CEnter Of MAss & MOMENTUM! - Smcisd

MOMENTUM!

Momentum A measure of how hard it is to stopan object which is moving. Related to both mass and velocity.Momentum mass x velocityp m v(in kg m/s)

Why is momentum important?The same momentum existsbefore and after a collision.Momentum lets us predictcollisions .and explosions!

Elastic Collisions(objects bounce and don’t stick) Linear momentum is conserved Total energy is conserved Total kinetic energy is conservedTKEin TKEout

Inelastic Collisions(objects collide and stick) Linear momentum is conserved Total energy is conserved Kinetic energy is not conservedTKEin TKEout

Momentum vs. Inertia Inertia is a property of mass that resists changes in velocity;however, inertia depends only on mass. Inertia is a scalar quantity (no direction). Momentum is a property of moving mass that resists changes ina moving object’s velocity. Momentum is a vector quantity that depends on both massand velocity (has direction).

EXAMPLE: Momentum practice problem.If a football player’s momentum is 400 kg m/s,and he has a weight of 1200 N, what is hisvelocity (in m/s)?Fw mg𝐅m 𝐠𝐖m 𝟏𝟐𝟎𝟎 𝑵𝒎𝟏𝟎 𝟐𝒔Ans. p m v𝐩 𝐯𝐦𝐦𝟒𝟎𝟎 𝐤𝐠𝐬𝟏𝟐𝟎 𝐤𝐠 𝐯Ans. 3.33 m/s v

P.O.D. 1: Find the momentum of the following.Car: m 1800 kg; v 288 km /hrBus: m 9000 kg; v 16 m /sTrain: m 3.6 104 kg; v 4 m /s

Momentum vs. Kinetic Energy Kinetic energy and momentum are different quantities, eventhough both depend on mass and velocity. Kinetic energy (TKE ½mv2) is a scalar quantity. Mass is always and when you square the velocity you always get a answer.Kinetic Energy doesn’t depend on direction.Kinetic EnergyABMomentum ½(4 kg)( 1 m/s)2 2 J ½(4 kg)(1 m/s)2 2 JAB

Momentum vs. Kinetic EnergyMomentum (p mv)is a vector, so it always depends on direction.Sometimes momentum is if velocity is in the direction andsometimes momentum is if the velocity is in the direction.ABKinetic EnergyMomentum ½(4 kg)( 1 m/s)2 2 J 4 kg 1 m/s 4 kg m/s ½(4 kg)(1 m/s)2 2 J 4 kg 1 m/s 4 kg m/sABTwo balls with the same mass and speed have the samekinetic energy but opposite momentum.

Conservation of MomentumThe law of conservation of momentum states whena system of interacting objects is not influenced byoutside forces (like friction), the total momentum of thesystem cannot change.

Collisionsv1v2v3m1m2m1v1m1v2m2v4m2v3m1m2

EXAMPLE: Elastic collisionsTwo 0.165 kg billiard balls roll toward each other and collidehead-on. Initially, the 5-ball has a velocity of 0.5 m/s. The 10-ballhas an initial velocity of -0.7 m/s. The collision is elastic and the10-ball rebounds with a velocity of 0.4 m/s, reversing its direction.What is the velocity of the 5-ball after thecollision?

Ans. G.U.E.S.S.G. ivens: m1 m2 0.165 kg, v1 0.5 m/s, v2 0.7 m/s, v4 0.4 m/sU. known: v3E. quation: m1v1 m2v2 m1v3 m2v4S. olve:m1v1 m2v2 m1v3 m2v4 m2v4 m2v4m1v1 m2v2 m2v4 m1v3𝐦𝟏 𝐯𝟏 𝐦𝟐 𝐯𝟐 𝐦𝟐 𝐯𝟒𝐦𝟏S. ubstitute: 𝐯𝟑𝒎𝒔𝒎𝒔𝒎𝒔(𝟎.𝟏𝟔𝟓 𝒌𝒈)(𝟎.𝟓 ) (𝟎.𝟏𝟔𝟓 𝒌𝒈)( 𝟎.𝟕 ) (𝟎.𝟏𝟔𝟓 𝒌𝒈)(𝟎.𝟒 )𝟎.𝟏𝟔𝟓 𝐤𝐠 𝐯𝟑 0.6 m/s v3

P.O.D. 2: A 200 kg football player moves at 5 m/s towards a150 kg player moving at 7 m/s. They collide and bounce offeach other in opposite directions elastically. If the 200 kgplayer is moving at 3 m/s after the impact, how fast (in m/s) isthe 150 kg player moving?

EXAMPLE: Inelastic collisionsA train car moving to the right at 10 m/s collides with aparked train car. They stick together and roll along the track.If the moving car has a mass of 8,000 kg and the parked carhas a mass of2,000 kg, what is their combinedvelocity after the collision?

