L9 Momentum PHYS101 - UNIVERSE OF ALI OVGUN
Physics 101Lecture 9Linear Momentum andCollisionsDr. Ali ÖVGÜNEMU Physics Departmentwww.aovgun.com
Linear Momentumand CollisionsqqqqqqqConservationof EnergyMomentumImpulseConservationof Momentum1-D Collisions2-D CollisionsThe Center of MassFebruary 13, 2017
Conservation of EnergyqqΔ E Δ K Δ U 0 if conservative forces are the onlyforces that do work on the system.The total amount of energy in the system is constant.1 2111mv f mgy f kx 2f mvi2 mgyi kxi22222qqΔ E Δ K Δ U -fkd if friction forces are doing workon the system.The total amount of energy in the system is stillconstant, but the change in mechanical energy goesinto “internal energy” or heat.11 1 1 f k d mv 2f mgy f kx 2f mvi2 mgyi kxi2 22 2 2 February 13, 2017
Linear MomentumqqThis is a new fundamental quantity, like force, energy.It is a vector quantity (points in same direction asvelocity).The linear momentum p of an object of mass m movingwith a velocity v is defined to be the product of themass and velocity:!!p mvqqThe terms momentum and linear momentum will beused interchangeably in the textMomentum depend on an object’s mass and velocityFebruary 13, 2017
Linear Momentumrrq Linear momentum is a vector quantity p mvIts direction is the same as the direction ofthe velocityq The dimensions of momentum are ML/Tq The SI units of momentum are kg m / sq Momentum can be expressed in componentform:px mvx py mvy pz mvznFebruary 13, 2017
Newton’s Law andMomentumqNewton’s Second Law can be used to relate themomentum of an object to the resultant forceacting on it!!!!Δv Δ(mv )Fnet ma m ΔtΔtqThe change in an object’s momentum divided bythe elapsed time equals the constant net forceacting on the object!Δp change in momentum ! FnetΔttime intervalFebruary 13, 2017
ImpulseqWhen a single, constant force acts on theobject, there is an impulse delivered to the! !objectnnnnI FΔtrI is defined as the impulseThe equality is true even if the force is not constantVector quantity, the direction is the same as thedirection of the force!Δp change in momentum ! FnetΔttime intervalFebruary 13, 2017
Impulse-MomentumTheoremqThe theorem statesthat the impulseacting on a system isequal to the changein momentum of thesystem!! !Δp Fnet Δt I!!!!I Δp mv f mviFebruary 13, 2017
Calculating the Change ofMomentumr rrΔp pafter pbefore mvafter mvbefore m(vafter vbefore )For the teddy bearΔp m [0 ( v)] mvFor the bouncing ballΔp m [v ( v)] 2mvFebruary 13, 2017
Ex1: How Good Are the Bumpers?In a crash test, a car of mass 1.5 103 kg collides with a wall andrebounds as in figure. The initial and final velocities of the car are vi -15m/s and vf 2.6 m/s, respectively. If the collision lasts for 0.15 s, find(a) the impulse delivered to the car due to the collision(b) the size and direction of the average force exerted on the carqFebruary 13, 2017
How Good Are the Bumpers?In a crash test, a car of mass 1.5 103 kg collides with a wall andrebounds as in figure. The initial and final velocities of the car are vi -15m/s and vf 2.6 m/s, respectively. If the collision lasts for 0.15 s, find(a) the impulse delivered to the car due to the collision(b) the size and direction of the average force exerted on the carqpi mvi (1.5 10 3 kg )( 15m / s ) 2.25 10 4 kg m / sp f mv f (1.5 103 kg )( 2.6m / s) 0.39 10 4 kg m / sI p f pi mv f mvi (0.39 10 4 kg m / s) ( 2.25 10 4 kg m / s) 2.64 10 4 kg m / sΔp I 2.64 10 4 kg m / sFav 1.76 105 NΔt Δt0.15sFebruary 13, 2017
Ex2: Impulse-MomentumTheoremqA child bounces a 100 g superball on thesidewalk. The velocity of the superballchanges from 10 m/s downward to 10 m/supward. If the contact time with thesidewalk is 0.1s, what is the magnitude ofthe impulse imparted to the superball?(A)(B)(C)(D)(E)02 kg-m/s20 kg-m/s200 kg-m/s2000 kg-m/s!!!!I Δp mv f mviFebruary 13, 2017
Ex3: Impulse-MomentumTheorem 2qA child bounces a 100 g superball on thesidewalk. The velocity of the superballchanges from 10 m/s downward to 10 m/supward. If the contact time with thesidewalk is 0.1s, what is the magnitude ofthe force between the sidewalk and thesuperball?(A) 0!!!!! IΔp mv f mvi(B) 2 NF (C) 20 NΔt ΔtΔt(D) 200 N(E) 2000 NFebruary 13, 2017
