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the exam on wednesday
really cover
uh... first
five lectures
the first two
homework assignments
so i list here the
topics
the way we discuss them
course
it is not possible too
this cause all of them today books
make a selection
i recall that we
discussed skating
and we use the interesting example probably a bloody mary
animal
an animal has legs
and we defined
the overall of sites
of the animal as j big golden tasks
and then we said well
that is here
if you are
and a few more has legs
and thickness d
it was completely reasonable to say well
at l will have to be proportional to ask
animal ten times larger than the number
flags will be typically ten times longer
since the mats
of the animal
must be proportional
its eyes to the power three
will also be proportional
domains of the few more to the power three
and then
came in these ke argument mainly
you don't want
the bones to be crushed
which is called you'll be in physics
if i take a piece of concrete
block of concrete and i put too much pressure on its it starts to crumble
what got into a government may have had in mind
and in order to protect
animals who get bigger and bigger and bigger
against this
crushing
we argued i will not go through that argument now anymore
that the mass we'll have to be proportional
the script
which is the cross-section
of the people
and so you see immediately
that d square has to be proportional help from two-thirds
must be proportional
for the length
of the few who'd apollo one-and-a-half
so this would mean
if you compare an elephant with amal
elephants
overall slices about hunger pangs
larger than a month you would expect a few more to be about a hundred times
larger which is true
but you wouldn't expect if he were to be about a thousand times sticker
and that turns out to be
not from as we have seen
the teamwork offbeat
elephant is only a hundred times
so it's kills just pesticides
and then so i was in the fact that nature
doesn't have to protect against
of the bones
there is a much larger danger
which we call
and buckling
is the phenomenon that the bones
do this
and it now you puts too much pressure on it
the bones will break
and that's the case
remember
in fact all you have to do
if you have to scale d
proportional
at which is not intuitive
bethnal too easy to divide
but that's the case
and so did angel then nation nature
protects animals against is buckling
and then the buckling becomes
i would imagine
the bones
at some point in time
will be hooked up on friday
will break and that's what nature
tries to present
so that was a skating argument
and that's now
talk about
top quality but look there are scams a little bit in a random way over my
topics
so let's not
talk about
stop products
i have a vector any
any expands ex-royal which is the units vector eniac spelled action
loser in life
my roof
plus a g
these are good for you inspectors
he explains the direction of these artifacts component
light and easy component of the factory
i have another factor b
the effects
the of wine
why don't
beauty
now the dot product
eight
dot b also called skater product
is the same
and be dog eniku
and it is the find as any acts
the acts
goes any
why
delight
close eighty
damaging it is the number
it is a skater it just simple number
and sold his number
can be larger than zero
because of it can be put to zero panels of this moment
just
there's a lot of weight
that's you can defy you can call these
maktoob noble longish
prefer that
there's another way that you can
finder dot thought i could give you exactly the same results
if you have a vector
and you have that that could be
andy angle between them
data
then you can project be only
or ailene be for that matter makes no difference
and their projection
the length of this projection
is then of course be
coastline data
and so a dolby
and that is exactly the same you may want to go
through approval that's
it is the length of age
times the length of the
and cosign
a fake
and that will give you precisely the same themselves
what is interesting about this formulation which pissed
lax
that you can immediately see
it can be
too i had ninety degree angle sport two hundred seventy degrees for that matter
then the doll product is zero
but that's a incited to get through this mom
which you like through that other
let us take a
down to it
example
over the top product
supposedly
equals
reacts
and beyond
equals
to access
as to why
and i'm asking you what is the doc product
well you could turn use method number one we picture
this case he said
far too fast is beneath me
aac history
be accessed through to the dispute six
well
there is no
a blind
there is no easy to get him
just six
that's the top top
you could have done it that way
it's a little bit more complicated but i certainly want to show you
that it works
this is the expiration
and at the white election we don't have to look into disney direction because
there's no g component
then this will be
sector eighty
at this point
