Momentum AP Review - San Marcos CISD
MomentumAP ReviewMomentumHow hard it is to stop a moving object.Related to both mass and velocity.For one particlep mvFor a system of multiple particlesP pi miviUnits: N s or kg m/sMomentum is a vector!ANSWER KEYProblem: Impulse (1984)56. Two planets have the same size, but differentmasses, and no atmospheres. Which of thefollowing would be the same for objects withequal mass on the surfaces of the two planets?I. The rate at which each would fall freelyII. The amount of mass each would balanceon an equal-arm balanceIII. The amount of momentum each wouldacquire when given a certain impulseProblem: Momentum (1998)(A)(B)(C)(D)(E)I onlyIII onlyI and II onlyII and III onlyI, II, and IIIExplain your reasoning:43. The magnitude of the momentum of theobject is increasing in which of the cases?(A)(B)(C)(D)(E)II onlyIII onlyI and II onlyI and III onlyI, II, and IIIAns. Explain your reasoning:Graph III is the only graph where acceleration ishappening (as evidenced by a curved d vs. tgraph). This means that a net force is beingapplied to the object of mass m (by Newton’sSecond Law). In order for momentum toincrease, an impulse (J) needs to be applied (aForce F over a period of time, t). p J F tAns. The gravity would be different on bothplanets because gravity depends both on the massof the planet and its radius. Since gravity isdifferent on each planet, they would not fall at thesame rate. But if you put both masses on oppositesides of an equal-arm balance they would balancesince the masses are equal. Also, since they havethe same mass, the same amount of impulse (J F t) should produce the same results. In terms ofmomentum increase.Problem: Impulse (1998)57. A ball of mass 0.4 kg is initially at rest on theground. It is kicked and leaves the kicker's footwith a speed of 5.0 m/s in a direction 60 abovethe horizontal. The magnitude of the impulseimparted by the ball to the foot is most nearly(A) 1 N s(B)3 N s(C) 2 N s(D)2N s3(E) 4 N sImpulse (J)The product of an external force and time, whichresults in a change in momentumJ FtJ PUnits: N s or kg m/s1/29/2018Show your work:Ans. The impulse is simply J m v. Themass is 0.4 kg and the speed is 5.0 m/s. J 0.4kg 5.0 m/s 2 N s. The angle is irrelevanthere.Momentum-1Bertrand
Law of Conservation of MomentumIf the resultant external force on a system is zero,then the momentum of the system will remainconstant.The sum of the momentums before a collision isequal to the sum of the momentums after acollision. Pb PaProblem: Conservation of Momentum(1998)CollisionsFollow Newton’s Third Law which tells us thatthe force exerted by body A on body B in acollision is equal and opposite to the forceexerted on body B by body A.During a collision, external forces are ignored.The time frame of the collision is very short.The forces are impulsive forces (high force, shortduration).Collision TypesElastic: P is conserved, K is conservedInelastic: P is conserved, K is NOT conservedPerfectly Inelastic means the bodies stick togetherProblem: Collisions (1993)4. An open cart on a level surface is rolling withoutfrictional loss through a vertical downpour ofrain, as shown above. As the cart rolls, anappreciable amount of rainwater accumulates inthe cart. The speed of the cart will(A) increase because of conservation ofmomentum(B) increase because of conservation ofmechanical energy(C) decrease because of conservation ofmomentum(D) decrease because of conservation ofmechanical energy(E) remain the same because the raindropsare falling perpendicular to the directionof the cart's motionExplain your reasoning:Ans. Mechanical energy is not conserved, ingeneral. Total energy is, but mechanical energy isnot. Momentum is ALWAYS conserved! So,suppose you had a cart that has a mass of 10kgmoving at 5 m/s. It has momentum of 50 kg*m/s.Since friction does not act, the momentum willremain 50kg*m/s (Newton's law). Suppose atsome later time it has filled up with 10kg of rain,so now the cart has a mass of 20kg. It still has the50kg*m/s of momentum, so it must be moving at2.5m/s, which means it has decreased due toconservation of momentum.10. Which of the following is true when an object ofmass m moving on a horizontal frictionless surfacehits and sticks to an object of mass M m, which isinitially at rest on the surface?(A) The collision is elastic.(B) All of the initial kinetic energy of the less-massiveobject is lost.