Conservation of MomentumqIn an isolated and closed system,the total momentum of thesystem remains constant in time.nnnnIsolated system: no external forcesClosed system: no mass enters orleavesThe linear momentum of eachcolliding body may changeThe total momentum P of thesystem cannot change.February 13, 2017
Conservation of MomentumqStart from impulse-momentumtheorem!!!F21Δt m1v1 f m1v1i!!!F12 Δt m2v2 f m2v2i!!F21Δt F12 ΔtqSinceq!!!!Then m1v1 f m1v1i (m2 v2 f m2 v2i )qSo!!!!m1v1i m2 v2i m1v1 f m2 v2 fFebruary 13, 2017
Conservation of MomentumqqqqWhen no external forces act on a system consisting oftwo objects that collide with each other, the totalmomentum of the system remains constant in time!! !!Fnet Δt Δp p f pi!!When Fnet 0 then Δp 0For an isolated system!!p f piSpecifically, the total momentum before the collision willequal the total momentum after the collision!!!!m1v1i m2 v2i m1v1 f m2 v2 fFebruary 13, 2017
Ex4: The ArcherAn archer stands at rest on frictionless ice and fires a 0.5-kg arrowhorizontally at 50.0 m/s. The combined mass of the archer and bow is60.0 kg. With what velocity does the archer move across the ice afterfiring the arrow?qpi p fm1v1i m2 v2i m1v1 f m2 v2 fm1 60.0kg , m2 0.5kg , v1i v2i 0, v2 f 50 m / s, v1 f ?0 m1v1 f m2 v2 fm20.5kgv1 f v2 f (50.0m / s) 0.417m / sm160.0kgFebruary 13, 2017
Ex5: Conservation ofMomentumqA 100 kg man and 50 kg woman on iceskates stand facing each other. If the womanpushes the man backwards so that his finalspeed is 1 m/s, at what speed does she recoil?(A) 0(B) 0.5 m/s(C) 1 m/s(D) 1.414 m/s(E) 2 m/sFebruary 13, 2017
Types of CollisionsMomentum is conserved in any collisionq Inelastic collisions: rubber ball and hard ballqnnqKinetic energy is not conservedPerfectly inelastic collisions occur when the objectsstick togetherElastic collisions: billiard ballnboth momentum and kinetic energy are conservedFebruary 13, 2017
Collisions SummaryqqqqqIn an elastic collision, both momentum and kineticenergy are conservedIn a non-perfect inelastic collision, momentum isconserved but kinetic energy is not. Moreover, theobjects do not stick togetherIn a perfectly inelastic collision, momentum is conserved,kinetic energy is not, and the two objects stick togetherafter the collision, so their final velocities are the sameElastic and perfectly inelastic collisions are limiting cases,most actual collisions fall in between these two typesMomentum is conserved in all collisionsFebruary 13, 2017
More about Perfectly InelasticCollisionsqqWhen two objects stick togetherafter the collision, they haveundergone a perfectly inelasticcollisionConservation of momentumm1v1i m2 v2i (m1 m2 )v fm1v1i m2 v2ivf m1 m2qKinetic energy is NOT conservedFebruary 13, 2017
Ex6: An SUV Versus a CompactqAn SUV with mass 1.80 103 kg is travelling eastboundat 15.0 m/s, while a compact car with mass 9.00 102kg is travelling westbound at -15.0 m/s. The cars collidehead-on, becoming entangled.(a)(b)(c)Find the speed of the entangledcars after the collision.Find the change in the velocityof each car.Find the change in the kineticenergy of the system consistingof both cars.February 13, 2017
An SUV Versus a Compact(a)Find the speed of the entangled m 1.80 103 kg , v 15m / s11icars after the collision.2m2 9.00 10 kg , v2i 15m / spi p fm1v1i m2 v2i (m1 m2 )v fm1v1i m2 v2ivf m1 m2v f 5.00 m / sFebruary 13, 2017
An SUV Versus a Compact(b)Find the change in the velocityof each car.v f 5.00 m / sm1 1.80 10 3 kg , v1i 15m / sm2 9.00 10 2 kg , v2i 15m / sΔv1 v f v1i 10.0m / sΔv2 v f v2i 20.0m / sm1Δv1 m1 (v f v1i ) 1.8 10 4 kg m / sm2 Δv2 m2 (v f v2i ) 1.8 10 4 kg m / sm1Δv1 m2Δv2 0February 13, 2017
An SUV Versus a Compact(c)Find the change in the kinetic3m 1.80 10kg , v1i 15m / senergy of the system consisting 1m2 9.00 10 2 kg , v2i 15m / sof both cars.v f 5.00 m / s112KEi m1v1i m2v22i 3.04 10 5 J22112KE f m1v1 f m2v22 f 3.38 10 4 J22ΔKE KE f KEi 2.70 105 JFebruary 13, 2017
