Chapter 7 Linear Momentum

Chapter 7Linear Momentum

Units of Chapter 7 Momentum and Its Relation to Force Conservation of Momentum Collisions and Impulse Conservation of Energy and Momentum inCollisions Elastic Collisions in One Dimension

Units of Chapter 7 Inelastic Collisions Center of Mass (CM) CM for the Human Body Center of Mass and Translational Motion Torque

Momentum The sports announcer says, "Going into the allstar break, the Chicago White Sox have themomentum." The headlines declare "ChicagoBulls Gaining Momentum." The coach pumpsup his team at half-time, saying "You have themomentum; the critical need is that you usethat momentum and bury them in this thirdquarter."

Momentum in Sports Momentum is a commonly used term insports. A team that has the momentum is onthe move and is going to take some effort tostop. A team that has a lot of momentum isreally on the move and is going to be hard tostop. Momentum refersto the quantity ofmotion that an object has.

Momentum Momentum can be definedas "mass in motion.“ The momentum of an object is equal to themass of the object times the velocity of theobject. Momentum mass x velocity p mv p - momentum Units of momentumkg x m/s

Momentum, Mass and Speed An object has a large momentum if either itsmass or its velocity is large. If a car and a truck moveat the same speed,the truck has moremomentum since it has larger mass. But a fast car can have more momentum thana slowly moving truck. p 20 kg m/s2m / s10 kg5m / s p 20 kg m/s 4kg And the momentum of a truck at rest 0.

Momentum and Direction Momentum is a vector quantity. It has magnitude and direction. The direction of the momentum vector is thesame as the direction of the velocity of theball. What is the momentum of a 2-kg bowling ballmoving eastward at 5 m/s? p 10 kg x m/s, eastward

Practice Problems 1. Determine the momentum of a .a. 60-kg halfback moving eastward at 9 m/s.b. 1000-kg car moving northward at 20 m/s.c. 40-kg freshman moving southward at 2 m/s.a. p mv (60kg)(9m/s) 540 kg m/s, eastwardb. p mv (1000kg)(20m/s) 20000 kg m/s,northward c. p mv (40 kg)(2 m/s) 80 kg m/s, southward

2. A car possesses 20 000 units of momentum.What would be the car's new momentum if .a. its velocity was doubled.b. its velocity was tripled.c. its mass was doubled (by adding more passengersand a greater load)d. both its velocity was doubled and its mass wasdoubled.a. p 40,000 kg m/sb. p 60,000 kg m/sc. p 40, 000 kg m/sd. p 80,000 kg m/s

Questions: 1. Define momentum. 2. What letter represents momentum? 3. What units of measurement does momentumhave? 4. Is momentum a vector or scalar quantity? 5. How can you increase the momentum of anobject? 6. What has a greater momentum a 2.0-kg cartmoving at a speed of 5.0 m/s, or a 1.0- kg cartmoving at a speed of 8.0 m/s.

Conservation Law Momentum: p mvClosed isolated system: No particles enter or leaveNo net external forceIn a closed isolated system the total momentum is constant p const ptotal const

Conservation of MomentumDuring a collision, measurements show that thetotal momentum does not change:(7-3)

Conservation of Momentum p A pB p A p B pB p B p A p A ( p B pB ) p A p A Momentum is pB p Atransferred.

Conservation of MomentumDuring a collision, the reason that the totalmomentum is conserved is that momentumis transferred from one object to the other!

7-2 Conservation of MomentumMore formally, the law of conservation ofmomentum states:The total momentum of an isolated system ofobjects remains constant.

7-2 Conservation of MomentumMomentum conservation works for a rocket aslong as we consider the rocket and its fuel tobe one system, and account for the mass lossof the rocket.

Demonstration - Impulseand Change in Momentum1. Describe the velocity of the cartafter you have given it a push.Neglecting friction, the cart will roll at constantvelocity.2. How the velocity will change when its plungerstrikes the fixed force sensor.The cart will reverse direction.The cart will accelerate smoothly while the plungeris in contact with the sensor and return to itsoriginal position with a relatively constant, lowerspeed.

Increasing Momentum

Impulse- Momentum ChangeTheorem vFnet ma m t v t Fnet m t t Fnet t m v Fnet t pImpulse change in momentumThe quantity force x time is called impulse.