would be a three
this would be too
these would be too
so this would be defect to be
i did to me to keep you know that this angle
forty five degrees photos from
between the two
so if we don't quite
method number two
it'll be fun
first the length of age
at three
times the length of b
that is to
crimes is critical to
this is too busy stewardesses to scramble to
front because i know
forty five degrees
which is one haha
screw to any ends with six
note is that this crew took two and a spirit of to either of these two
and you get sick
at the same answer of course
but would be a dumb thing
since it can be done
so much
uh... on cross products
i don't want to go through the formalism of cross products
the way we did that with the determinant
are just want to
remind you that if you have across products
directors
it is minus
because any
and that would be
the magnitude
of c
is the length of a
i'm going to be
either side of the angle
the factor c
dot products across products
is always perpendicular
composi
and perpendicular to be
know it's it's perpendicular to the plane
of the two factors
now the probe indicate to the plane and in that case it's perpendicular to
blackboard you have two choices if you're coming at you perpendicular
which commands fight straight into the blackboard
and now
everyone has to his own way of doing it
culture was caught the right time corkscrew
you take the first one it is mentioned in this case any
and you rotate it
over the shortest angle to be
when you do that
rotate you corkscrew
scene from where you sit in comfort clockwise and the courts will comes to
you
and so the direction
the factory
is such that you will see the people of the vector post coming straight
the blackboard so that gives you then
the direction
now i will give you
the position
x oven
object as a from turned off
and then we're going to ask ourselves a lot of questions
sausages
accelerations
sort of everything you can
finkel everything we could
covered speeds
i will cover here fort seconds off time so this is the time actions
in seconds
and we will cover four seconds
sold at fifty one
four
well and let the object
the exposition
plus six
this is my
x-axis this is going to object is actually moving
is history
minus three and this let's say if you need us
let's make a little grades but it's easier for me too
put in the
wells
career
uh...
uh...
uh...
all right
malcolm's
acts as a function of
time he calls
he second
object is here
and it came from there
and this part
is a parabola
and this parabola here is horizontal
important
you have to know that uh... this is a parabola
and this year
historical
so dulbecco's from plus six two three
then it goes from miami three
then it stays there for one second
and then it goes back
in one second
to plastics
as a one-dimensional problem
emotion
only x
will be ecstatic
let's analyze all these difference seconds that occurred
but first a
the first seconds
if we're stuck
jim this is a parabola
you know that the
acceleration is constant
so i hope you will conclude immediately
eight
must be approved meeting is a constant
the position acts as a function of time
should change of sports
x zero
v zero t
was one-half
he's great
ike specter to know that he creation
very often do i give you
equations
but the exam
and that may well happen during the second and third exam but that will not
indicates this time
you creations are all
very fundamental
you have to make them part of you world
for this immigration
you would have to remember
all right
what is the the velocity
year
velocity starts out two b zero
and the velocity here
is not theory
if i look at
bunty equals one
then i have your x zero four six
it starts out with what it is the order to give in
and i get plus one half
times eighteen square it with it is only one seconds
and so
when it is at three
i have six
levanthal
times once created
until you find that eighty
equals minus six
readers
second's christmas during this
first second the acceleration might have six meters
the second screen
and the velocity
as a function of time
reading for this moment
zero
plus eighty
incurable zero
and so that is minus
six times from velocities changing
in the manufactured
well right now
about the ends
of the first second
well i can say that access three
what about all of the velocity wilder velocities
minors
six
when nobody
i don't know about it
it's truly doing this for a second
days minus six meters per second square but it's changes abruptly at this point
so she'll find at this point by contextually
known physical so i really don't know
exactly at the end
what acceleration
let's not go to the second circuit
a nazi
what happens there
second second
and first
let's look
and then we'll look at
situation again
doing that second second
it is clear
and this is a straight line
that the velocity remains constant
handed remains minus six meters per second
but that's exactly what it was at this point at the end
you can see it go six meters
publicity to minus three
in one second