(C) The momentum of the objects that are stuck togetherhas a smaller magnitude than the initial momentum ofthe less-massive object.(D) The speed of the objects that are stuck togetherwill be less than the initial speed of the less-massiveobject.(E) The direction of motion of the objects that are stucktogether depends on whether the hit is a head-oncollision.Explain your reasoning:Ans. Again, by conservation of momentum, if themass increases, the velocity must decrease. pin poutProblem: Collisions (1993)11. Two objects having the same mass traveltoward each other on a flat surface, each with aspeed of 10 meter per second relative to thesurface. The objects collide head-on and arereported to rebound after the collision, eachwith a speed of 20 meters per second relativeto the surface. Which of the followingassessments of this report is most accurate?(A) Momentum was not conserved, therefore the report is false.(B) If potential energy was released to the objectsduring the collision, the report could be true.(C) If the objects had different masses, the report could be true.(D) If the surface was inclined, the report could be true.(E) If there was no friction between the objects and thesurface, the report could be true.Explain your reasoning:Ans. Momentum is conserved provided that anexternal force not be applied. If the objects havepotential energy stored, this could provide the forceand alter the momentum balance.1/29/2018Momentum-2Krummell
Problem: Collision (1988)Problem: Collision (1998)3. A railroad car of mass m is moving at speed when it collides with a second railroad car ofmass M which is at rest. The two cars locktogether instantaneously and move along thetrack. What is the speed of the carsimmediately after the collision?(A) (B) m M(C) M m(D) (m M) m(E) mvm M41. Two objects of mass 0.2 kg and 0.1 kg,respectively, move parallel to the x-axis, asshown above. The 0.2 kg object overtakes andcollides with the 0. 1 kg object. Immediately afterthe collision, the y-component of the velocity ofthe 0.2 kg object is 1 m/s upward. What is they-component of the velocity of the 0.1 kg objectimmediately after the collision?(A)(B)(C)(D)(E)Show your work:Ans. This is an elastic collision so the formulam1v1 m2v2 (m1 m2)v3 applies.Substituting our given values:m M(0) (m M)v3m v3m MProblem: Momentum Change (1988)2 m/s downward0.5 m/s downward0 m/s0.5 m/s upward2 m/s upwardShow your work:Ans. Has to be 2 m/s downwards. The verticalmomentum is zero initially. So at the end, italso has to be zero.Pyin pyoutm vyi m vyf0.2 kg 1 m/s 0.1 kg vyf2 m/s vyf7. A tennis ball of mass m rebounds from aracquet with the same speed v as it hadinitially, as shown above. The magnitude ofthe momentum change of the ball is(A) 0(B) mv(C) 2mv(D) 2mv sin (E) 2mv cos Show your work:Ans. The horizontal componentscancel each other out because they are in opposite directions. The incident component goes to the right andcancels out the reflected component which goes to the left.The verticalcomponents however add up since they are bothupwards. The vertical components are1/29/2018Momentum-3Krummell
3 m v sin m v1.Similarly in the y-directionpy 3 m v cos m v1We, can see from this thatpx py3 m v sin 3 m v cos sin cos 45 .ExplosionMathematically, handled just like an ordinaryperfectly inelastic collision.Momentum is conserved, kinetic energy is not.Problem: Explosion (1998)So now that we know the angle we can find thevelocity.3 m v cos m v13 v cos v167. A stationary object explodes, breaking intothree pieces of masses m, m, and 3m. The twopieces of mass m move off at right angles toeach other with the same magnitude ofmomentum mV, as shown in the diagram above.What are the magnitude and direction of thevelocity of the piece having mass 3m ?v v13 cos θ v13 cos 45o v1 223 2v13 2 2v13Show your work: 2vAns. The magnitude of the velocity is, and3the direction is 45 degrees to the left of straightdown. To answer the question you need to useconservation of momentum. Before the explosionthere is no momentum, so there must be no netmomentum after the explosion.The two small pieces have momentum vectorspointing right and up, so in order for the largepiece to cancel both of these out it must beheading down and to the left. So we take the angle as being the angle to the left of straight down.This eliminates answers A and C.Looking at just the x-direction (left-right)direction to start with, the momentum of the largepiece in the x-direction ispx 3 m v sin .And this has to be equal to the momentum of thesmall piece heading right.px 3 m v sin .1/29/2018Momentum-4Krummell
