Conservation

Conservation

Theorems

Theorems

:

:

Momentum

Momentum

Luis

Luis

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Originally introduced by Newton as the quantity of motion

Originally introduced by Newton as the quantity of motion

the momentum

the momentum

is

is

defined

defined

as p =

as p =

mv

mv

Momentum

Momentum

of

of

a

a

particle

particle

Using

Using

Newton

Newton

´

´

s

s

second

second

law

law

we

we

can relate

can relate

the

the

momentum

momentum

of

of

a

a

particle

particle

to

to

the force

the force

acting

acting

on

on

the

the

particle

particle

Substituing

Substituing

the

the

force

force

F by

F by

ma

ma

net

net

Luis

Luis

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The

The

net

net

force

force

acting

acting

on

on

a

a

particle

particle

equals

equals

the

the

time

time

rate

rate

of

of

change

change

of

of

the

the

particle

particle

´

´

s

s

momentum

momentum

In

In

his

his

famous

famous

treatise

treatise

Principia (1687)

Principia (1687)

Newton

Newton

presents

presents

the

the

second

second

law

law

of

of

motion

motion

in

in

this

this

form

form

Conservation

Conservation

of

of

Momentum

Momentum

The

The

total

total

momentum

momentum

of

of

a

a

system

system

of

of

particles

particles

reads

reads

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Luis

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Anchordoqui

According

According

to

to

Newton

Newton

´

´

s

s

second

second

law

law

If

If

constant

constant

If

If

the

the

sum

sum

of

of

the

the

external

external

forces

forces

on

on

a

a

system

system

remains

remains

zero

zero

the

the

total

total

momentum

momentum

of

of

the

the

system

system

is

is

conserved

conserved

Luis

Luis

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Anchordoqui

During

During

repair

repair

of

of

the

the

Hubble

Hubble

Space

Space

Telescope

Telescope

,

,

an

an

astronaut

astronaut

replaces

replaces

a

a

damaged

damaged

solar panel

solar panel

during

during

a

a

spacewalk

spacewalk

Pushing

Pushing

the

the

detached

detached

panel

panel

away

away

into

into

space

space

,

,

she

she

is

is

propoelled

propoelled

in

in

the

the

opposite

opposite

direction

direction

.

.

Luis

Luis

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Anchordoqui

Before

Before

After

After

A

A

Skateboard

Skateboard

workout

workout

A 40

A 40

kg

kg

skateboarder

skateboarder

on

on

a 3

a 3

kg

kg

board

board

is

is

trainning

trainning

with

with

two

two

5

5

kg

kg

weights

weights

.

.

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Luis

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Beginning

Beginning

from

from

rest

rest

,

,

she

she

throws

throws

the

the

weights

weights

horizontally

horizontally

,

,

one

one

at

at

a time,

a time,

from

from

her

her

board

board

.

.

The

The

speed

speed

of

of

each

each

weight

weight

is

is

7 m/s

7 m/s

relative

relative

to

to

her

her

after

after

it

it

is

is

thrown

thrown

.

.

Assume

Assume

the

the

board

board

rolls

rolls

without

without

friction

friction

.

.

V = 0.66 m/s

V = 0.66 m/s

V = 1.39 m/s

V = 1.39 m/s

Collisions and Impulse

Collisions and Impulse

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Luis

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When

When

two

two

objects

objects

collide

collide

they usually

they usually

exert

exert

very

very

large

large

forces

forces

on

on

each

each

other

other

for

for

very

very

brief

brief

time

time

The

The

impulse

impulse

of

of

a

a

force

force

exerted

exerted

during

during

a

a

time

time

Δ

Δ

t = t

t = t

C

C

t

t

is

is

a vector

a vector

defined

defined

as

as

f

f

i

i

Impulse

Impulse

is

is

a

a

measure

measure

of

of

both

both

strength

strength

and

and

duration

duration

of

of

the

the

collision

collision

force

force

Impulse

Impulse

momentum

momentum

theorem

theorem

for

for

a

a

particle

particle

Impulse

Impulse

momentum

momentum

theorem

theorem

for

for

a

a

system

system

Average force

Average force

R = 190 m

R = 190 m

to

to

hole

hole

Hitting

Hitting

a golf

a golf

ball

ball

0

0

Perfectly

Elastic

and

Perfectly

Inelastic

Collisions

Luis

Luis

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In

In

elastic

elastic

collisions

collisions

the

the

kinetic

kinetic

energy

energy

of

of

the

the

system

system

is

is

conserved

conserved

Luis

Luis

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Anchordoqui

A bullet travelling 850

A bullet travelling 850

m/s

m/s

collides

collides

inelastically

inelastically

with an apple,

with an apple,

which disintegrates completely moments later.

which disintegrates completely moments later.

Exposure time is less than a

Exposure time is less than a

millionth

millionth

of

of

a

a

second

second

.

.

(a)

(a)

What

What

was

was

the

the

Earth's

Earth's

recoil

recoil

speed

speed

?

?

