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The exam on Wednesday will cover
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our first five lectures and the
first two homework assignments.
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And so I list here the topics
the way we discussed them.
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Of course, it is not possible
to discuss all of them today
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but I will make a selection.
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I recall
that we discussed scaling
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and we used the interesting
example of Galileo Galilei--
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an animal,
and the animal has legs.
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And we defined the overall size
of the animal as yea big--
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we called that "s."
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And then we said,
well, there is here the femur
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and the femur has length l
and thickness d.
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It was completely reasonable
to say
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well... that l will have
to be proportional to S.
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If an animal is
ten times larger than another
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its legs will be typically
ten times longer.
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Since the mass of the animal
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must be proportional to its size
to the power three
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it will also be proportional
to the length of the femur
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to the power three,
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and then came in
this key argument--
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namely, you don't want
the bones to be crushed.
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Which is called "yielding"
in physics.
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If I take a piece of concrete,
a block of concrete,
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and I put too much pressure
on it, it starts to crumble.
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And that's what Galileo Galilei
may have had in mind.
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And in order to protect animals
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who get bigger
and bigger and bigger
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against this crushing,
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we argued-- and I will not go
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through that argument
now anymore--
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that the mass will have to be
proportional to d squared,
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which is the cross-section
of the femur.
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And so, you see immediately
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that d squared has to be
proportional to l to the third
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so d must be proportional
to the length of the femur
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to the power one and a half.
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So this would mean that if you
compare an elephant with a mouse
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the elephant's overall size is
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about 100 times larger
than a mouse.
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You would expect the femur
to be about 100 times larger,
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which is true.
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But you would then expect
the femur to be
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about 1,000 times thicker
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and that turns out to be
not true, as we have seen.
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In fact, the femur
of the elephant
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is only 100 times thicker,
so it scales just as the size.
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And the answer lies in the fact
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that nature doesn't have
to protect
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against crumbling of the bones.
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There is a much larger danger,
which we call "buckling."
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And buckling is the phenomenon
that the bones do this
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and if now you put too much
pressure on it
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the bones will break.
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And if that's the case,
you remember that, in fact,
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all you have to do is you have
to scale d proportional to l,
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which is not intuitive--
that's not so easy to derive--
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but that's the case.
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And so the danger, then, that
nature protects animals against
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is this buckling, and when
the buckling becomes too much
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then, I would imagine, the
bones, at some point in time--
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well, these are tough bones,
aren't they?--
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(snaps )
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will break, and that's
what nature tries to prevent.
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So that was a scaling argument.
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And let's now talk
about dot products.
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If I look there...
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I scan it a little bit
in a random way over my topics,
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so let's now talk
about dot products.
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I have a vector A...
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Ax times x roof, which is the
unit vector in the x direction,
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plus Ay y roof plus Az Z roof.
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So these are
the three unit vectors
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in the x, y and z direction.
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And these are the x components,
y and the z component
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of the vector A.
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I have another vector, B.
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B of x, x roof, B of y, y roof,
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B of z, z roof.
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Now, the dot product...
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A dot B-- also called
the scalar product--
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is the same as B dot A
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and it is defined as Ax Bx
plus Ay By plus Az Bz.
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And it's a number.
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It is a scalar,
it is a simple number.
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And so this number can be larger
than zero-- it can be positive--
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it can be equal to zero, it can
also be smaller than zero.
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They're just dumb numbers.
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There is another way
that you can define...
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You can call this method number
one, if you prefer that.
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There is another way that
you can find the dot product.
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It would give you exactly
the same result.
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If you have a vector A
and you have the vector B
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and the angle
between them is theta,
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then you can project B on A--
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or A on B, for that matter,
it makes no difference--
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and that projection...
the length of this projection
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is then, of course,
B cosine theta.
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And so A dot B...
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and that is exactly the same.
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You may want to go
through a proof of that.
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It is the length of A
times the length of B
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times cosine of theta.
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And that will give you
precisely the same result.
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What is interesting about this
formulation, which this lacks,
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that you can immediately see
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that if the two are
at 90-degree angles
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or 270 degrees, for that matter,
then the dot product is zero.
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So that's an insight
that you get through this one
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which you lack
through that other method.
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Let us take a down-to-earth
example of a dot product.
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Suppose A equals 3x
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and B equals 2x plus 2y,
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and I am asking you,
what is the dot product?
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Well, you could use method
number one, which, in this case
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is by far the fastest,
believe me.
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Ax is 3 and Bx is 2,
so that gives me a 6.
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There is no Ay, there is no Az,
so that's the answer.