ADVANCED TOPIC Center of MassWhere all the mass can be considered to existFor uniform objects, the center of mass resides atgeometric center.For collection of points, use these equationsxcm mixi / miycm miyi / mizcm mizi / miwhere xcm, ycm, and zcm are the coordinates of thecenter of mass, and mi is the total mass of thesystem.Problem: Center of Mass (1993)8. The two spheres pictured above have equaldensities and are subject only to their mutualgravitational attraction. Which of the followingquantities must have the same magnitude forboth spheres?(A) Acceleration(B) Velocity(C) Kinetic energy(D) Displacement from the center of mass(E) Gravitational forceProblem: Center of Mass (1998)63. Two people of unequal mass are initiallystanding still on ice with negligible friction. Theythen simultaneously push each other horizontally.Afterward, which of the following is true?(A) The kinetic energies of the two people areequal.(B) The speeds of the two people are equal.(C) The momenta of the two people are ofequal magnitude.(D) The center of mass of the two-person systemmoves in the direction of the less massiveperson.(E) The less massive person has a smaller initialacceleration than the more massive person.Show your work or explain yourreasoning:Ans. the gravitational force between themwould be the same because of Newton’s ThirdLaw. For every action there is an equal andopposite reaction. The center of mass would becloser to the larger sphere. The acceleration dueto gravity would be greater for the larger sphere.Explain your reasoning:Ans. The kinetic energies won’t be the same(Ans. A) because their velocities are squared andtheir masses aren’t the same either. By the law ofconservation of momentum though, theirmomenta must equal zero after the collision astheir momenta is zero before. Ans D, the centerof mass, doesn’t make sense. The center of masswould move towards the heavier person ifanything.1/29/2018Momentum-5Krummell
MomentumFor one particlep mvFor a system of multiple particlesP pi miviIf center of mass is definedP MvcmMomentum is a vector!Momentum and CollisionsThe same momentum exists before and after acollision.Law of Conservation of MomentumIf the resultant external force on a system is zero,then the vector sum of the momenta of theobjects will remain constant. pb paProblem: Energy and Momentum (CM 1984)Problem: Conservation of Momentum (CM 1998)Questions 13-1431. An object having an initial momentum thatmay be represented by the vector abovestrikes an object that is initially at rest.Which of the following sets of vectors mayrepresent the momenta of the two objectsafter the collision?A system consists of two objects having massesml and m2 (ml m2). The objects are connectedby a massless string, hung over a pulley as shownabove, and then released.(A)(B)(C)13. When the speed of each object is v, themagnitude of the total linear momentum ofthe system is(A) (m1 m2) v(B) (m2 - m1) v(C)½(ml m2)v(D) ½(m2 - m1)v(E) m2v14. When the object of mass m2 has descended adistance h, the potential energy of the systemhas decreased by(A) (m2 - ml)gh(B) m2gh(C) (m1 m2)gh(D) ½(ml m2)gh(E) 0State your reasoning:Ans.Probl. 13. One object is rising (m2) and the otheris falling (m1). The momenta don’t cancel outbecause the masses are different. The resultantmomentum is p p1 – p2 m2v – m1v if weallow downwards to be the positive direction.Probl. 14. As a system, the change in thepotential energy would be the difference betweenthe potential energies of the two masses. Oneblock’s potential energy goes up by m1gh whilethe other’s potential energy goes down by m2gh.Therefore, the decrease in potential energy has amagnitude given by (m2 – m1)gh.1/29/2018(D)(E)State your reasoning:Ans. Because of momentum conservation, themomentum before has to equal the momentumafterwards. pi pf.The momentum initially is tothe right and has a large magnitude. The onlyanswer that will give a resultant of largemagnitude to the right is Ans. E. Answer A givesa resultant of zero. Answer B gives a resultantthat is double the initial momentum. Answer Cgives a resultant that is going left. Answer Dmight give the right magnitude but the direction iswrong.Momentum-6Krummell
