Conservation of Momentum 8.01t Oct 25, 2004Momentum and Impulse G • momentum p = mv G G m• change in momentum ∆p G =∆v GG • average impulse I = F ∆ =∆ p G tave avePRS Question Identical constant forces push two identical objects A and B continuously from a starting line to a finish line. If A is initially at rest and B is initially moving to the right, 1. Object A has the largerchange in momentum. 2. Object B has the largerchange in momentum. 3. Both objects have the samechange in momentum 4. Not enough information isgiven to decide.Non-Constant Force and Impulse • Cart colliding with a spring = = ttfG iN G G )I = lim ∑Fi∆ti ≡∫ F(tdt ∆ti→0 i=1 t=0PRS Question If a Mack truck and a Ford Escort have a head-on collision, which vehicle will experience the greater impulse? 1. the Mack truck2. the Ford Escort3. both experience the same impulsePRS Question If a Mack truck and a Ford Escort have a head-on collision, which vehicle will experience the greater acceleration? 1. the Mack truck2. the Ford Escort3. both experience the same accelerationConservation of Momentum • The total change in momentum of a system and its surroundings between the final state and the initial state is zero, G G∆psystem +∆ psurroundings = 0External force on a System • The total internal force on a system is zero • Newton’s Third Law: internal forces cancel in pairs • total force acting on the system is the sum the external forces G G G Gtotal total total total F = Fext + Fint = FextExternal Force and Momentum Change G iN G iNdp G dp G total= = total =∑Ftotal iF i =∑ = i=1 i=1 dt dt GG total total F = Fext F G dp G total total = ext dtCenter of Mass • Position of center of mass iN G= ∑ m riiG R = i=1 cm iN= ∑ mi i=1 • Velocity of center of mass =iN G∑ m v =V G ii G total i=1 p= =cm iN total ∑ mi m i=1Translational Motion of the Center of Mass Gtotal • Total momentum p Gtotal = m Vcm • External force and acceleration of center of mass GG dp G total Gtotal = totaldVcm total AFext = m = m cm dt dtConservation of Momentum • completely isolate system from the surroundings total = ext G F G0 • change in momentum of the system is also zero ∆ G p = system G0PRS Question Drop a stone from the top of a high cliff.Consider the earth and a the stone as a system. As the stone falls, the momentumof the system 1. increases in the downward direction. 2. decreases in the downward direction.3. stays the same.4. not enough information to decide.Modeling: External Forces and Conservation of Momentum 1. Identify the objects that compose the system 2. Identify your initial and final states of the system 3. Identify any external forces acting on the system 4. Decide whether momentum is conserved or not in each direction.Modeling: Instantaneous Collisions • Decide whether or not the collision is instantaneous. • External impulse changes the momentum of the system. GG GIexternal = Fext ∆tcol =∆ psystem • if the collision time is approximately zero, themomentum is approximately conserved. G • If ∆tcol ≅ 0 then ∆p G system ≅ 0Momentum Diagram Draw a momentum diagram for the initial and finalstates. a) Choose symbols to identify each mass and velocity in the system. a) Identify a set of positive directions and unit vectors for each state. c) Decide whether you are using components ormagnitudes for your velocity symbols.Momentum Diagram Since momentum is a vector quantity, identify the initial and final vector components of the total momentum GInitial State p Gtotal = m v1,0 + m v G + ...0 1 2 2,0 total x-comp: p = m1 ( v + m2 ( v + ...x,0 x x)1,0 )2,0 total y-comp: py,0 = m1 ( vy )1,0 + m2 ( vy )2,0 + ... Final State Gtotal = m v GG vp f 11, f + m2 2, f + ... totalx-comp: p , = m1 ( v + m2 ( v + ...xf x x)1, f )2, f totaly-comp: pyf = m1 ( vy )1, f + m2 + ... , ( vy )2, fConservation of Momentum When the total external force on a system is zero, then the total initial momentum of the system equals the total final momentum of the system, Gtotal Gtotal p0 = p fStrategies: Conservation of Momentum • If momentum is conserved, write down the conservation equations for momentum in each direction total total p x f x,0 = p , mv +m2 (v +=mv + m2 (v +...x x x1 ()1,0 )2,0 ... 1 (x )1, f )2, f total total = p ,py,0 y f ... 1 (mvy +m2 +=mvy )+ m2 (vy )+...1 ()1,0 (vy )2,0 1, f 2, fClass Problem 1: Acrobat and Clown An acrobat of mass m1 jumps upwards offa trampoline with aninitial velocity v0 . At a height h0, the acrobat grabs aclown of mass m2. Assume that the time the acrobat takes to grab the clown isnegligibly small.Problem 1: Acrobat and Clown (con’t) a) What is the velocity of the acrobat immediately before grabbing the clown? b) What is the velocity of the acrobat immediatelyafter grabbing the clown? c) What high do the acrobat and clown rise? d) If the acrobat had missed the clown, how highwould have the acrobat risen?PRS Question Suppose a golf ball is hurled at a heavy bowling ballinitially at rest and bounces elastically from the bowlingball. After the collision, 1. The golf ball has the greater momentum and the greater kinetic energy. 2. The bowling ball has the greater momentum and the greater kinetic energy. 3. The golf ball has the greater momentum but has thesmaller kinetic energy. 4. The bowling ball has the greater momentum but has the smaller kinetic energy.Class Problem 2: Exploding Projectile An instrument-carrying projectile of mass m1 accidentally explodes at the top of its trajectory. The horizontal distance between launch point and theexplosion is x0. The projectile breaks into two pieceswhich fly apart horizontally. The larger piece, m3, hasthree times the mass of the smaller piece, m2. To the surprise of the scientist in charge, the smaller piecereturns to earth at the launching station. Neglect airresistance and effects due to the earth’s curvature.Problem 2: Exploding Projectile (con’t) a) What is the velocity of the projectile at the top of its flight just before the collision? b) What is the velocity of the smaller piece just after the collision? c) What is the velocity of the larger piece just after the collision? d) How far away does the larger piece land from the station?PRS Question Consider the exothermic reaction (final kinetic energy is greater than the initial kinetic energy). H + H → H2 + 5ev Two hydrogen atoms
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