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It's just 6--
that's the dot product.
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You could have done it that way.
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It's a little bit
more complicated
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but I certainly want
to show you that it works.
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If this is the x direction
and this is the y direction--
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we don't have to look
into the z direction
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because there is
no z component--
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then this would be vector A
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and this point would be at 3.
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B... this would be 2,
and this would be 2
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and so this would be
the vector B.
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And it's immediately clear now
that this angle... 45 degrees.
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That follows
from the 2 and the 2.
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So if we now apply
method number two, A dot B.
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First the length of A, that's 3,
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times the length of B,
that is 2,
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times the square root of 2--
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this is 2, this is 2,
this is 2... square root 2--
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times the cosine of 45 degrees,
which is one-half square root 2,
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and the answer is 6.
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Notice that
this square root of 2
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and this square root of 2
equal just 2, and you get 6.
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You get the same answer,
of course.
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But it would be
a dumb thing to do it
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since it can be done
so much easier.
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On cross products...
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I don't want to go
through the formalism
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of cross products the way we
did that with the determinant.
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I just want to remind you
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that if you have a cross product
of two vectors,
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that is minus B cross A,
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and that the magnitude of C
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is the length of A
times the length of B
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times the sine
of the angle between them.
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The vector C, the dot product...
the cross product
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is always perpendicular to
both A and perpendicular to B.
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In other words,
it's perpendicular to the plane
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of the two vectors.
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Now, if it's perpendicular
to the plane,
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then in that case, it's
perpendicular to the blackboard.
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You have two choices:
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it's either coming
at you perpendicular
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or it's coming right
straight into the blackboard.
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And now everyone has
his own way of doing it.
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I taught you what's called
"the right-hand corkscrew" rule.
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You take the first one that is
mentioned-- in this case, A--
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and you rotate it
over the shortest angle to B.
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When you do that,
you rotate your corkscrew--
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seen from where you're sitting--
counterclockwise.
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Then the corkscrew comes to you.
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And so the direction
of the vector
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is such that you will see
the tip of the vector
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as though it's coming
straight out of the blackboard.
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And so that gives you,
then, the direction.
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Now I will give you
the position x of an object
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as a function of time
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and then we're going to ask
ourselves a lot of questions
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about velocities, accelerations,
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sort of everything
you can think of,
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everything we have covered--
speeds...
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And I will cover here
four seconds of time.
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So this is the time axis
in seconds
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and we will cover four seconds.
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So let this be
one, two, three, four.
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And let the object be
at position plus six.
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This is my x-axis, this is where
the object is actually moving,
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and this is three,
and here is minus three
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and this, let's say,
is in meters.
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Let's make a little grid
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so that's easier for me
to put in the curve.
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All right, so now comes
x as a function of t.
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The time t equals t seconds.
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The object is here
and it came from there.
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And this part is a parabola
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and this parabola
here is horizontal.
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It's important,
you have to know that,
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so this is a parabola
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and this, here, is horizontal.
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So the object goes
from plus six to three,
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then it goes to minus three,
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then it stays there
for one second
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and then it goes back
in one second to plus six.
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It's a one-dimensional problem.
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The motion is only in the
x-axis, along the x direction.
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Well, let's analyze all these
different seconds that occur.
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Let's first take the first
second, during the first second.
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Since this is a parabola,
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you know that
the acceleration is constant
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so I hope
you will conclude immediately
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that a must be a constant.
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If a is a constant, the
position x as a function of time
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should change as follows:
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x zero plus v zero t
plus one-half a t squared.
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I expect you to know
this equation.
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Very often do I give you
equations at the exam
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and that may well happen during
the second and the third exam,
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but it will not be the case
this time.
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The equations are
all very fundamental
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and you have to make them
part of your world.
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So this is an equation
that you will have to remember.
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All right,
what is the velocity here?
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The velocity starts out
to be zero
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and the velocity here is
not zero anymore.
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If I look at time t equals one,
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then I have here x zero or six.
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It starts out with
velocity zero-- that's a given.
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And I get plus one-half
times a t squared
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but this is only one second,
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and so when it is at three,
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I have six plus one-half a
times one squared,
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and so you find
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that a equals minus six meters
per second squared.
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So during this first second
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the acceleration is minus six
meters per second squared.
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And the velocity, v,
as a function of time,
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is the derivative of this one,
is v zero plus at.
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V zero was zero, and so
that is minus six times t.
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So the velocity is changing
in a linear fashion.
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What do I know about
the end of the first second?
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Well, I can say that x is three.
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What do I know
about the velocity?
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Well, the velocity is minus six.