Ans. G.U.E.S.S.G. ivens: m1 8000 kg m2 2000 kg, v1 10 m/s, v2 0 m/sU. known: v3E. quation: m1v1 m2v2 (m1 m2)v3𝐦𝟏 𝐯𝟏 𝐦𝟐 𝐯𝟐𝐦𝟏 𝐦𝟐S. olve: 𝐯𝟑S. ubstitute:𝒎𝒎(𝟖𝟎𝟎𝟎 𝒌𝒈)(𝟏𝟎 𝒔 ) (𝟐𝟎𝟎𝟎 𝒌𝒈)(𝟎 𝒔 )𝟖𝟎𝟎𝟎 𝐤𝐠 𝟐𝟎𝟎𝟎 𝐤𝐠 𝐯𝟑8 m/s v3

P.O.D. 3: A 2000 kg bus rear ends a 2500 kg bus which ismoving at 5 m/s. If the 2000 kg bus was moving at 30 m/sinitially, how fast (in m/s) would the two buses move forwardtogether after the collision?

Force is the Rate of Change of Momentum Momentum changes whena net force is applied. The inverse is also true:– If momentum changes,forces are created. If momentum changesquickly, large forces areinvolved.This means that force and momentum aredirectly proportional

Force and Momentum ChangeThe relationship between force and motion follows directlyfrom Newton's Second Law.D𝐯F m a F m F D t m D vD𝐭F D t D𝒑Force (N)F D pD tChange in time (sec)Change in momentum(kg m/sec)

Impulse DefinedThe product of F Dt from the last slide is called Impulse.The symbol for impulse is J. So, by definition:J F Dt

Example : A 2000-kg cartraveling at 90 km/h crashesinto a concrete wall that doesnot give at all.(a) Estimate the time ofcollision, assuming thatthe car decelerates 30 m/sper second.(b) Estimate the average forceexerted by the wall on thecar.Ans. 90 km/hr is 25 m/s(a) The time of deceleration is given by vf vi –at𝒎So t 𝐯𝐟 𝐯𝐢𝐚 t 𝐦𝟎 𝒔 𝟐𝟓 𝐬𝐦 𝟑𝟎 𝟐𝒔𝒎𝐢(b) Fnet t m Dv Fnet t m vt 𝟐𝟓 𝒔𝐦 𝟑𝟎 𝟐𝒔𝒎 t Fnet 𝟐𝟓 𝒔𝐦 0.83 s 𝟑𝟎 𝟐𝒔m(2000 kg)( 25 s )0.83 s 60,240.96 N

P.O.D. 4A sucker of mass 80 kg feels an impulse of 3000 kg m/s for 0.5 secondswhen he gets kicked by a buddy.How much force (in N) does he feel?What is his acceleration (in m/s2)?With what velocity (in m/s) does he move away from the impact?

Variable ImpulseFind the area under the F(t) graph for a variable force to findthe impulse, or integrate ( )EX: What is theimpulse in thefirst fourseconds?Ans. A ½bhA ½(4 s)(4 N)A 8 N s

INTEGRATION REFRESHER:Finding the “area under the curve” isthe same thing as integrating.The trick for integrating is:If f(x) kxn then f(x) dx 𝐱 𝐧 𝟏k𝐧 𝟏

P.O.D. 5The force acting on a farm hand by a horse’s kick is given byF(t) 9t2 – 4t 6. The farm hand is initially at rest at t 0 s.Find the impulse on the farm hand over the first 40milliseconds.

Conservation of Momentum in 2-DTo handle a collision in 2-D, we conserve momentum in each dimensionseparately.Choosing down & right as positive:m1 1 2v1 am1vam2m2v2before:px m1 v1 cos 1 m2 v2 cos 2py m1 v1 sin 1 m2 v2 sin 2 bafter:px m1 va cos a m2 vb cos bvbpy m1 va sin a m2 vb sin bConservation of momentum equations:pbefore pafterm1 v1 cos 1 m2 v2 cos 2 m1 va cos a m2 vb cos bm1 v1 sin 1 m2 v2 sin 2 m1 va sin a m2 vb sin b

Example34 m/s152 gbefore40 5 m/s0.3 kgA mean, old dart strikes an innocentmango that was just passing byminding its own business. Whichway and how fast do they move offtogether?Working in grams and taking left & down as :md vd (md mm) vf152 (34) sin 40 452 v sin 452 g152 (34) cos 40 - 300 (5) 452 v cos Dividing equations : 1.35097 tan vSubstituting into either of the first twoequations :v 9.14 m/safter 53.4908

P.O.D. 6:A pool player hits a cue ball in the x-direction at 0.80m/s. The cue ball knocks into the 8-ball, which movesat a speed of 0.30 m/s at an angle of 35o angle abovethe x-axis. Determine the angle of deflection of thecue ball. Assume the masses of the balls are thesame.

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