More About ElasticCollisionsqBoth momentum and kinetic energyare conservedm1v1i m2 v2i m1v1 f m2 v2 f1111222m1v1i m2 v2i m1v1 f m2 v22 f2222qqTypically have two unknownsMomentum is a vector quantitynnqDirection is importantBe sure to have the correct signsSolve the equations simultaneouslyFebruary 13, 2017
Elastic CollisionsqA simpler equation can be used in place of the KEequation1111222m1v1i m2 v2i m1v1 f m2 v22 f2222m1 (v12i v12f ) m2 (v22 f v22i )v v ( v v )m1 (v11i i v1 f )( v21ii v1 f ) m21(fv2 f v22i )(f v2 f v2i )m1v1i m2 v2i m1v1 f m2 v2 fm1 (v1i v1 f ) m2 (v2 f v2i )v1i v1 f v2 f v2im1v1i m2 v2i m1v1 f m2 v2 fFebruary 13, 2017
Summary of Types ofCollisionsq In an elastic collision, both momentum and kineticenergy are conservedv1i v1 f v2 f v2iqm1v1i m2 v2i m1v1 f m2 v2 fIn an inelastic collision, momentum is conserved butkinetic energy is notm1v1i m2 v2i m1v1 f m2 v2 fqIn a perfectly inelastic collision, momentum is conserved,kinetic energy is not, and the two objects stick togetherafter the collision, so their final velocities are the samem1v1i m2 v2i (m1 m2 )v fFebruary 13, 2017
Ex7: Conservation ofMomentumqAn object of mass m moves to the right with aspeed v. It collides head-on with an object ofmass 3m moving with speed v/3 in the oppositedirection. If the two objects stick together, whatis the speed of the combined object, of mass 4m,after the collision?(A)(B)(C)(D)(E)0v/2v2v4vFebruary 13, 2017
Problem Solving for 1DCollisions, 1qCoordinates: Set up acoordinate axis and definethe velocities with respectto this axisnqIt is convenient to makeyour axis coincide with oneof the initial velocitiesDiagram: In your sketch,draw all the velocityvectors and label thevelocities and the massesFebruary 13, 2017
Problem Solving for 1DCollisions, 2qConservation ofMomentum: Write ageneral expression for thetotal momentum of thesystem before and afterthe collisionnnEquate the two totalmomentum expressionsFill in the known valuesm1v1i m2 v2i m1v1 f m2 v2 fFebruary 13, 2017
Problem Solving for 1DCollisions, 3qConservation of Energy:If the collision is elastic,write a second equationfor conservation of KE, orthe alternative equationnThis only applies to perfectlyelastic collisionsv1i v1 f v2 f v2iqSolve: the resultingequations simultaneouslyFebruary 13, 2017
One-Dimension vs TwoDimensionFebruary 13, 2017
Two-Dimensional CollisionsqFor a general collision of two objects in twodimensional space, the conservation of momentumprinciple implies that the total momentum of thesystem in each direction is conservedm1v1ix m2 v2ix m1v1 fx m2 v2 fxm1v1iy m2v2iy m1v1 fy m2 v2 fyFebruary 13, 2017
Two-Dimensional CollisionsThe momentum is conserved in all directionsm1v1ix m2 v2ix m1v1 fx m2 v2 fxq Use subscripts forqnnnqIdentifying the objectm1v1iy m2v2iy m1v1 fy m2 v2 fyIndicating initial or final valuesThe velocity componentsIf the collision is elastic, use conservation ofkinetic energy as a second equationnRemember, the simpler equation can only be usedfor one-dimensional situationsv1i v1 f v2 f v2iFebruary 13, 2017
Glancing CollisionsqqqThe “after” velocities have x and y componentsMomentum is conserved in the x direction and in they directionApply conservation of momentum separately to eachdirectionmv m v mv m v1 1ix2 2 ix1 1 fx2 2 fxm1v1iy m2v2iy m1v1 fy m2 v2 fyFebruary 13, 2017
2-D Collision, exampleParticle 1r is moving atvelocity v1i andparticle 2 is at restq In the x-direction, theinitial momentum ism1v1iq In the y-direction, theinitial momentum is 0qFebruary 13, 2017
2-D Collision, example contqqAfter the collision, themomentum in the x-direction ism1v1f cos θ m2v2f cos φAfter the collision, themomentum in the y-direction ism1v1f sin θ m2v2f sin φm1v1i 0 m1v1 f cos θ m2v2 f cos φ0 0 m1v1 f sin θ m2 v2 f sin φqIf the collision is elastic, applythe kinetic energy equation111m1v12i m1v12f m2 v22 f222February 13, 2017
