Momentum And Collisions - University Of Western Ontario

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Force, Time and Momentum We now consider the time for which a forceacts. This leads to the concept of momentum.rrp mv5/14/2008Momentum - 12

Flea m 4.5 x 10-7 kg v 1.0 m/s p 4.5 x 10-7 kg·m/s5/14/2008Momentum - 13

Linear Momentum 1 Linear Momentum Mass x VelocityMassVelocity 0.145 kg 44 m/sMomentum 6.44 kg·m/s5/14/2008Momentum - 14

Linear Momentum 5 Linear Momentum is avector. It has magnitude. It has direction.Are the momenta the same?5/14/2008Momentum - 15

Two MassesBrian Wettlaufer 2001Tyrone Williams 1989 Two football players, m1, m2,have the same linear momentum.Which has the larger kinetic energy?5/14/2008Momentum - 16

m1v1 m2 v2v1 m2 v2 m1121221 122 2mvK1 K2mv.m1 is the lighter playerThe lighterplayer hasthe greaterKE!5/14/2008m1v1 v1 m2 v2 v2m2 1m1Momentum - 17

Change in MomentumV 0mmInitial momentum mvFinal momentum 0Change in momentum Final momentum – Initial momentum (0) - ( mv) - mv5/14/2008Momentum - 18

Change in Momentum Throw a rubber ball at the wall. What is the change in momentum? V-VInitial momentum mvFinal momentum - mvChange in momentum pf-pi ml5/14/2008Momentum - 19

Force, Time and MomentumrFext macmrrdvFext dt m dtdtrr Fext dt mdvrr mv f mvirrr Fext dt p f pi5/14/2008Momentum - 110

Force, Time and MomentumrrrFdt p pfi A force acting over time causes a change in momentum. The area under the force-time graph equals the change inlinear momentum. We can change momentum by exerting a little force for along time, or a big force for a short time.5/14/2008Momentum - 111

Collision 1FA on BFB on AABIn a collision, cart A exerts a force on Cart B, and CartB exerts a force on cart A. How are these twoforces related?(A) FA on B FB on A(B) FA on B FB on A(C) FA on B FB on A5/14/2008Momentum - 112

Collision 2FA on BFB on AABIs the time that FB on A acts(A) longer than(B) the same(C) less than5/14/2008the time FA on B acts?Momentum - 113

Collisions 3FA on BFB on AABWhat can we say about the (force timestime for cart A) and the (force timestime for cart B)?5/14/2008Momentum - 114

Collisions 4FA on BFB on AABForce exerted on B x time it acts -(force exerted on car A time it acts)Change in linear momentum of B -(change in linear momentum of A)If the momentum of B goes up 3 units of momentum,the momentum of A goes down 3 units of momentum.In a collision,Linear Momentum is Conserved (LMIC)5/14/2008Momentum - 115

Collisions 5FA on BFB on AABTake the two cars as the system: Fext 0In a collision,Linear Momentum Is Conserved.One of the most famous conservation laws in Physics.It holds always, even in the realm of the very fast or very smallwhere Newton’s Laws fail.5/14/2008Momentum - 116

Decision Steps for Collision ProblemsPictorial Model Diagram Frame of Reference– x 0, y 0, U 05/14/2008Momentum - 117

Physical Model Identify the system Free Body diagramMathematical ModelCollisionLinear momentum is always conservedWe will expand this model later.5/14/2008Momentum - 118

Example A boy, M 70 kg, and a girl m 35 kg, onskates push off each other. The girl goes East at5m/s. Describe the motion of the boy.Pictorial ModelBoyGirl y x5/14/2008Momentum - 119

Physical ModelInitial Statepi 05/14/2008Final StatepB -MvBMomentum - 1pA mvA20

Mathematical Modelrrpi p fp x ,i p x , f0 mvG MvBmvGvB M 2.5m / s5/14/2008Momentum - 121

Summaryrrp mv Linear Momentum Change in Momemtum Relation between force, time and linearmomentum:rrrFdt p pfi Linear Momentum is conserved in collisions5/14/2008Momentum - 122

Conservation of MomentumrrrFdt p pfi Area under F vs tChange of MomentumMomentum is conserved when Area under F vs t is zero.Why is this the case in Collisions and Explosions?Area under F vs t is called Impulse5/14/2008Momentum - 123

Summaryrrp mv1. Momentum2. Area3. This area is called Impulse4. Impulse - Momentum5/14/2008rF.dt rF.dt rrrFdt p pfi Momentum - 124