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What do I know about a?
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I don't... I don't know about a.
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It's true that
during this first second
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a is minus six meters
per second squared,
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but it changes abruptly
at this point
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so it's ill-defined
at this point.
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In fact, it's
actually nonphysical.
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So I really don't know
exactly at the end
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what the acceleration is.
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Let's now go
to the second second
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and let's see
what happens there.
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The second second.
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And first let's look during,
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and then we'll look
at the situation at the end.
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During the second second,
it is clear--
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since this is a straight line--
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that the velocity
remains constant
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and it remains
minus six meters per second.
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That is exactly what it was
at this point at the end.
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You can see it go six meters--
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from plus three
to minus three-- in one second
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so the velocity is
minus six meters per second.
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The acceleration is
therefore zero.
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You see that the acceleration
changes abruptly
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from minus six meters
per second squared to zero
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so I can't tell you what it is
exactly at this moment in time.
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So that's the situation
during the second second.
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And what is the situation
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at the end of the second
section... second second?
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At the end, I know
that x equals minus three.
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What is the velocity?
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I don't know, because
it changes abruptly here
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from minus six to zero,
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so I don't know exactly
what it is at that point.
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It's a nonphysical thing,
it's a very abrupt change.
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And the acceleration, yeah,
that's also a very tricky thing,
286
00:18:33 --> 00:18:39
because if the velocity
287
00:18:35 --> 00:18:41
is minus six on this side
of the two seconds
288
00:18:38 --> 00:18:44
and here becomes zero, and if
that happens in a split second,
289
00:18:43 --> 00:18:49
there must be
ahuge acceleration
290
00:18:45 --> 00:18:51
just at that point
which is nonphysical.
291
00:18:49 --> 00:18:55
So I would also put a question
mark at the a...
292
00:18:52 --> 00:18:58
I don't know what the a is.
293
00:18:55 --> 00:19:01
So we'll go to the
third second... this part.
294
00:18:58 --> 00:19:04
295
00:19:01 --> 00:19:07
Let's first look during
the third second.
296
00:19:06 --> 00:19:12
Well, the object
isn't going anywhere,
297
00:19:09 --> 00:19:15
it's just sitting there.
298
00:19:11 --> 00:19:17
x remains minus three
299
00:19:16 --> 00:19:22
and the velocity is zero
and a is zero--
300
00:19:21 --> 00:19:27
We can agree on that.
301
00:19:24 --> 00:19:30
What is the situation
at the end of the third second?
302
00:19:28 --> 00:19:34
That means t equals three.
303
00:19:31 --> 00:19:37
Well, all I know is
that x is minus three.
304
00:19:33 --> 00:19:39
That's nonnegotiable.
305
00:19:36 --> 00:19:42
What the velocity is,
I don't know,
306
00:19:38 --> 00:19:44
because it's changing abruptly
from zero to a positive value.
307
00:19:43 --> 00:19:49
So that's ill-defined
308
00:19:46 --> 00:19:52
and the same is true
for the acceleration.
309
00:19:48 --> 00:19:54
There is a sudden change
in velocity.
310
00:19:50 --> 00:19:56
That means there must be
a huge acceleration.
311
00:19:52 --> 00:19:58
It's unknown, ill-defined
312
00:19:55 --> 00:20:01
because this curve is,
of course, not very physical.
313
00:20:00 --> 00:20:06
Let's now look
at the last second.
314
00:20:03 --> 00:20:09
This is the fourth second.
315
00:20:07 --> 00:20:13
First, during.
316
00:20:09 --> 00:20:15
317
00:20:10 --> 00:20:16
Well, it's going
from minus three to plus six
318
00:20:13 --> 00:20:19
and it's a straight line,
so the velocity is constant.
319
00:20:16 --> 00:20:22
If the velocity is constant
320
00:20:17 --> 00:20:23
then you can immediately
conclude that a is zero--
321
00:20:20 --> 00:20:26
there is no acceleration--
322
00:20:21 --> 00:20:27
and it goes nine meters
in a time span of one second.
323
00:20:25 --> 00:20:31
But it's now plus--
plus nine meters per second.
324
00:20:31 --> 00:20:37
So the object first went
from positive values of x
325
00:20:36 --> 00:20:42
to zero
and to negative values for x.
326
00:20:39 --> 00:20:45
During all that time,
the velocity was negative
327
00:20:43 --> 00:20:49
by our sign convention
328
00:20:44 --> 00:20:50
and now the velocity,
it goes back to plus six.
329
00:20:48 --> 00:20:54
The velocity becomes
plus nine meters per second.