(b)

(b)

What

What

fraction

fraction

of

of

the

the

meteor's

meteor's

kinetic

kinetic

energy

energy

was

was

transformed

transformed

to

to

kinetic

kinetic

energy

energy

of

of

the

the

Earth

Earth

?

?

(c) By how

(c) By how

much

much

did

did

the

the

Earth

Earth

kinetic

kinetic

energy

energy

change

change

as a

as a

result

result

of

of

this

this

collision

collision

?

?

A meteorwhosemasswasabout10 kgstrucktheEarth

(M = 6 x 10 kg) witha speedofabout15 km/s

andcametorestin Earth

8

8

24

24

v = 2.5 x 10 m/s

v = 2.5 x 10 m/s

Luis

Luis

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C

C

13

13

∆

∆

K = 0.19 J

K = 0.19 J

Earth

Earth

final

final

Earth

Earth

initial

initial

meteor

meteor

= 1.7 x 10

= 1.7 x 10

K

K

C

C

17

17

K

K

Conservation of Energy and Momentum in Collisions

Conservation of Energy and Momentum in Collisions

Momentum is conserved in all collisions

Momentum is conserved in all collisions

Collisions in which kinetic energy

Collisions in which kinetic energy

is conserved as well are called elastic collisions

is conserved as well are called elastic collisions

and those in which it is not are called inelastic

and those in which it is not are called inelastic

Approach

Approach

Collision

Collision

If

If

elastic

elastic

If

If

inelastic

inelastic

Luis

Luis

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Anchordoqui

Here we have two objects colliding elastically

Here we have two objects colliding elastically

We know the masses and the initial speeds

We know the masses and the initial speeds

This allows us to solve for the two unknown final

This allows us to solve for the two unknown final

speeds

speeds

Luis

Luis

Anchordoqui

Anchordoqui

Elastic Collisions in One Dimension

Since both momentum and kinetic energy are conserved

Since both momentum and kinetic energy are conserved

we can write two equations

we can write two equations

Luis

Luis

Anchordoqui

Anchordoqui

Coefficient of Restitution

Coefficient of Restitution

where is the relative velocity of the two objects afte

where is the relative velocity of the two objects afte

r the

r the

collision and is their relative velocity before it.

collision and is their relative velocity before it.

(a) Show that e = 1 for a perfectly elastic collision

(a) Show that e = 1 for a perfectly elastic collision

e = 0 for a complete inelastic collision.

e = 0 for a complete inelastic collision.

for a perfectly elastic collision e = 1

for a perfectly elastic collision e = 1

for a complete inelastic collision e = 0

for a complete inelastic collision e = 0

A measure of inelasticity in a head

A measure of inelasticity in a head

C

C

on collision of two objects is the

on collision of two objects is the

coefficient of restitution defined as

coefficient of restitution defined as

Luis

Luis

Anchordoqui

Anchordoqui

Determine a formula fore in terms of the original height h and t

Determine a formula fore in terms of the original height h and t

he

he

maximum height

maximum height

h'reached

h'reached

after one collision.

after one collision.

Collisions in Two or Three Dimensions

Collisions in Two or Three Dimensions

Conservation of energy and momentum can also be used to analyze

Conservation of energy and momentum can also be used to analyze

collisions in two or three dimensions

collisions in two or three dimensions

but unless the situation is very simple

but unless the situation is very simple

the math quickly becomes unwieldy

the math quickly becomes unwieldy

Luis

Luis

Anchordoqui

Anchordoqui

In anelasticcollisionbetweentwoobjectsofequalmasswiththetargetobject

initiallyatresttheanglebetweenthefinal velocitiesoftheobjectsis90º

Momentum

Momentum

conservation

conservation

m

m

Assumethatthetargetballishit correctlyso thatitgoesin the

pocket

Fromthegeometryoftherighttriangle

Becausetheballswillseparateat90ºifthetargetballgoesin the

pocketthisdoesappeartobe a goodpossibilityofa scratchshot

LuisAnchordoqui

θ

θ

3 m

1 m

√

3 m

1

2

Fromthegeometryofthelefttriangle

Eruption

Eruption

of

of

a

a

large

large

volcano

volcano

on

on

Jupiter

Jupiter

´

´

s

s

moon

moon

When

When

the

the

volcano

volcano

erupts

erupts

the

the

speed

speed

of

of

the

the

effluence

effluence

exceeds

exceeds

the

the

escape

escape

speed

speed

of

of

Io

Io

and

and

so a

so a

stream

stream

of

of

particles

particles

is

is

projected

projected

into

into

space

space

The

The

material in

material in

the

the

stream

stream

can

can

collide

collide

with

with

and

and

sticks

sticks

to

to

the

the

surface

surface

of

of

an

an

asteroid

asteroid

passing

passing

through

through

the

the

stream

stream

We

We

now

now

consider

consider

the

the

effect

effect

of

of

the

the

impact

impact

of

of

this

this

material

material

on

on

the

the

motion

motion

of

of

the

the

asteroid

asteroid

Luis

Luis

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Anchordoqui

Dividing

Dividing

by

by

∆

∆

t

t

Taking

Taking

the

the

limit

limit

∆

∆

t 0

t 0

that

that

also

also

means

means

∆

∆

M 0

M 0

and

and

∆

∆

v 0

v 0

Rocket

Propulsion

Themassoftherocketchangescontinously

as itburnsfuel andexpelsexhaustgas

Luis

Luis

Anchordoqui

Anchordoqui

Rocket

Propulsion

(

cont

’

d

)