sort of a lost city
is minus six
yeah meters per second
the acceleration is therefore zero
if you get the acceleration changes abruptly from minus six meters per
second square
two zero so
i can tell you what it is exactly at this moment in time
but that's the situation during
second
second one of the situation at the ended
of the second section
at the end i know that exit polls minus three
was a velocity
because it changes abruptly beautiful minus six to zero so i don't know
exactly what was at that point it soon
non physical things
very abrupt change
the acceleration
yeah that's also a very tricky thing
because if the velocity
is minus sixty on the site of the two seconds
and here becomes zero
and if that happens at it in a split second to let me know
deceleration just at that point which is not physical
so i would also put a question mark
at the end o'neal keys
so we go to that effect
let's first look
if it's
well
the object isn't going anywhere just sitting there
x remains minus three
and
the velocity
zero
hazier we can agree on that
one of the situation at the end
of the fifth second the people three
well i'd always excuse my next week
that's non-negotiable
what the velocities
i don't know because it's changing abruptly
on zero
to a positive value
so detailed to find
and the same is true for the acceleration business southern changing
velocity that means there must be a huge acceleration
ill-defined
because this group
is of course not very physical
let's not look
velocity four seconds
chris during
is going from one two three two plus six is a straight line sort of a lawsuit is
constant
is the velocity is constant then you can immediately conclude that eighty zero
there's no acceleration
and it goes not only does in a time span a one second
but it's now plus
plus nine meters per second
but you object first
audited values of x
zero into negative values for external all the time
the velocity walls
by replying convention
and now
philosophy
goes back to plus six
velocity becomes plus nine
for a second
one of the story at the end of the four seconds
though i can say is that x equals plus six
i don't know much more
i don't know what the velocities near there were no with the acceleration is
the false hopes there anyhow
now i would
think that it is reasonable to have to forward question
was the aboriginal also difference lynn between time zero and part of fort
ever to blossom
redefined average velocity
as the position
at time four seconds
minus two position at times euro divided by four that is our definition
adc zero david pasig
exporters of politics
photo upstairs in zero
sold the average velocity
doing this for second trip
it's zero
you may not like that
it may go against intuition
of course i couldn't agree more mission but that's the way we define philosophy
each is defined differently
each is the magnitude off the velocity vector and the speech therefore always
has a positive value another show you know what is the average
pge
between
zero and four
that is the distant but it has travelled in these four seconds
let's first go for the first second it goes from plus the six-two plus three
so little ready troubles
three meters
that in the second
second it goes from plus three to minus three so we're troubles
a lot of six meters
and then in the third seconds
it's lazier doesn't do anything
the distance traveled zero
and then in the last one second it gets very active
and it's travels
nine leaders
knowledge you don't issue plus signs here
dunno minus sign that would make no sense
and this
authors in four seconds
so
that is
four-and-a-half
for the everett speakers four-and-a-half meters per second
but the average velocity
is zero
record now
the plot
albee
philosophy
as a function off time
let me put here
four-and-a-half
but i just have enough room here
to make the velocity plz
as a function of time
i'll make a new won't
this is my contacts list
and this
velocity
zero
one second
two seconds
three seconds
foresight
and his lawsuit is in need us
per second
i go up here
plus ten
and here
minus five
yours mine is six
so what do i don't know i know that the
velocity during the first second
minus sixty
fotolia
and saudi during the first second
this is the
falsity
as a function of time it starts at zero you can see that
and when it is here because of a lot of killed minus six meters
during the second
seconded remains minus six meters per second
dole's during the second circuit
philosophy is not changing
doing just that second
the velocity jobs all of us not into zero you see homes and utilities
and so all of us total doing two thirds seconded becomes zero
but i have to somehow connection of course between the two to make this
physical
so in in very small amount of time that will have to walk her that's why did you
check celebration at that point
which also get an acceleration here at this point because also changes below
and then delete a four seconds
the velocities plus nine meters per second
and so we jump off
squall plus nine
uh...