330
00:20:53 --> 00:20:59
What is the story at the end
of the fourth second?
331
00:20:56 --> 00:21:02
Well, all I can say is
that x equals plus six.
332
00:21:00 --> 00:21:06
I don't know much more.
333
00:21:02 --> 00:21:08
I don't know
what the velocity is.
334
00:21:03 --> 00:21:09
Neither do I know
what the acceleration is.
335
00:21:05 --> 00:21:11
The plot stops there, anyhow.
336
00:21:08 --> 00:21:14
337
00:21:11 --> 00:21:17
Now, I would think
338
00:21:13 --> 00:21:19
that it is reasonable
to ask the following question:
339
00:21:15 --> 00:21:21
What is the average velocity,
for instance,
340
00:21:18 --> 00:21:24
between time zero and time four?
341
00:21:21 --> 00:21:27
Average velocity.
342
00:21:23 --> 00:21:29
We define average velocity
343
00:21:25 --> 00:21:31
as the position
at time four seconds
344
00:21:29 --> 00:21:35
minus the position at time zero,
divided by four.
345
00:21:32 --> 00:21:38
That is our definition.
346
00:21:34 --> 00:21:40
At zero, it is at plus six,
at four, it is at plus six.
347
00:21:38 --> 00:21:44
So the upstairs is zero,
348
00:21:40 --> 00:21:46
so the average velocity during
this four-second trip is zero.
349
00:21:47 --> 00:21:53
You may not like that, it may go
against your intuition.
350
00:21:50 --> 00:21:56
Of course!
I couldn't agree more with you
351
00:21:52 --> 00:21:58
but that's the way
we define velocity.
352
00:21:55 --> 00:22:01
Speed is defined differently.
353
00:21:57 --> 00:22:03
Speed is the magnitude
of the velocity vector
354
00:22:01 --> 00:22:07
and the speed, therefore,
always has a positive value.
355
00:22:04 --> 00:22:10
And I will show you now
356
00:22:05 --> 00:22:11
what is the average speed
between time zero and four.
357
00:22:13 --> 00:22:19
That is the distance
358
00:22:14 --> 00:22:20
that it has traveled
in these four seconds.
359
00:22:17 --> 00:22:23
Well, let's first go
through the first second.
360
00:22:19 --> 00:22:25
It goes from plus six
to plus three.
361
00:22:22 --> 00:22:28
So it already travels
three meters.
362
00:22:25 --> 00:22:31
Then in the second second
363
00:22:27 --> 00:22:33
it goes from plus three
to minus three
364
00:22:29 --> 00:22:35
so it travels
another six meters.
365
00:22:32 --> 00:22:38
And then in the third second
366
00:22:34 --> 00:22:40
it's lazy,
it doesn't do anything,
367
00:22:36 --> 00:22:42
so the distance traveled
is zero.
368
00:22:40 --> 00:22:46
And then in the last one second
369
00:22:41 --> 00:22:47
it gets very active
and it travels nine meters.
370
00:22:46 --> 00:22:52
Notice you only see
plus signs here.
371
00:22:48 --> 00:22:54
There are no minus signs,
it would make no sense.
372
00:22:51 --> 00:22:57
And this occurs in four seconds,
373
00:22:54 --> 00:23:00
so that is 4.5 meters
per second.
374
00:23:02 --> 00:23:08
So the average speed is
4.5 meters per second,
375
00:23:05 --> 00:23:11
but the average velocity
is zero.
376
00:23:10 --> 00:23:16
We could now make a plot of the
velocity as a function of time.
377
00:23:20 --> 00:23:26
Let me put here the 4.5.
378
00:23:24 --> 00:23:30
I just have enough room here
to make the velocity plot
379
00:23:28 --> 00:23:34
as a function of time.
380
00:23:31 --> 00:23:37
I'll make a new one.
381
00:23:35 --> 00:23:41
This is my time axis,
and this is the velocity.
382
00:23:45 --> 00:23:51
This is zero.
383
00:23:47 --> 00:23:53
One second, two seconds,
three seconds, four seconds.
384
00:23:54 --> 00:24:00
And this velocity is
in meters per second.
385
00:24:01 --> 00:24:07
I go up here to plus ten
386
00:24:08 --> 00:24:14
and here is minus five,
here is minus six.
387
00:24:16 --> 00:24:22
So, what do I do now?
388
00:24:18 --> 00:24:24
I know that the velocity during
the first second is minus six t
389
00:24:23 --> 00:24:29
so it's linear.
