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Linear Momentumand Impulse

Linear Momentum Linear momentum (p)is a vectorp is parallel to vUnit: kg·m/sThe net momentum ofa collection of objectsis the vector sum ofthe momentum of each objectp p mv p1 p2 p3 . m1v1 m2v2 m3v3 .

Momentum problemA bullet of mass 0.005 kg moving at a speed of 100m/s lodges within a 1-kg block of wood restingon a frictionless surface and attached to ahorizontal spring of k 50N/m.m1v1 m2 v2 m1v1' m2 v2'a)What is the velocityof the block theinstant after thebullet strikes it?m1v1 0 (m1 m2 )v 'm1v1v m1 m2'(0.005kg)(100m / s) 0.005kg 1.00kg 0.50m / s

Momentum problemA bullet of mass 0.005 kg moving at a speed of100 m/s lodges within a 1-kg block of woodresting on a frictionless surface and attachedto a horizontal spring of k 50N/m.ET ETb)What is the maximumcompression of thespring?12mv2 12 kx2mv 2x k(1.005kg)(0.5m / s ) 2 50 N / m 0.07m

Force and momentumThe time rate of changeof the momentum of aparticle is equal to thenet force acting onthe particle. dpFnet dt“Let the time rate of change ofmomentum be with you.”

Force and momentum In general, force andmomentum are related as: dp ddv dmFnet (mv ) m vdt dtdtdt When the mass isconstant, thisbecomes the familiarequation: dvFnet m madt

Non-constant mass Mass changes for rockets, bags of sand(with holes in them), and other specialcases.See below for a changing mass:http://eepybird.com/dcm1.html

Changing mass problemA rocket whose initial mass is 850kg consumesfuel at the rate of 2.3kg/s. The speed of theexhaust gases relative to the rocket engine is2800m/s. dpdv dmFnet m vWhat thrust does thedtdtdt dmrocket engine provide? 0 vdt (2800m / s )(2.3kg / s ) 6440 N

ImpulseImpulse (J) is a change inmomentum. dp Start with Newton’s second Law: F dtWhich can be written as: dp F (t )dtIntegrate with respect to time to find the change inpftf momentum:J dp F (t )dt pi ti

ImpulseA 2.0 kg toy car travels at 0.50m/s East before aturn and at 0.40m/s South after the turn.What is the impulse (change in momentum) of thecar due to the turn? J p p f pi mv f mvi m(v f vi ) (2.0kg) ( 0.40m / s ) ˆj (0.50m / s)iˆ (1.0iˆ 0.8 ˆj )kg m / s

Conservation of LinearMomentumThe linear momentum (p) of a system isconserved (does not change) unless thesystem experiences an external force.

ProblemAn 80-kg lumberjack stands at one end of afloating 400-kg log that is at rest relative to theshore of a lake.If the lumberjack jogs to the other end of the log at2m/s relative to the shore, what happens to thelog while he is moving?

Elastic collisionIn an elastic collision objects typically bounce, andNO energy is lost Momentum is conserved Kinetic energy is conserved pbefore pafterKbefore K after

Inelastic collisionIn a totally inelastic collision, objects stick together. Momentum is conserved Energy is lost to sound, sparks, mechanicaldeformation, etc. pbefore pafterKbefore K after

Example of Inelastic collision:Would you be safer in an old (heavy) 1959Chevrolet Bel Air or in a newer (lighter)2009 Chevy Malibu?http://www.youtube.com/watch?v fPF4fBGNK0U

Collision review Momentum is conserved in all collisionsElastic collisions: no deformation occurs Inelastic collisions: deformation occurs Kinetic energy is “lost”Perfectly inelastic collisions Kinetic energy is also conservedObjects stick together; kinetic energy is “lost”Explosions Reverse of perfectly inelastic collisions;kinetic energy is “gained”

Collision problemTwo skaters collide and embrace, in a completelyinelastic collision. Dean, of mass 83 kg, isinitially moving east with speed 6.2 m/s. Torvill,of mass 55 kg, is initially traveling north withspeed 7.8 m/s.What are the skaters’ speed and direction after thecollision?v 4.86m / s 4.9m / s 39.8 40

Momentum is conserved in all collisions Elastic collisions: no deformation occurs Kinetic energy is also conserved Inelastic collisions: deformation occurs Kinetic energy is "lost" Perfectly inelastic collisions Objects stick together; kinetic energy is "lost" Explosions Reverse of perfectly inelastic collisions;

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