Lecture 9 Momentum, Impulse

Lecture 9Momentum, impulseand energyPre-reading: KJF §9.1 and 9.2

MOMENTUM AND IMPULSEKJF chapter 9

beforeafterCOLLISIONcomplex interaction3

Linear Momentum of a BodyWe define the momentum of an object as:p mvwhere m mass and v velocity.p is a vector and is in the same direction as v.(Don’t confuse p with power or pressure.)Units: kg.m.s–1KJF §9.24

Momentum and Newton’s 2nd LawIf Fnet and m are constant, thenFnet ma m Δv/Δt Δmv/Δt Δp/ΔtNewton originally expressed his second law interms of momentum.If m or F are NOT constant then:5Fnet dp/dt

Momentum of a system of particlesTotal momentum p is the vector sum of individualmomenta:p pi mi vi(Vector sum!)Now consider the particles as one system:N2L becomes: Fext dp/dtwhere Fext net external forcei.e. not forces between particles in the system.6

Conservation of MomentumSo if Fext 0 for a systemthen dp/dt 0 total momentum is constantWhen the particles interact (e.g. billiard ballcollision, explosion etc.), if net external force iszero then total momentum before interactionequals total momentum after i.e. momentumis conservedpinitial pfinal( mivi )initial ( mivi )final mtot vcmKJF §9.47

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ExampleCalculate the recoil speed of a pistol (mass 0.90 kg)given that the bullet has mass 8.0 g and emergesfrom the pistol with a speed of 352 ms–1. Assumethe momentum of the exhaust gas is negligible.vBv P [3.1 ms–1]10

ImpulseThe impulse J of a force is defined as the changein momentum Δp caused by that force.From Newton’s Second Law, if F is constantF Δp/ΔtThenF Δt Δp JExample: 1.0 kg object falls under gravity. Calculate theimpulse the object experiences due to its weight afterfalling for 10 s.[98 kg m s–1 downwards]KJF §9.111

Impulse (2)However, if F is not constantJ Fav t or J F(t) dti.e. impulse area under F vs t curveRemember that if F is constant Fav FConsider a ball hitting a wall:vivfKJF §9.1F acting for time twall12J ΔpFwall on ballTime

Impulse (3)J Δp Fav tYou can minimise the average force Fav during animpact, by increasing the impact time te.g. seat belts, driver's air bag, wicket-keeper’sglove, thick landing mat for high jumper.13

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Force on a high-jumperConsider a 50 kg high-jumperfalling 2.0 m.– What is her momentum justbefore impact?– What force will she experience ifshe slows down in– 0.01 s–1s15

Example: Hitting a cricket ballA 150g cricket ball is bowled with aspeed of 20 ms–1. The batsman hits itstraight back to the bowler at 40 ms–1,and the impulsive force of bat on ballhas the shape as shown.(a) What is the maximum force the batexerts on the ball?(b) What is the average force the batexerts on the ball?16[30 kN, 15 kN]

ENERGYKJF chapter 10

What is Energy?Energy is needed to do useful work.Energy can move things, heat things up, cool them down,join things, break things, cut things, make noise, make light,and power our electronics, etc.Energy is a scalar (no direction, not a vector — easy maths!)S.I. unit of energy is the joule, JExamples of energy? Energy of motion - "kinetic energy"Stored energy - "potential energy": gravitational, elastic, chemicalEnergy in hot objects - "thermal energy"KJF §10.1–10.218

Kinetic EnergySimplest form of energy is energy of motion –kinetic energyK ½ mv2where m is mass (kg) and v is magnitude ofvelocity (ms–1).Unit definition: 1 J 1 joule 1 kg.(m.s–1)2 1 kg.m2.s–2Example: A 1.0 kg mass moves @ 2.0 ms–1. Find K.E.[K ½ 1.0 kg 4.0 m2.s–2 2.0 J]KJF §10.519

Gravitational Potential EnergyStored energy due to height in a gravitational field:G.P.E. or U mghwhere m is mass (kg) and h is height above theorigin level (m).The origin position (h 0) can be freely chosenU is always relative to some reference level or position.Example: A 1.0 kg mass is held 10 m above the ground.Find its G.P.E. relative to the ground.[U 1.0 kg 9.8 ms–2 10 m 98 J]KJF §10.620

Mechanical EnergyKinetic energy and potential energy addedtogether are called Mechanical Energy.Potential energy is stored energy resulting fromany force which depends only on position (e.g.gravity, force in a spring, electrostatic attraction).Gravitational potential energy is only oneexample of this.KJF §10.321

Law of Conservation of EnergyEnergy cannot be created or destroyed(i.e. it is "conserved")It can only be changed from one form to anotherORIn an isolated system — one where there is noenergy transfer into or out of the system — thetotal energy Etot is conserved.KJF §10.322

Questions

An explosion splits an object initially at rest into twopieces of unequal mass. Which piece moves at greaterspeed?1. The more massive piece.2. The less massive piece.3. They both move at the same speed.4. There is not enough information to tell.

An explosion splits an object initially at rest into twopieces of unequal mass. Which piece has greater kineticenergy?1. The more massive piece.2. The less massive piece.3. They both have the same kinetic energy.4. There is not enough information to tell.

Compared to the amount of energy required toaccelerate a car from rest to 10 kmh–1, the amount ofenergy required to accelerate the same car from 10kmh–1 to 20 kmh–1 is1. the same2. twice as much3. three times as much4. four times as much

NEXT LECTUREWork, power and potential energyRead: KJF §9.1, 9.2

Energy is a scalar (no direction, not a vector — easy maths!) S.I. unit of energy is the joule, J Examples of energy? Energy of motion - "kinetic energy" Stored energy - "potential energy": gravitational, elastic, chemical Energy in hot o