and so
we have here
during the last second
and again this is not physical but i have to be somehow
transition
until you see
the velocity as a function of time
malcolm's it interesting question
is it possible
if i gave you this
but this is a given
you can see that
good you can put this back to that
and the answer is yes provided
but i tell you
what the position is
at p zero at three equal zero
actual
imports
plus six
and that's is sufficient for you
do you think information
and to reconstruct
it's an interesting thing to do
and if you feel like it's
i would say given the shot
alright so far
about speeds and edwards velocities
and accelerations
let's not go to
projectors
three-dimensional
projector trajectories
thank goodness
how old was never three-dimensional
they're always two-dimensional because the trajectory itself is in a vertical
plane
and so we normally
when we throw up an object
gravitational field
the trajectory in a plane
so we'll going to
have wondered trajectory
bisbee
ecstatic shin
and let this be
wider action
increasing values
white increasing values
i think an object
and alright
with an initial velocity
the zero
what is the object going to do
you're going to get a proud or not
into a frontal gravity
and it comes down here again
and where we have this kind of a problem
we will decompose its
to one-dimensional motions while in the actual action
and warm
the white house action
we already decompose right away
the orthotic
time can you call zero
into
component which i call the zero x
and that of course is the zero
times the coastline of all file
the angle
sulphide
and the
velocity in the white election
a time people zero
i will call that v zero
in the white election
and that is the zero
i'm assignable
and now
i have to know how
the object moving the expert action as a function of time and how we behave as a
function
what kind of the white action soaking up on the equations
forty ecstatic
acts as a function of the time
people external
plus v zero ext
contracting
that's all there is no acceleration
the velocity in the excel reaction as a function of time
simply
the zero ext
it never changes
so thats
eccentric
now we take the white election
why as a function of the time equals y zero
the zero ***
plus one-half eighteen square
development that i'm going to be used
is always positive even ninety four dates meters per second career
or sometimes i make it easy
miami's always positive
and scenes in this case
i have chosen this to be increasing value wide only reason why i don't know
hooding
might as well have
gps clear
marked as some of you think because the
acceleration is doin' that's not a reason
because i could have called
direction increasing why they would have been plus
one-half g square
through the content of my choosing this the direction in which white increases
the plus one haha
eighties granted that you would normally see
i'm going to replace that now
by minus
one-half
gps quick
then the velocity in the white abduction had functional time would be considered
to be the case
that is the zero
why
minus
g_t_
andy acceleration dot
equals minus
so these are two recreation said go from the motion
vitality
holds if there is no
ever had no pre-trial anytime that is very unrealistic
if we're in the worst but when we are far away from where it
as we
where is the case he one thirty five
rituals flying at an altitude of about thirty thousand feet
that of course it is a little bit more
realistic
and therefore the example that i have fixed throw up an object is the one
well by the k_c_-one thirty-five
at an altitude somewhere around twenty five to thirty thousand feet
company litespeed
of four hundred and twenty five miles for work
does the engines all
then for the remaining
whatever it was about
thirty seconds
everyone including the airplane
has no wait
that's the case
that i now want to work out
qualitatively with you
in the case of the
casey thirty-five will take an angle fall fall forty five degrees
and will take three zero
which was
about four hundred and twenty five miles per hour remember that
that lecture
four hundred and twenty five
miles per hour
translates
into about hundred an eighty nine
meters
the sec
and sort of news that be
velocity v zero y n v zero x eyeballs and saying
because of the forty five degree angle
and that is of course the hundred eighty nine
divided by the square root of two
matters about
one hundred thirty three
meters per second
both are positive
keep in mind because this is what i call the increase in value four y
dvd increasingly both positive values finds don't matter
is a given on top of the first question that i could ask you one example
when is the plane
at its highest point
of his trajectory
howell
how you think about the point
where it started when it through the engines off when it went into a
free-fall
so when is it here
and what is this
well wedded to bear
that's when the wall street in the white election
become zero it is positive
get smaller and smaller because of the dedication of federation comes to a
whole becomes zero
so i have to disinclination
when are you zero
this is the one i pick
slices euro
equals
plus
one thirty three
minus ten consti
you may think that the gravitation acceleration at an altitude of thirty
