# Conservation Theorems: Momentum - UWM Center for Gravitation ...

Ηλεκτρονική - Συσκευές

8 Οκτ 2013 (πριν από 4 χρόνια και 9 μήνες)

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Conservation
Conservation
Theorems
Theorems
:
:
Momentum
Momentum
Luis
Luis
Anchordoqui
Anchordoqui
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
Anchordoqui
Anchordoqui
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
Luis
Luis
Anchordoqui
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
Anchordoqui
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
Anchordoqui
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
.
.
Luis
Luis
Anchordoqui
Anchordoqui
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
Luis
Luis
Anchordoqui
Anchordoqui
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
Anchordoqui
Anchordoqui
In
In
elastic
elastic
collisions
collisions
the
the
kinetic
kinetic
energy
energy
of
of
the
the
system
system
is
is
conserved
conserved
Luis
Luis
Anchordoqui
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
?
?
(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
Anchordoqui
Anchordoqui
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
Anchordoqui
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
Anchordoqui
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
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