Momentum & Energy conservation

Momentum

Newton’s 2 nd law (shorthand version) F = ma a = change in v time F = m change in v time

FcFc FtFt F c = m c change in v c time F c t = m c change in v c F t = m t change in v t time F t t = m t change in v t F c t + F t t = m c change in v c + m t change in v t (F c + F t )t =change in m c v c + change in m t v t Car truck collision (F c + F t )t =change in (m c v c + m t v t )

FcFc FtFt Car truck collision (F c + F t )t =change in (m c v c + m t v t ) Newton’s 3 rd law: F c = -F t (F c + F t )t = 0 0 = change in (m c v c + m t v t ) m c v c + m t v t stays constant!

Momentum = mv m c v c = momentum of car M t v t = momentum of thruck m c v c + m t v t = total momentum this changes this stays constant Momentum is conserved! Before = -40 After = -40

True for all collisions before =+20 after =+20 visit

Revisit the canoe at the dock Initial momentum canoe = 0 boy = 0 Total = 0 final momentum canoe = m c v c boy = m b v b Total = 0

Momentum is a vector: mv collision in 2 dimensions

Finding nemo eating

Billiard balls 2 before 1 p tot 1 2 after p tot

Conservation of momentum on a sub-atomic level  - meson proton  p  p before after p tot

Rocket travel before P0P0 after P 0 + p exhaust p

Rifle recoil mVmVmVmV

Machine-gun granny

Work and Energy

Physicist’s definition of “work” dist Work = F x dist ∥ A scalar (not a vector) dist ∥

Atlas holds up the Earth But he doesn’t move, dist ∥ = 0 Work= F x dist ∥ = 0 He doesn’t do any work!

Garcon does work when he picks up the tray but not while he carries it around the room dist is not zero, but dist ∥ is 0

Why this definition? Newton’s 2 nd law: F=m a Definition of work + a little calculus Work= change in ½mv 2 A scalar equation A vector equation This scalar quantity is given a special name: kinetic energy

Concept of Kinetic Energy Emilie du Châtelet ( ) Brilliant mathematician One of Voltaire’s lovers

Work = change in KE This is called: the Work-Energy Theorem

Units again… Kinetic Energy = ½mv 2 kg m2s2m2s2 work = F x dist ∥ Nm=kg ms2ms2 m =1Joule same!

Work done by gravity start end distdist ∥ W=mg Work = F x dist ∥ = - mg x change in height = - change in mg h change in vertical height

Gravitational Potential Energy Work grav = -change in mgh This is called: “Gravitational Potential Energy” (or PE grav ) Work grav = -change in PE grav change in PE grav = -Work grav

If gravity is the only force doing work…. -change in mgh = change in ½ mv 2 0 = change in mgh + change in ½ mv 2 change in (mgh + ½ mv 2 ) = 0 mgh + ½ mv 2 = constant Work-energy theorem:

Conservation of energy mgh + ½ mv 2 = constant Gravitational Potential energy Kinetic energy If gravity is the only force that does work: PE + KE = constant Energy is conserved

Free fall (reminder) V 0 = 0 t = 0s V 1 = 10m/s t = 1s V 2 = 20m/s t = 2s V 3 = 30m/s t = 3s V 4 = 40m/s t = 4s 75m 60m 35m 0m height 80m

m=1kg free falls from 80m V 0 = 0 h 0 =80m t = 0s V 1 = 10m/s; h 1 =75m t = 1s V 2 = 20m/s; h 2 =60m 600J 200J 800J t = 2s V 3 = 30m/s; h 3 =35m 350J 450J 800J t = 3s V 4 = 40m/s; h 4 = J 800J t = 4s mgh ½ mv 2 sum 800J 0 800J 750J 50J 800J

pendulum W=mg T Two forces: T and W T is always ┴ to the motion (& does no work)

Pendulum conserves energy h max E tot =mgh max E tot =1/2 m(v max ) 2

Roller coaster

Work done by a spring Relaxed Position F=0 F x I compress the spring (I do + work; spring does -work) Work done by spring = - change in ½ kx 2

If spring is the only force doing work…. -change in ½ kx 2 = change in ½ mv 2 0 = change in ½ kx 2 + change in ½ mv 2 change in ( ½ kx 2 + ½ mv 2 ) = 0 ½ kx 2 + ½ mv 2 = constant Work-energy theorem: potential energy in the spring

Conservation of energy springs & gravity mgh + ½ kx 2 + ½ mv 2 = constant Gravitational potential energy Kinetic energy If elastic force & gravity are the only forces doing work: PE grav + PE spring + KE = constant Energy is conserved spring potential energy

example KineticE Spring PE grav PE

Two types of forces: “Conservative” forces forces that do + & – work Gravity Elastic (springs, etc) Electrical forces … “Dissipative” forces forces that only do – work Friction Viscosity …. -work  change in PE -work  heat (no potential energy.)

(-)Work done by friction  heat

Thermal atomic motion Heat energy = KE and PE associated with the random thermal motion of atoms Airsolid

Work-energy theorem (all forces) Work fric = change in (PE+KE) Work done dissipative Forces (always -) Kinetic energy -Work fric = change in heat energy potential energy From all Conservative forces - change in Heat Energy = change in (PE+KE) Work fric = -change in heat energy

Work – Energy Theorem (all forces) 0 = change in Heat Energy + change in (PE+KE) 0 = change in ( Heat Energy +PE+KE) Heat Energy + PE + KE = constant Law of Conservation of Energy

Energy conversion while skiing Friction: energy gets converted to heat Potential energy Potential energy  kinetic energy

Units again Heat units: 1 calorie = heat energy required to raise the temp of 1 gram of H 2 O by 1 o C 1 calorie= 4.18 Joules Kg m 2 /s 2

Food Calories 1 Calorie = 1000 calories = 1Kcalorie 1 Calorie= 4.18x10 3 Joules The Calories you read on food labels 8 x 10 5 J 7 x 10 6 J 2 x 10 6 J

Power Rate of using energy: amout of energy elapsed time Units: Joule second 1= 1 Watt Power = A 100 W light bulb consumes 100 J of electrical energy each second to produce light

Other units Over a full day, a work-horse can have an average work output of more than 750 Joules each second 1 Horsepower = 750 Watts

Kilowatt hours energy time Power =  energy = power x time  power unit x time unit = energy unit Elec companies use: Kilowatts (10 3 W) hours (3600 s ) 1 kilowatt-hour = 1kW-hr = 10 3 W x 3.6x10 3 s = 3.6x10 6 WsJ x

In Hawaii electrical energy costs about 25cents /kW-hr about 300 won What is the cost in Seoul?