Considera rocketmovingstaightup withvelocityv relativetoEarthand

assumethefuel isburnedatconstantrate

α

Therocket’smassattime t is

m = m C

α

t

0

Initialmassoftherocket

Theexhaustgases leavetherocketenginewith

velocityu relativetotherocket

Wechoosetherocketincludingunspentfuel as thesystem

Neglectingairdragtheonlyexternalforceonthesystemisthatofgravity

LuisAnchordoqui

ex

F = mg

next,ext

Therocketequationisthen

Thequantity–

αu isthethrustforceexertedontherocketby theexhaust

gases

ex

mgC

αu = m

ex

dv

dt

F = C

αu = Cu

th

ex

dmdt

ex

Rocket

Propulsion

(

cont

’

d

)

Integration

Integration

leads

leads

to

to

The

The

acceleration

acceleration

is

is

then

then

Choosing

Choosing

upward

upward

as

as

the

the

positive y

positive y

direction

direction

the

the

direction

direction

of

of

u

u

is

is

downward

downward

so u =

so u =

C

C

u

u

Substituting

Substituting

gives

gives

ex

ex

ex,y

ex,y

ex

ex

LuisAnchordoqui

Saturn

Saturn

V:

V:

America

America

’

’

s

s

Moon

Moon

Rocket

Rocket

Saturn

Saturn

V

V

C

C

developed

developed

at

at

NASA

NASA

’

’

s

s

Marshall

Marshall

Space

Space

Flight

Flight

Center

Center

C

C

was

was

the

the

largest

largest

in a

in a

family

family

of

of

liquid

liquid

C

C

propellant

propellant

rockets

rockets

that

that

solved

solved

the

the

problem

problem

of

of

getting

getting

to

to

the

the

Moon

Moon

32

32

Saturns

Saturns

were

were

launched

launched

not

not

one

one

failed

failed

!!!

!!!

The

The

Saturn

Saturn

V

V

was

was

flight

flight

C

C

tested

tested

twice

twice

without

without

crew,

crew,

the

the

first

first

manned

manned

Saturn

Saturn

V

V

sent

sent

the

the

Apollo

Apollo

8

8

astronauts

astronauts

into

into

orbit

orbit

around

around

the

the

Moon

Moon

in

in

December

December

1968

1968

After

After

two

two

more

more

missions

missions

to

to

test

test

the

the

lunar

lunar

landing

landing

vehicle

vehicle

in

in

July

July

16th 1969 a

16th 1969 a

Saturn

Saturn

V

V

launched

launched

the

the

crew

crew

of

of

Apollo

Apollo

11

11

to

to

the

the

first

first

manned

manned

landing

landing

on

on

the

the

Moon

Moon

Apollo

Apollo

11

11

rocket

rocket

blast

blast

off

off

Estimate

Estimate

the

the

final

final

speed

speed

of

of

the

the

first

first

stage

stage

at

at

burnout

burnout

and

and

its

its

vertical

vertical

height

height

Luis

Luis

Anchordoqui

Anchordoqui

m

m

≈

≈≈≈

≈

≈≈≈

2.8 x 10

2.8 x 10

kg

kg

mean

mean

thrust

thrust

≈

≈≈≈

≈

≈≈≈

37 x 10 N

37 x 10 N

mass

mass

of

of

first

first

stage

stage

fuel

fuel

≈

≈≈≈

≈

≈≈≈

2.1 x 10

2.1 x 10

kg

kg

u

u

≈

≈≈≈

≈

≈≈≈

2600 m/s

2600 m/s

0

0

6

6

6

6

6

6

Saturn

Saturn

V:

V:

America

America

’

’

s

s

Moon

Moon

Rocket

Rocket

(

(

cont

cont

’

’

d

d

)

)

Luis

Luis

Anchordoqui

Anchordoqui

Saturn

V:

America

’

s

Moon

Rocket

(

cont

’

d

)

The

The

final

final

rocket

rocket

mass

mass

m= 2.8 x 10

m= 2.8 x 10

kg

kg

–

–

2.1 x 10

2.1 x 10

kg

kg

= 0.7 x 10

= 0.7 x 10

kg

kg

6

6

6

6

6

6

Thespeedofthespaceshipatburnout

Time toburnout

Saturn

V:

America

’

s

Moon

Rocket

(

cont

’

d

)

LuisAnchordoqui

SaturnV: America’sMoonRocket(cont’d)

EvaluateC frominitialconditions@t = 0 y = 0 andm = m

0

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