390
00:24:26 --> 00:24:32
And so during the first second
391
00:24:31 --> 00:24:37
this is the velocity
as a function of time.
392
00:24:35 --> 00:24:41
It starts at zero,
you can see that,
393
00:24:39 --> 00:24:45
and when it is here
394
00:24:40 --> 00:24:46
it has a velocity
of minus six meters per second.
395
00:24:44 --> 00:24:50
During the second second
396
00:24:48 --> 00:24:54
it remains
minus six meters per second.
397
00:24:51 --> 00:24:57
So during the second second,
the velocity is not changing.
398
00:24:59 --> 00:25:05
It stays there.
399
00:25:01 --> 00:25:07
During the third second
400
00:25:04 --> 00:25:10
the velocity jumps
all of a sudden to zero--
401
00:25:07 --> 00:25:13
you see
how nonphysical that is.
402
00:25:09 --> 00:25:15
And so all of a sudden,
during the third second
403
00:25:11 --> 00:25:17
it becomes zero.
404
00:25:14 --> 00:25:20
So there has to be somehow
405
00:25:15 --> 00:25:21
a connection, of course,
between the two
406
00:25:17 --> 00:25:23
to make this physical.
407
00:25:18 --> 00:25:24
So in a very small amount
of time that will have to occur.
408
00:25:21 --> 00:25:27
That's why you get ahuge
acceleration here at that point.
409
00:25:25 --> 00:25:31
Of course, you also get an
acceleration here at this point,
410
00:25:28 --> 00:25:34
because there's also
a change in velocity.
411
00:25:30 --> 00:25:36
And then,
during the fourth second,
412
00:25:33 --> 00:25:39
the velocity is
plus nine meters per second,
413
00:25:36 --> 00:25:42
and so we jump up.
414
00:25:37 --> 00:25:43
415
00:25:39 --> 00:25:45
Let's make this plus nine.
416
00:25:43 --> 00:25:49
417
00:25:48 --> 00:25:54
And so we have here
during the last second...
418
00:25:54 --> 00:26:00
And again, this is nonphysical,
419
00:25:55 --> 00:26:01
so there has to be
somehow a transition.
420
00:25:57 --> 00:26:03
And so here you see the velocity
as a function of time.
421
00:26:03 --> 00:26:09
Now comes
an interesting question.
422
00:26:05 --> 00:26:11
Is it possible,
if I gave you this--
423
00:26:10 --> 00:26:16
so this is a given,
you can't see that--
424
00:26:13 --> 00:26:19
could you convert
this back to that?
425
00:26:16 --> 00:26:22
And the answer is yes,
426
00:26:18 --> 00:26:24
provided that I tell you
what the position is at t zero.
427
00:26:22 --> 00:26:28
At t equals zero,
x equals plus six
428
00:26:28 --> 00:26:34
and that is sufficient
429
00:26:29 --> 00:26:35
for you to use this information
and to reconstruct that.
430
00:26:34 --> 00:26:40
It's an interesting thing to do,
and if you feel like it
431
00:26:37 --> 00:26:43
I would say, give it a shot.
432
00:26:40 --> 00:26:46
All right, so far, about speeds
433
00:26:44 --> 00:26:50
and average velocities
and accelerations.
434
00:26:49 --> 00:26:55
Let's now go to trajectories,
three-dimensional trajectories.
435
00:26:57 --> 00:27:03
Trajectories, thank goodness,
436
00:26:59 --> 00:27:05
are almost
never three-dimensional.
437
00:27:03 --> 00:27:09
They're always two-dimensional,
438
00:27:05 --> 00:27:11
because the trajectory itself
is in a vertical plane
439
00:27:08 --> 00:27:14
and so we normally...
440
00:27:10 --> 00:27:16
When we throw up an object
in a gravitational field,
441
00:27:13 --> 00:27:19
you have the trajectory
in a plane.
442
00:27:17 --> 00:27:23
443
00:27:22 --> 00:27:28
So we're going to have
one trajectory.
444
00:27:28 --> 00:27:34
Let this be the x direction
and let this be the y direction.
445
00:27:36 --> 00:27:42
Increasing values of y,
increasing values of x.
446
00:27:43 --> 00:27:49
I take an object
447
00:27:45 --> 00:27:51
and I throw it up with
an initial velocity v zero.
448
00:27:52 --> 00:27:58
And what is the object
going to do?
449
00:27:54 --> 00:28:00
You're going to get a parabola
under the influence of gravity
450
00:27:59 --> 00:28:05
and it comes down here again.