Ex8: Collision at anIntersectionA car with mass 1.5 103 kg travelingeast at a speed of 25 m/s collides atan intersection with a 2.5 103 kg vantraveling north at a speed of 20 m/s.Find the magnitude and direction ofthe velocity of the wreckage after thecollision, assuming that the vehiclesundergo a perfectly inelastic collisionand assuming that friction between thevehicles and the road can beneglected.qmc 1.5 103 kg , mv 2.5 103 kgvcix 25m / s, vviy 20m / s, v f ?θ ?February 13, 2017
Collision at an Intersectionmc 1.5 103 kg, mv 2.5 103 kgvcix 25 m/s, vviy 20 m/s, v f ?θ ? p pxi mc vcix mv vvix mc vcix 3.75 10 4 kg m/sxf mc vcfx mv vvfx (mc mv )v f cosθ3.75 10 4 kg m/s (4.00 103 kg)v f cosθ4p mv mv mv 5.00 10kg m/s yi c ciy v viy v viy pyf mc vcfy mv vvfy (mc mv )v f sin θ5.00 10 4 kg m/s (4.00 103 kg)v f sin θFebruary 13, 2017
Collision at an Intersectionmc 1.5 103 kg , mv 2.5 103 kgvcix 25m / s, vviy 20m / s, v f ?θ ?5.00 10 4 kg m/s (4.00 103 kg)v f sin θ3.75 10 4 kg m/s (4.00 103 kg)v f cosθ5.00 10 4 kg m / stan θ 1.3343.75 10 kg m / sθ tan 1 (1.33) 53.1!5.00 10 4 kg m/svf 15.6 m/s3!(4.00 10 kg ) sin 53.1February 13, 2017
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Linear Momentum and Collisions Dr. Ali ÖVGÜN EMU Physics Department www.aovgun.com. February 13, 2017 Linear Momentum and Collisions q Conservation of Energy q Momentum . Ex3: Impulse-Momentum Theorem 2 q A child bounces a 100 g superball on the sidewalk. The velocity of the superball changes from 10 m/s downward to 10 m/s
CHAPTER 3 MOMENTUM AND IMPULSE prepared by Yew Sze Ling@Fiona, KML 2 3.1 Momentum and Impulse Momentum The linear momentum (or "momentum" for short) of an object is defined as the product of its mass and its velocity. p mv & & SI unit of momentum: kgms-1 or Ns Momentum is vector quantity that has the same direction as the velocity.
Momentum ANSWER KEY AP Review 1/29/2018 Momentum-1 Bertrand Momentum How hard it is to stop a moving object. Related to both mass and velocity. For one particle p mv For a system of multiple particles P p i m i v Units: N s or kg m/s Momentum is a vector! Problem: Momentum (1998) 43. The magnitude of the momentum of the
momentum is kg·m/s. Linear Momentum Linear momentum is defined as the product of a system's mass multiplied by its velocity: p mv. (8.2) Example 8.1Calculating Momentum: A Football Player and a Football (a) Calculate the momentum of a 110-kg football player running at 8.00 m/s. (b) Compare the player's momentum with the
1. Impulse and Momentum: You should understand impulse and linear momentum so you can: a. Relate mass, velocity, and linear momentum for a moving body, and calculate the total linear momentum of a system of bodies. Just use the good old momentum equation. b. Relate impulse to the change in linear momentum and the average force acting on a body.
Momentum: Unit 1 Notes Level 1: Introduction to Momentum The Definition Momentum is a word we sometime use in everyday language. When we say someone has a lot of momentum, it means they are on a roll, difficult to stop, really moving forward. In physics, momentum means "mass in motion". The more mass an object has, the more momentum it has.
Momentum (p mv)is a vector, so it always depends on direction. Sometimes momentum is if velocity is in the direction and sometimes momentum is if the velocity is in the direction. Two balls with the same mass and speed have the same kinetic energy but opposite momentum. Momentum vs. Kinetic Energy A B Kinetic Energy Momentum
momentum), it is optimal for investors to hold less or may short the momentum asset and hence sufier less or even beneflt from momentum crashes. Key words: Portfolio selection, momentum crashes, dynamic optimal momentum strategy. JEL Classiflcation: C32, G11 Date: March 11, 2016. 1
“ASTM D 4255/D 4255M The standard test method for in-plane shear properties of polymer matrix composite materials by the rail shear method ”. This set-up, however, requires drilling holes through the specimen. In this study, a new design based on friction and geometrical gripping, without the need of drilling holes through the composite specimen is presented. Quasi-static tests have been .