thousand feet
could be substantially less
then the canonical number of ten
it is a little less because you're a little bit foot away from the earth
but it's only point three percent less
and so we'll just accepted ten
too easy to work with
and so
renders it at the highest point credit twenty
equals
thirteen point three seconds
but that's about how long it takes
to get there
but i gave a lecture last time i said it's about fifteen seconds because i
make the numbers
around the mall starting point three seconds
what does this this does age now abdullah have to go to this equation
archaic
equals zero because i'm going to define the point where the planes starting to
project we are called that one zero zero one free to do that
eighty four zero
plus
one hundred and thirty three
the speech
finds
fifteen
point three seconds
minus won't have time gene and spliced
fifteen
point three
that is what age
and that turned out to be about
eight hundred an eighty five commuters
because told you last night at about nine hundred close enough
so we know how long it takes to reach peak
and we know
boppy vertical distance
and the whole trip
back to this starting point if we call the sort of starting point starting out
the truth
this whole trip will take twice the amount of time
get back to this point
when the engines are restart it
is about
haha twenty seven seconds
how far has the plane travel ban in horizontal direction
well now i go back to
this equation
sonali say aha
extend
what it is back at this point
must be xterra which i've conveniently to zero
plus eleven thirty three meters per second which is the velocity accel
rection
which never changes
when the plane is here
that velocity accel ection is the same on the thirty three
meters per second as it was here
which by the way is about three hundred miles per hour
that never changes if there is no hair
directly there
friction of any kind
so we get plus two hundred and thirty three
finds the time of the whole trip takes
twenty-six
point six seconds
and that
if you convert that to columbus is about three-and-a-half cool
now you could ask yourself the question
what used to be velocity off that plane
at that point s
and no
you may want to abandon all these
well no mention ideal x and while you may say welk
there is a prevalent is completely symmetric
the plane comes up here is four hundred and twenty five miles per hour at an
angle forty five degrees
that obviously it comes down here at an angle forty five degrees
and to speak much again before within twenty five miles for
and you would support
on the percent of course clear
of all you do
appreciate however that
i could continue
to think of this as
to one-dimensional motions
and i can therefore
calculates
what the velocity in the acts that action is
s
and what the velocity of the white election the status
toward the velocity
eniac spelled action
at four deaths
i go to equation the second equation there
that is indeed a relaxed
that is plus hundreds and thirty-three
readers
sect
what are the velocity in the white house
i have to go to this equation l
these you know why mine is dat
so i get
plus one of them thirty three
minus ten
times the twenty six point six seconds to reach that point asp
i want to find
minus one hundred
and thirty-three
needles
the velocity in the white house reaction
started off
plus under thirty three
now yes minus one to thirty three you see these assigned sensitive
his wonderful with the great thing about treated that way
so you know no
that it comes in was a velocity
of hundreds and thirty-three
in the ecstatic shinde
was a bit
harder than thirty-three
into my despite election
and so what is the
net
the some of the two vectors
that of course this vector
and no surprise lending two forty five degrees
and this one
is the square root of two parent home to thirty-three
and that of course if you back two hundred eighty nine the just a second
underpinning nine
not be just a second
and that is four hundred
and twenty five
cqu
are not recommending that you would do these chorus is perfectly
reasonable to immediately come to that conclusion
because of the symmetry
parabola
that's not true
uniform circular motion
uniform circular motion
of curls when an object
goes around in a circle
and when the sbi
never changes
if this page doesn't change
and of what's happening
or dosti
but the direction
changes all the time
the speech does not
so here we have
or circle
disney resort
and
at this moment in time the object is here
has a certain velocity
ninety degrees
and later in time
objectives here
speed is the same
but the direction has changed
ninety degrees
inspectors
at the statement
in a situation like this that we have uniform
circular motion
or you're afforded
circular
*** side of the fire
what we call
the period
in seconds
that's the time to go around
then we identify what we call the frequency
that is how many times it goes around per second
i prefer the latter f
but our book users the greek letter new
i find a new often very confusing because your philosophy that's why i
prefer the etc
it is one of the team
and so do you know it's our second
minus one
with most businesses
we'll call that hurts
that hurts means
to go ten times around
and then we identify only got
angular
