Momentum & Energy conservation

Momentum

Newton’s 2nd law (shorthand version)

F = ma

a = change in v

time

F = mchange in vtime

Fc Ft

Fc = mc change in vc time

Fc t = mc change in vc

Ft = mt change in vt time

Ft t = mt change in vt

Fc t + Ft t = mc change in vc + mt change in vt

(Fc + Ft)t =change in mcvc + change in mtvt

Car truck collision

(Fc + Ft)t =change in (mcvc + mtvt)

Fc Ft

Car truck collision

(Fc + Ft)t =change in (mcvc + mtvt)

Newton’s 3rd law: Fc = -Ft (Fc + Ft)t = 0

0 = change in (mcvc + mtvt)

mcvc + mtvt stays constant!

Momentum = mvmcvc = momentum of car

Mtvt= momentum of thruck

mcvc + mtvt = total momentum

this changesthis changes

this stays constant

Momentum is conserved!

Before = -40

After = -40

True for all collisions

before =+20

after =+20

visit www.physicsclassroom.com/mmedia/index.html

Revisit the canoe at the dockInitial momentumcanoe = 0boy = 0Total = 0

final momentumcanoe = mcvc

boy = mbvb

Total = 0

Momentum is a vector: mvcollision in 2 dimensions

Finding nemoeating

Billiard balls

2

before

1ptot

1

2

after

ptot

Conservation of momentumon a sub-atomic level

- meson- meson

proton

p

p

before

after

ptot

ptot

Rocket travel

before

P0

after

P0 + pexhaust

p

Rifle recoil

mVmV

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= Fx dist∥ = 0

He doesn’t do any work!

Garcon does work whenhe picks up the tray

but not while hecarries it around

the room

dist is not zero,but dist∥ is 0

Why this definition?

Newton’s 2nd law: F=m a

Definition of work+ a little calculus

Work= change in ½mv2

A scalar

equation

A vector

equation

This scalar quantity is givena special name: kinetic energy

Concept of Kinetic Energy

Emilie du Châtelet (1706-1749)

Brilliant mathematician

One of Voltaire’s lovers

Work = change in KE

This is called:

the Work-Energy Theorem

Units again…

Kinetic Energy = ½mv2

kg

m2

s2

work = F x dist∥

N m =kg

ms2

m=1Joule

same!

Work done by gravity

start

end

dist dist∥

W=mg

Work = F x dist∥

= -mg x change in height

= -change in mg h

change in vertical height

Gravitational Potential Energy

Workgrav = -change in mgh

This is called:“Gravitational Potential Energy” (or PEgrav)

Workgrav = -change in PEgravchange in PEgrav = -Workgrav

If gravity is the only force doing work….

-change in mgh = change in ½ mv2

0 = change in mgh + change in ½ mv2

change in (mgh + ½ mv2) = 0

mgh + ½ mv2 = constant

Work-energy theorem:

Conservation of energy

mgh + ½ mv2 = constant

GravitationalPotential energy

Kinetic energy

If gravity is the only force that does work:

PE + KE = constant

Energy is conserved

Free fall (reminder)

V0 = 0

t = 0s

V1 = 10m/s

t = 1s

V2 = 20m/s

t = 2s

V3 = 30m/s

t = 3s

V4 = 40m/s

t = 4s

75m

60m

35m

0m

height

80m

m=1kg free falls from 80m

V0 = 0 h0=80m

t = 0s

V1 = 10m/s; h1=75m

t = 1s

V2 = 20m/s; h2=60m 600J 200J 800J

t = 2s

V3 = 30m/s; h3=35m 350J 450J 800J

t = 3s

V4 = 40m/s; h4=0 0 800J 800J

t = 4s

mgh ½ mv2 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

hmax

Etot=mghmaxEtot=mghmax

Etot=1/2 m(vmax)2

Roller coaster

Work done by a spring

RelaxedPosition

F=0

Fx

I compressthe spring

(I do + work;spring does

-work)

Work done by spring = - change in ½ kx2

If spring is the only force doing work….

-change in ½ kx2 = change in ½ mv2

0 = change in ½ kx2 + change in ½ mv2

change in ( ½ kx2 + ½ mv2) = 0

½ kx2 + ½ mv2 = constant

Work-energy theorem:

potential energy in the spring

Conservation of energysprings & gravity

mgh + ½ kx2 + ½ mv2 = constant

Gravitationalpotential energy

Kinetic energy

If elastic force & gravity are the only forces doing work:

PEgrav + PEspring + KE = constant

Energy is conserved

springpotential energy

example

KineticE

Spring PE

grav PE

Two types of forces:

“Conservative” forcesforces that do + & – work

•Gravity

•Elastic (springs, etc)

•Electrical forces

•…

“Dissipative” forcesforces that only do – work

•Friction

•Viscosity

•….

-work change in PE

-work heat(no potential energy.)

(-)Work done by frictionheat

Thermal atomic motion

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

Air solid

Work-energy theorem(all forces)

Workfric = change in (PE+KE)Work done

dissipative

Forces(always -)

Kinetic energy

-Workfric = change in heat energy

potential energyFrom all

Conservative forces

-change in Heat Energy = change in (PE+KE)

Workfric = -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 energykinetic energy

Units againHeat units:

1 calorie = heat energy required to raise the temp of 1 gram of H2O by 1o C

1 calorie= 4.18 JoulesKg m2/s2

Food Calories

1 Calorie = 1000 calories = 1Kcalorie

1 Calorie= 4.18x103 JoulesThe Calories you read on food labels

8 x 105 J

7 x 106 J

2 x 106 J

Power

Rate of using energy:amout of energy

elapsed time

Units:Joule

second 1 = 1 Watt

Power =

A 100 W light bulbconsumes 100 J of

electrical energy eachsecond 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 hoursenergytimePower = energy = power x time

power unit x time unit = energy unit

Elec companies use:Kilowatts(103 W)

hours(3600 s)

1 kilowatt-hour = 1kW-hr

= 103 W x 3.6x103

s = 3.6x106 WsJ

x

In Hawaii electrical energy costs about 25cents /kW-hrabout 300 won

What is the cost in Seoul?