7-3 Collisions and ImpulseDuring a collision, objects aredeformed due to the largeforces involved.Since, we canwrite(7-5)The definition of impulse: J F tJ impulse

Impulse To change an object’s momentum requires anoutside force which will must act for some timeperiod. J Ft Δp pfinal-pinitial So to cause a greater change in an object’smomentum you need either apply a bigger forceor apply this force for a longer period of time orboth. This idea has applications in the world of sportsand also the world of firearms.

Increasing Momentum The pitcher exerts a force ona ball over as long a distance andfor as time as possible.The greater Ft, the larger p, higher the releasespeed. The pitcher exerts a force on a ball over as long adistance and for as long as possible.Hitting a ball. In order to increase the momentum of the ball off the teethe most, the hitter should follow throughwhen hitting the ball off the tee.

7-3 Collisions and ImpulseSince the time of the collision is very short, weneed not worry about the exact time dependenceof the force, and can use the average force.

Impulse Examplea. What is the momentum of a 8-kg bowling ball rolling at2 m/s?b. If the bowling ball rolls into a pillow and stops in 0.5 s,calculate the average force the pillow exerts on the ball.c. What average force does the ball exert on the pillow?a. p mv (8kg)(2m/s) 16 kg m/sb. Ft p p 0-16 -16 kg m/sF p/t -16/0.5 -32Nc. 32N

Practice ProblemA 100-kg car, moving at 50 m/s is slowed to astop in 25.0 s. What was the impulse applied tothe car? What force was required?Ft p p pfinal-pinitial 0-mv 0 – 100(50) -5000 kg m/sF p/t -5000/25.0 -200N

Question: You can’t throw a raw egg against a wallwithout braking it, but you can throw it atthe same speed into a sagging sheetwithout it. Explain.

Questions:1. Define impulse.2. State the impulse-momentum theorem.3. How can you increase the change inmomentum of an object? Where doesthat apply?4. How can you decrease the impact forceon an object? Where does that apply?

Do NowA 0.05 kg. tennis ball is moving to the left at 5 m/swhen it is hit by a tennis racquet. The force vs.time graph for this collision is shown to the right.After the collision with the racquet, the ball ismoving to the right with what velocity?

Questions:1. Newton’s Second law states that if no netforce is exerted on the system, noacceleration occurs. Does it follow that nochange in momentum occurs?2. Newton’s Third Law states that the forcea cannon exerts on a cannonball is equaland opposite to the force the cannonball exerts on a cannon. FConCB FCBonC Does it follow thatJ ConCB J CBonC

7-4 Conservation of Energy and Momentumin CollisionsMomentum is conservedin all collisions.Collisions in whichkinetic energy isconserved as well arecalled elastic collisions,and those in which it isnot are called inelastic.

7-5 Elastic Collisions in One DimensionHere we have two objectscolliding elastically. Weknow the masses and theinitial speeds.Since both momentumand kinetic energy areconserved, we can writetwo equations. Thisallows us to solve for thetwo unknown finalspeeds.

7-6 Inelastic CollisionsWith inelastic collisions, some ofthe initial kinetic energy is lost tothermal or potential energy. Itmay also be gained duringexplosions, as there is theaddition of chemical or nuclearenergy.A completely inelastic collision isone where the objects sticktogether afterwards, so there isonly one final velocity.

7-7 Collisions in Two or Three DimensionsConservation of energy and momentum can alsobe used to analyze collisions in two or threedimensions, but unless the situation is verysimple, the math quickly becomes unwieldy.Here, a moving objectcollides with an objectinitially at rest. Knowingthe masses and initialvelocities is not enough;we need to know theangles as well in order tofind the final velocities.

7-7 Collisions in Two or Three DimensionsProblem solving:1. Choose the system. If it is complex,subsystems may be chosen where one ormore conservation laws apply.2. Is there an external force? If so, is thecollision time short enough that you canignore it?3. Draw diagrams of the initial and finalsituations, with momentum vectors labeled.4. Choose a coordinate system.

7-7 Collisions in Two or Three Dimensions5. Apply momentum conservation; there will beone equation for each dimension.6. If the collision is elastic, apply conservationof kinetic energy as well.7. Solve.8. Check units and magnitudes of result.

7-3 Collisions and Impulse The impulse tells us that we can get the same change in momentum with (LARGE FORCE x short time) large force acting for a short time, or (small force x LONG TIME) small force acting for a longer time. This is why you should bend your knees when you land; why airbags work; and why landing on a pillow hurts less