Force, Time and MomentumrrrFdt p pfi The area under the force-time graph equals thechange in linear momentum. If the LHS of thisequation is zero, then: Linear Momentum Is Conserved (LMIC)5/14/2008Momentum - 125

Rocket ExampleWorked problem: A space vehicle is traveling inspace and the exhausted rocket motor isdisengaged. Determine the speed of the nosecone with respect to the earth after separation.Lets work out this problem.5/14/2008Momentum - 126

Kinetic Energy & Collisions 1: Elastic Collisions No kinetic energy is lost. The contact forces are springs, gravity orelectromagnetic. Demonstration: Air cars with springs.5/14/2008Momentum - 127

Inelastic Collisions Some kinetic energy is lost. The contact forcesare not springs. If the cars stick together, the maximum possiblekinetic energy is lost Demonstration: Air cars with putty.5/14/2008Momentum - 128

Physical Model Identify three stages:– Before the collision– Collision– After the collision Free Body diagram Is the collision force a spring, gravity orelectromagnetic?– If yes, Kinetic Energy is conserved.5/14/2008Momentum - 129

Mathematical Model Pre-collision. – Apply Physics Collision– Linear momentum is always conserved– Is the contact force gravity or a spring?– If YES then Kinetic Energy is conserved Post-collision – Apply Physics Solve equations5/14/2008Momentum - 130

Pictorial ModelsCollision xTwo carts of equal mass m 2.0 kg collide, and stick together.V1i 0.5 m/s; v2i 0.Find the final speed. Now we’ll solve this problem.5/14/2008Momentum - 131

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Summary Collisions– Linear Momentum is always conserved Is the collision force gravity or springs?––––If YES, then Kinetic Energy K is conserved also.Elastic – K is conservedInelastic – K is not conserved.Perfectly Inelastic – maximum K is lost5/14/2008Momentum - 133

Elastic CollisionsPictorial ntum - 134

Elastic CollisionsPhysical ModelAfterBeforev1iv2iv1fv2fm1m2m1m2System is the two blocksrLinearMomentumisconservedF 0 extCollision is via a spring – KE is conserved5/14/2008Momentum - 135

Elastic CollisionsMathematical ModelLinear Momentum is conserved:m1v 1i m2v 2i m1v1f m2v 2 fKinetic Energy is conserved:125/14/200821 1i1222 2i1221 1f1222 2fm v m v mv m vMomentum - 136

Elastic CollisionsMathematical ModelRe-arrange the two equations:m1(v1i v1 f ) m2 (v2 f v2i )21i21f22f(1)22fm1(v v ) m2 (v v )m1(v1i v1 f )(v1i v1 f ) m2 (v2 f v2i )(v2 f v2i )(2)Divide (2) by (1)v1i v1 f v2 f v2i5/14/2008Momentum - 1(3)37

Mathematical ModelElastic CollisionsSolve (1) and (3)v1 f2m2m1 m 2 v1i v2im1 m 2m1 m 2v2 f2 m1m 2 m1 v1i v2im1 m 2m1 m 2You do not have to remember these equations5/14/2008Momentum - 138

Weighted Average Cookies in the StoreBags Mass100.400150.405200.4105/14/2008Weighted Average:10 0.400 15 0.405 20 0.410WA 10 15 20 0.406 kgMomentum - 139

Centre of Massy10 kg20 kgx1xcm5/14/20082310 2.0 20 3.0 10 20 2.66 mMomentum - 140

Centre of Massym2m1m3xx1xcm5/14/2008x2x3m1 x1 m2 x2 m3 x3 . m1 m2 m3 .Momentum - 141

yTwo Dimensions24kg15kg3kgx123xcm3 0 4 1 5 2 1.17 m3 4 5ycm3 0 4 1 5 0 0.33 m3 4 55/14/2008Momentum - 142

Motion of the Centre of Mass.The motion of the centre of mass is special.Starting from N2 for a point particle, we can showthat for any system Fext MacmThe proof of this is in the text in section 8.6.We are not going over this in detail.5/14/2008Momentum - 143

Centre of Mass The Centre of Massmoves in a parabola5/14/2008Momentum - 144

4. Impulse - Momentum 3. This area is called Impulse F.dt r. 5/14/2008 Momentum - 1 25 Force, Time and Momentum The area under the force-time graph equals the change in linear momentum. If the LHS of this . Collisions - Linear Momentum is always conserved Is the collision force gravity or springs? - If YES, then Kinetic .