451
00:28:03 --> 00:28:09
And where we have
this kind of a problem
452
00:28:05 --> 00:28:11
we will decompose it
in two one-dimensional motions,
453
00:28:10 --> 00:28:16
one in the x direction
and one in the y direction.
454
00:28:16 --> 00:28:22
We already decompose
right away the velocity
455
00:28:22 --> 00:28:28
at time t equals zero
456
00:28:25 --> 00:28:31
into a component
which I call v zero x
457
00:28:29 --> 00:28:35
and that, of course, is v zero
times the cosine of alpha
458
00:28:33 --> 00:28:39
if the angle is alpha.
459
00:28:37 --> 00:28:43
And the velocity
in the y direction
460
00:28:40 --> 00:28:46
at time t equals zero--
461
00:28:42 --> 00:28:48
I will call that
v zero in the y direction
462
00:28:46 --> 00:28:52
and that is v zero
times the sine of alpha.
463
00:28:52 --> 00:28:58
And now I have to know
464
00:28:55 --> 00:29:01
how the object moves in the x
direction as a function of time
465
00:28:58 --> 00:29:04
and how it behaves as a function
of time in the y direction.
466
00:29:02 --> 00:29:08
So here come the equations
for the x direction.
467
00:29:07 --> 00:29:13
x as a function of time equals
x zero plus v zero x times t.
468
00:29:14 --> 00:29:20
That's all--
there is no acceleration.
469
00:29:19 --> 00:29:25
The velocity in the x direction
as a function of time
470
00:29:23 --> 00:29:29
is simply v zero x--
it never changes.
471
00:29:27 --> 00:29:33
So that's the x direction.
472
00:29:32 --> 00:29:38
Now we take the y direction.
473
00:29:36 --> 00:29:42
y as a function of time equals
474
00:29:38 --> 00:29:44
y zero plus v zero y t
plus one-half at squared.
475
00:29:49 --> 00:29:55
My g value that I'm going
to use is always positive--
476
00:29:53 --> 00:29:59
either 9.8 meters
per second squared
477
00:29:56 --> 00:30:02
or sometimes I make it
easy to use it, 10--
478
00:29:59 --> 00:30:05
but mine is always positive.
479
00:30:01 --> 00:30:07
And since in this case
480
00:30:04 --> 00:30:10
I have chosen this to be
the increasing value of y,
481
00:30:06 --> 00:30:12
that's the only reason
482
00:30:08 --> 00:30:14
why I would now have to put in
minus one-half gt squared--
483
00:30:12 --> 00:30:18
not, as some of you think,
484
00:30:14 --> 00:30:20
because the acceleration
is down.
485
00:30:16 --> 00:30:22
That's not a reason.
486
00:30:17 --> 00:30:23
Because I could have called
this direction increasing y.
487
00:30:21 --> 00:30:27
Then it would have been
plus one-half gt squared.
488
00:30:24 --> 00:30:30
So the consequence
of my choosing
489
00:30:26 --> 00:30:32
this the direction
in which y increases...
490
00:30:28 --> 00:30:34
Therefore, the
plus one-half at squared
491
00:30:32 --> 00:30:38
that you would normally see,
492
00:30:34 --> 00:30:40
I'm going to replace that now
by minus one-half gt squared.
493
00:30:42 --> 00:30:48
Then the velocity in the y
direction as a function of time
494
00:30:45 --> 00:30:51
would be this derivative,
that is, v zero y minus gt
495
00:30:52 --> 00:30:58
and the acceleration
equals minus g.
496
00:30:55 --> 00:31:01
So these are the three equations
497
00:30:57 --> 00:31:03
that govern the motion
in the y direction.
498
00:31:04 --> 00:31:10
This only holds
if there is no air drag,
499
00:31:09 --> 00:31:15
no friction of any kind.
500
00:31:10 --> 00:31:16
That is very unrealistic
if we are near Earth,
501
00:31:14 --> 00:31:20
but when we are
far away from Earth,
502
00:31:18 --> 00:31:24
as we were with the KC-135--
503
00:31:21 --> 00:31:27
which was flying at an altitude
of about 30,000 feet--
504
00:31:26 --> 00:31:32
that, of course, is
a little bit more realistic.
505
00:31:29 --> 00:31:35
And therefore the example
506
00:31:30 --> 00:31:36
that I have picked
to throw up an object
507
00:31:33 --> 00:31:39
is the one whereby the KC-135,
508
00:31:36 --> 00:31:42
at an altitude somewhere
around 25,000 or 30,000 feet,
509
00:31:39 --> 00:31:45
comes in at a speed of 425 miles
per hour, turns the engines off
510
00:31:45 --> 00:31:51
and then, for
the remaining whatever it was--
511
00:31:49 --> 00:31:55
about 30 seconds--
512
00:31:50 --> 00:31:56
everyone, including
the airplane, has no weight.
