L9 Momentum F1819 - UNIVERSE OF ALI OVGUN
Physics 101Lecture 9Linear Momentum andCollisionsAssist. Prof. Dr. Ali ÖVGÜNEMU Physics Departmentwww.aovgun.com
Linear Momentumand CollisionsqqqqqqqConservationof EnergyMomentumImpulseConservationof Momentum1-D Collisions2-D CollisionsThe Center of MassDecember 18, 2018
Conservation of EnergyqqD E D K D 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 kxi22222qqD E D K D 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.11æ1ö æ1ö- f k d ç mv 2f mgy f kx 2f - ç mvi2 mgyi kxi2 22è2ø è2øDecember 18, 2018
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 velocityDecember 18, 2018
Linear Momentum!!q 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 mvznDecember 18, 2018
Newton’s Law andMomentumqNewton’s Second Law can be used to relate themomentum of an object to the resultant forceacting on it!!!!Dv D(mv )Fnet ma m DtDtqThe change in an object’s momentum divided bythe elapsed time equals the constant net forceacting on the object!Dp change in momentum ! FnetDttime intervalDecember 18, 2018
ImpulseqWhen a single, constant force acts on theobject, there is an impulse delivered to the! !objectnnnnI FDt!I 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!Dp change in momentum ! FnetDttime intervalDecember 18, 2018
Impulse-MomentumTheoremqThe theorem statesthat the impulseacting on a system isequal to the changein momentum of thesystem!! !Dp Fnet Dt I!!!!I Dp mv f - mviDecember 18, 2018
Calculating the Change ofMomentum! !!Dp pafter - pbefore mvafter - mvbefore m(vafter - vbefore )For the teddy bearDp m [0 - (-v)] mvFor the bouncing ballDp m [v - (-v)] 2mvDecember 18, 2018
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 carqDecember 18, 2018
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 / sDp I 2.64 10 4 kg m / sFav 1.76 105 NDt Dt0.15sDecember 18, 2018
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 Dp mv f - mviDecember 18, 2018
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!!!!! IDp mv f - mvi(B) 2 N F (C) 20 NDt DtDt(D) 200 N(E) 2000 NDecember 18, 2018
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.December 18, 2018
Conservation of MomentumqStart from impulse-momentumtheorem!!!F21Dt m1v1 f - m1v1i!!!F12 Dt m2v2 f - m2v2i!!F21Dt - F12 DtqSinceq!!!!Then m1v1 f - m1v1i -(m2 v2 f - m2 v2i )qSo!!!!m1v1i m2 v2i m1v1 f m2 v2 fDecember 18, 2018
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 Dt Dp p f - pi!!When Fnet 0 then Dp 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 fDecember 18, 2018
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 50m / s, v1 f ?0 m1v1 f m2 v2 fm20.5kgv1 f v2 f (50.0m / s) -0.417m / sm160.0kgDecember 18, 2018
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/sDecember 18, 2018
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 conservedDecember 18, 2018
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 collisionsDecember 18, 2018
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 conservedDecember 18, 2018
Ex6: An SUV Versus a CompactqAn SUV with mass 1.80 103 kg is travelling eastbound at 15.0 m/s, while a compact car with mass 9.00 102 kgis 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.December 18, 2018
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 / sDecember 18, 2018
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 / sDv1 v f - v1i -10.0m / sDv2 v f - v2i 20.0m / sm1Dv1 m1 (v f - v1i ) -1.8 10 4 kg m / sm2 Dv2 m2 (v f - v2i ) 1.8 10 4 kg m / sm1Dv1 m2Dv2 0December 18, 2018
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 J22DKE KE f - KEi -2.70 105 JDecember 18, 2018
More About ElasticCollisionsqBoth momentum and kinetic energyare conservedm1v1i m2v2i 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 simultaneouslyDecember 18, 2018
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 fDecember 18, 2018
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 fDecember 18, 2018
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/2v2v4vDecember 18, 2018
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 massesDecember 18, 2018
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 fDecember 18, 2018
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 simultaneouslyDecember 18, 2018
One-Dimension vs TwoDimensionDecember 18, 2018
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 fyDecember 18, 2018
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 v2iDecember 18, 2018
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 fyDecember 18, 2018
2-D Collision, exampleParticle 1! is moving atvelocity v1i andparticle 2 is at restq In the x-direction, theinitial momentum isqm1v1iqIn the y-direction, theinitial momentum is 0December 18, 2018
2-D Collision, example contqqAfter the collision, themomentum in the x-direction ism1v1f cos q m2v2f cos fAfter the collision, themomentum in the y-direction ism1v1f sin q m2v2f sin fm1v1i 0 m1v1 f cos q m2v2 f cos f0 0 m1v1 f sin q - m2 v2 f sin fqIf the collision is elastic, applythe kinetic energy equation111m1v12i m1v12f m2 v22 f222December 18, 2018
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 ?q ?December 18, 2018
Collision at an Intersectionmc 1.5 103 kg, mv 2.5 103 kgvcix 25 m/s, vviy 20 m/s, v f ?q ?åpåpxi mc vcix mv vvix mc vcix 3.75 10 4 kg m/sxf mc vcfx mv vvfx (mc mv )v f cosq3.75 10 4 kg m/s (4.00 103 kg)v f cosq4p mv mv mv 5.00 10kg m/så yi c ciy v viy v viyåpyf mc vcfy mv vvfy (mc mv )v f sin q5.00 10 4 kg m/s (4.00 103 kg)v f sin qDecember 18, 2018
Collision at an Intersectionmc 1.5 103 kg , mv 2.5 103 kgvcix 25m / s, vviy 20m / s, v f ?q ?5.00 10 4 kg m/s (4.00 103 kg)v f sin q3.75 10 4 kg m/s (4.00 103 kg)v f cosq5.00 10 4 kg m / stan q 1.3343.75 10 kg m / sq tan -1 (1.33) 53.1!5.00 10 4 kg m/svf 15.6 m/s3!(4.00 10 kg ) sin 53.1December 18, 2018
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Linear Momentum q This is a new fundamental quantity, like force, energy. It is a vector quantity (points in same direction as velocity). q The linear momentum p of an object of mass mmoving with a velocity v is defined to be the product of the mass and velocity: q The terms momentum and linear momentum will be used interchangeably in the text
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
A Pre-Revolution Time Line Directions: Using the list in the box, fill in the events and laws that led up to the American Revolution. Write the event or law below each year. You may need to do some online research to complete this exercise. Boston Tea Party, Stamp Act Congress, Intolerable Acts, The French and Indian