velocity
omega
which is in radians
per second
since it takes
few seconds to go around
comply radiance omega is to apply divided by
already we have
the speed
which we can also think will continue
lineal velocity
how many meters per second is linear as opposed to
how many
radians per second
which is angular velocity exhibit is unreal velocity isn't angular velocity
and that many of the lawsuit in which in this case is really the speed
is of course
the circumference of the circle
divided by how many seconds it takes to go around
and that is also
omi gardens
and that is now in need of success
all this
is open possible
if there is
acceleration
and the acceleration this cool percent typical acceleration
it is always appointed
the center
a center pickle
eight
centered
anderson typical acceleration
the magnitude
is the script
divided by horror
which is therefore also
omega square art and that of course is meters
the second square
i want to
workout a specific example
an example
that our children
is
uelman centrifuge that these are used
why not
in houston
experiments
on humans
to see how they do u with
strong
acceleration
this is that
centrifuge
reviews
from there
axis of rotation
the axis of rotation
if you have
and the distance from here to view the or you may not think stories about fifty
meters
so he asked when i was going here
and then the thing goes wrong
so i would like to
workout
this with some
some numbers
the radius are
of if you like back
made myself for you
raises fifteen meters
defense of course a little bit on where the person is located in that speed
it goes around twenty four
revolutions per minute
and that translates
into all points for hertz
soda period to go around
rotation
to have sex
goes around
to have
the angular velocity omega
which is to quite divided by t
if you could confide in divided by two and a half
the truth comes out to be roughly two-and-a-half
particularly accident
activated
don't ever think that that have to be the same of course
just happens to come out that way for the mentions
tomader is about to have arabians
saket
andy
speeds
then e_s_p_
tangential speech if you want to call it is only too hard
that comes out to be
about thirty five
thirty seven point seven
no
meters per second
and that translates into about eighty five miles flowers with responsible
well of course the goal is
from outside
what is this antibiotic celebration
that is only desk reallinks
or if you prefer to take this clip divided by heart
you find of course exactly the same answer
if you have a date
and that is ninety five meters per second squid
and that is about ten times
medication acceleration on earth which is really phenomenal
if you add to the fact
that the direction is changing all the time when you go around
so you feel depend g
in this direction and then you can give you a different i can't imagine how
people
can actually
supplied that
not fate
most people like you and me if we were to be accelerated along a straight line
not even
a cynical read a direction changes at moma straight-line most of us
when we'd get close to six p
and it was a reason for that
well you get problems with your blood circulation and but not enough oxygen
goes through your brains
and that's one thing
how these astronauts can do it ten g and
the direction changing all the time
it beats me
if you take a boeing seven forty-seven
it takes thirty seconds from the moment that it starts on the runway
until it kicks off you should find that when you get a chance it's a very close
to thirty seconds
and by that time the plane has reached a speed of about two hundred fifty miles
per hour
and if you calculation assume that the acceleration is constant it's an easy
calculation it turns out that the acceleration is only two meters per
second squeak
there's only one fifth of the gravitational acceleration
feel sort of god wanted very comparable manuscript take an opportunity to meet a
specifically
these poor people
men and women nineteen five leaders per second script
i would like to address something that's
is not part of the exam
but that is something that i want you to think about something that is fun
and it's always nice to do something that is
it has to do is my last election
i have to clean my hands
first
for it to work like well
but you are at stake here
and i'm going to put a yardstick
on my hands
on my two fingers disheveled in front of me
yours
is resting on my two fingers
and i'm going to
move my two fingers towards each other
one of them begins to slide first of course
i can tell you which one
something very
strange will happen
this one starts to slide first it comes to his faults and then the other one
starts this life and it comes to personal and then this one starts to
slide and so on
and that is very strange
this is something you should be able to
explain
certainly after
lecture we had last time
look at this
cc alteration of the with the little faster
left is going quite is growing left is going right going like story once more
growing going that alternate
give this some thought
and you know pivots has an option that you can discuss
problems with other students
so make use of this discussion but then and see whether you can come through
an explanation
good luck on your exam
next friday