513
00:31:54 --> 00:32:00
That's the case
514
00:31:55 --> 00:32:01
that I now want to work out
quantitatively with you.
515
00:31:59 --> 00:32:05
In the case of the KC-135,
516
00:32:01 --> 00:32:07
we will take an angle for alpha
of 45 degrees
517
00:32:04 --> 00:32:10
and we will take v zero, which
was about 425 miles per hour.
518
00:32:10 --> 00:32:16
You may remember that
from that lecture.
519
00:32:14 --> 00:32:20
425 miles per hour translates
520
00:32:18 --> 00:32:24
into about 189 meters
per second.
521
00:32:24 --> 00:32:30
And so that means that the
velocity v zero y and v zero x
522
00:32:29 --> 00:32:35
are both the same
because of the 45-degree angle,
523
00:32:35 --> 00:32:41
and that is, of course, the 189
divided by the square root of 2.
524
00:32:40 --> 00:32:46
And that is
about 133 meters per second.
525
00:32:45 --> 00:32:51
Both are positive--
keep that in mind
526
00:32:47 --> 00:32:53
because this is what I call
the increasing value for y
527
00:32:50 --> 00:32:56
and this is
the increasing value of x.
528
00:32:52 --> 00:32:58
They are both positive values.
529
00:32:54 --> 00:33:00
Signs do matter.
530
00:32:56 --> 00:33:02
This is a given now.
531
00:32:57 --> 00:33:03
And now comes the first question
that I could ask you on an exam.
532
00:33:00 --> 00:33:06
When is the plane at its highest
point of its trajectory
533
00:33:07 --> 00:33:13
and how high is it
above the point
534
00:33:10 --> 00:33:16
where it started
when it turned the engines off
535
00:33:12 --> 00:33:18
when it went into free fall?
536
00:33:14 --> 00:33:20
So when is it here
and what is this distance?
537
00:33:18 --> 00:33:24
Well, when is it there?
538
00:33:21 --> 00:33:27
That's when the velocity
in the y direction becomes zero.
539
00:33:25 --> 00:33:31
It is positive.
540
00:33:26 --> 00:33:32
It gets smaller and smaller
541
00:33:28 --> 00:33:34
because of
the gravitational acceleration,
542
00:33:30 --> 00:33:36
comes to a halt
and becomes zero.
543
00:33:31 --> 00:33:37
So I ask this equation,
when are you zero?
544
00:33:36 --> 00:33:42
This is the one I pick
545
00:33:38 --> 00:33:44
and so I say, zero equals
plus 133 minus 10 times t.
546
00:33:47 --> 00:33:53
You may think that the
gravitational acceleration
547
00:33:50 --> 00:33:56
at an altitude of 30,000 feet
could be substantially less
548
00:33:54 --> 00:34:00
than the canonical number of 10.
549
00:33:56 --> 00:34:02
It is a little less
550
00:33:58 --> 00:34:04
because you're a little bit
further away from the Earth,
551
00:34:01 --> 00:34:07
but it's only 0.3 percent less,
and so we'll just accept the 10.
552
00:34:04 --> 00:34:10
It's easy to work with.
553
00:34:06 --> 00:34:12
And so when is it
at the highest point?
554
00:34:09 --> 00:34:15
That is when t
equals 13.3 seconds.
555
00:34:13 --> 00:34:19
So that's about how long it
takes to get there.
556
00:34:18 --> 00:34:24
When I gave
the lecture last time,
557
00:34:20 --> 00:34:26
I said it's about 15 seconds,
558
00:34:21 --> 00:34:27
because I made the numbers...
I rounded them off.
559
00:34:24 --> 00:34:30
It's about 30.3 seconds.
560
00:34:28 --> 00:34:34
And what is this distance h now?
561
00:34:29 --> 00:34:35
Ah! Now I have to go
to this equation.
562
00:34:33 --> 00:34:39
I say h equals zero, because
I'm going to define the point
563
00:34:38 --> 00:34:44
where the plane
starts its trajectory.
564
00:34:40 --> 00:34:46
I call that y zero zero,
I'm free to do that.
565
00:34:44 --> 00:34:50
h equals zero plus 133--
that is the speed--
566
00:34:51 --> 00:34:57
times 13.3 seconds minus
one-half times g-- that is 5--
567
00:34:59 --> 00:35:05
times 13.3 squared.
