBv A1mAmAmBv A1mB

i-ClickerCarts A and B are shown just before they collide.Which (if any) of the following statements couldpossibly be correct?I. “After the collision, the carts will stick together and moveoff to the left due to Cart B having more speed.”II. “They’ll stick together and move off to the right becauseCart A is heavier.”III. “The speed and the mass compensate. For completelyinelastic collision, both carts are going to be at rest afterthe collision.”IV. “For an elastic collision, they will change theirdirections, so Cart A will be moving to the left at 3 m/s andCart B will be moving to the right at 4 m/s.”A.)C.)E.)IIIIIII & IVB.)D.)IIIV

Collision in two dimensions (horizontal plane)After:Before:Write separate momentum conservation equationsfor components:0Px : mAv Ax1 mB vBx1 mAv Ax2 mB vBx200Py : mAv Ay1 mB vBy1 mAv Ay 2 mB vBy 2If collision is elastic:KE1 KE2Three equations, can solve for a maximum ofthree unknowns:Momentum and energy conservation alone are notsufficient to determine the final state.

i-ClickerTwo identical steel balls, S and T, are shown at the instantthat they collide. The paths and velocities of the two ballsbefore and after the collision are indicated by the dashedlines and arrows. What is the direction of the impulseon ball S?A.B.C.D.E.ACannot be determined without the time t.CNone of the other answers.EpiI Δppf

i-Clicker Completely Inelastic Collision Momentum conserved(Objects stick together)KE not. (KE Internal)

MOMENTUM & IMPULSE NEWTON’S 2nd Law: Write differently: F ma * * 6 (mv) dt d dt dv F m * * * 6 Define Momentum: p mv * * (Units: kg m/s N s) dt dp F * * 6 Net force Rate of change of momentum Consider this relationship further: Define Impulse: Vector that equals change in momentum dp F d