Impulse p JJ impulseJ F dtProblem: Impulse and Momentum(CM 1993)17. If one knows only the constant resultant forceacting on an object and the time during whichthis force acts, one can determine the(A) change in momentum of the object(B) change in velocity of the object(C) change in kinetic energy of the object(D) mass of the object(E) acceleration of the objectCollision TypesElastic CollisionsLinear momentum is conservedTotal energy is conservedKinetic energy is conservedInelastic CollisionsLinear momentum is conservedTotal energy is conservedKinetic energy is not conservedExplosionsTreated as an inelastic collision2-D CollisionsMomentum change is analyzed by componentProblem: Impulse and Momentum (CM 1984)State your reasoning:Ans. The impulse (J F t) is the change ofmomentum.Problem: Impulse and Momentum (CM 1998)12. The graph above shows the force on an objectof mass M as a function of time. For the timeinterval 0 to 4 s, the total change in themomentum of the object is(A) 40 kg m/s(B) 20 kg m/s(C) 0 kg m/s(D) -20 kg m/s(E) indeterminable unless the mass M of theobject is knownShow your work:Ans. The object experiences a positive impulseof J F t 10 N 2 s 20 N s for the first twoseconds and a negative impulse of J F t 10N 2 s 20 N s from t 2 s to t 4 s. Theseimpulses cancel each other out and there is nochange in momentum.1/29/201817. Two particles of equal mass mo, moving withequal speeds vo along paths inclined at 60 tothe x-axis as shown above, collide and sticktogether. Their velocity after the collision hasmagnitudev043v 02(A)(B)v02(C)2v 02(D)(E) voShow your work:Ans. The vertical components of momentumcancel each other out since one particle is goingdownwards and one is going upwards and thespeeds vo are equal. The horizontal components,however, are given by:pin poutmo vx1 mo vx2 (mo mo)vfmo vo cos mo vo cos (mo mo)vfThe horizontal components before the collisionare the adjacent sides of the right triangles formedby vo, vx, and vy. vxmo vo cos 60 mo vo cos 60 (2mo)vfmo vo (½) mo vo (½) (2mo)vfmo vo (½) mo vo (½) (2mo)vfmo vo 2movfmo vo 2movf2mo2mo½vo vfMomentum-7Krummell
Center of Mass of system of particlesThe point at which all of the mass of an object orsystem may be considered to be concentrated.Center of Mass for collection f pointsxcm mixi / Mycm miyi / Mozcm mizi / MVelocity of Center of Massvx,cm mivxi / Mvy,cm mivyi / Mvz,cm mivzi / MAcceleration of Center of Massax,cm miaxi / May,cm miayi / Maz,cm miazi / MCenter of Mass for simple solid objectPick geometric center if uniform densityCenter of Mass for complicated solid objectsXcm 1/M x dmProblem: Center of Mass (CM 1984)31. Mass M1 is moving with speed v towardstationary mass M2. The speed of the centerof mass of the system is(A)M1vM2( B) M1 1 vM2 (C) M2 1 vM1 (D) M1 1 vM2 (E) M1 v M1 M 2 Show your work:ANS: EProblem: Center of Mass (CM 1984)L𝑣𝑐𝑚 𝑣29. The center of mass of a uniform wire, bent inthe shape shown above, is located closest topoint(A) A(B) B(C) C(D) D (E) EShow your work:Ans. B. Take center of mass of each of the 3 linesegments, and find the center of mass of those:We are only interested in the ycm in this case.The horizontal center of mass is not needed sincethere is symmetry in this plane.ycm (mL/2 mL/2)/3 mL/3.Point B is located closest to the distance L/3.𝑀1 0𝑀1 𝑀2Force and MomentumFext dP/dt d(mv)/dtFor variable mass systems, we get thisFext dP/dt d(mv)/dt maFor variable mass systems, we get thisFext mdv/dt vdm/dtWe can also plug in number. Assume the lengthof each wire is 4 units𝑚(2) 𝑚(2) 𝑚(0)4𝑦𝑐𝑚 3𝑀3or just above the 1 unit mark, but not as far as 2.0units1/29/2018Momentum-8Krummell
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
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.
Houston Mr. James D. Dannenbaum Houston Mr. Jaime U. Davila, Sr. Austin-San Marcos Mr. Michael W. Dawkins Austin-San Marcos Mr. Hector De Leon Austin-San Marcos Ms. Sylvia A. de Leon Washington, DC-MD-VA-WV Mr. Kenneth P. DeAngelis Austin-San Marcos The Honorable Linnet F. Deily Houston Mr. Wofford Denius Los Angeles-Long Beach Mr. Martin W .
As noted in the findings of fact, the PE procedures at CSU San Marcos did not include assigning an OR score at the time Section 20.21 of the addendum was signed. Accordingly, CSU San Marcos was one of the campuses required to develop procedures for the assignment of an OR score. Absent a waiver, CSU San Marcos would be required to bargain with CSEA
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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.
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