568
00:35:04 --> 00:35:10
That is what h must be.
569
00:35:07 --> 00:35:13
And that turns out to be
about 885 meters.
570
00:35:14 --> 00:35:20
I think I told you last time
it's about 900, close enough.
571
00:35:19 --> 00:35:25
So we know now
how long it takes to reach p
572
00:35:21 --> 00:35:27
and we know
what the vertical distance is.
573
00:35:25 --> 00:35:31
And the whole trip
back to this starting point--
574
00:35:28 --> 00:35:34
if we call this sort
of a starting point,
575
00:35:31 --> 00:35:37
starting altitude--
576
00:35:32 --> 00:35:38
this whole trip will take
twice the amount of time.
577
00:35:35 --> 00:35:41
To get back to this point
when the engines are restarted
578
00:35:38 --> 00:35:44
is about 26.5, 27 seconds.
579
00:35:43 --> 00:35:49
How far has the plane traveled,
then, in horizontal direction?
580
00:35:48 --> 00:35:54
Well, now I go
back to this equation.
581
00:35:53 --> 00:35:59
So now I say, aha!
582
00:35:55 --> 00:36:01
x then,
when it is back at this point,
583
00:35:59 --> 00:36:05
must be x zero-- which I
conveniently choose zero--
584
00:36:02 --> 00:36:08
plus 133 meters per second,
585
00:36:05 --> 00:36:11
which is the velocity in the x
direction, which never changes.
586
00:36:08 --> 00:36:14
When the plane is here,
that velocity in the x direction
587
00:36:11 --> 00:36:17
is the same 133 meters
per second as it was here,
588
00:36:16 --> 00:36:22
which, by the way,
is about 300 miles per hour.
589
00:36:18 --> 00:36:24
That never changes
590
00:36:20 --> 00:36:26
if there is no air drag
or air friction of any kind.
591
00:36:23 --> 00:36:29
So we get plus 133
times the time
592
00:36:27 --> 00:36:33
and the whole trip
takes 26.6 seconds,
593
00:36:31 --> 00:36:37
and that, if you convert
that to kilometers
594
00:36:35 --> 00:36:41
is about 3.5 kilometers.
595
00:36:38 --> 00:36:44
Now, you could ask yourself
the question:
596
00:36:43 --> 00:36:49
What is the velocity
of that plane
597
00:36:47 --> 00:36:53
when it is at that point s?
598
00:36:50 --> 00:36:56
599
00:36:51 --> 00:36:57
And now... you may want
to abandon now
600
00:36:56 --> 00:37:02
this one-dimensional idea
of x and y.
601
00:36:59 --> 00:37:05
You may say, "Well, look.
602
00:37:01 --> 00:37:07
"This is a parabola and
it is completely symmetric.
603
00:37:04 --> 00:37:10
"If the plane comes up here
604
00:37:06 --> 00:37:12
"with 425 miles per hour
at an angle of 45 degrees,
605
00:37:10 --> 00:37:16
"then obviously it comes down
here at an angle of 45 degrees
606
00:37:14 --> 00:37:20
and the speed must again be
425 miles per hour."
607
00:37:17 --> 00:37:23
And you would score 100 percent,
of course-- it's clear.
608
00:37:21 --> 00:37:27
I want you
to appreciate, however,
609
00:37:24 --> 00:37:30
that I could continue
to think of this
610
00:37:28 --> 00:37:34
as two one-dimensional motions.
611
00:37:31 --> 00:37:37
And I can therefore calculate
612
00:37:34 --> 00:37:40
what the velocity
in the x direction is at s
613
00:37:40 --> 00:37:46
and what the velocity
in the y direction is at s.
614
00:37:44 --> 00:37:50
So what is the velocity
in the x direction at point s?
615
00:37:49 --> 00:37:55
I go to equation...
the second equation there.
616
00:37:52 --> 00:37:58
That is v zero x, that is
plus 133 meters per second.
617
00:38:00 --> 00:38:06
What is the velocity
in the y direction?
618
00:38:03 --> 00:38:09
Ah, I have to go
to this equation now.
619
00:38:05 --> 00:38:11
v zero y minus gt.
620
00:38:09 --> 00:38:15
So I get plus 133 minus 10
621
00:38:14 --> 00:38:20
times the 26.6 seconds
to reach that point s.
622
00:38:20 --> 00:38:26
And what do I find?
623
00:38:21 --> 00:38:27
Minus 133 meters per second.
624
00:38:26 --> 00:38:32
The velocity in the y direction
started off plus 133,
625
00:38:31 --> 00:38:37
but now it is minus 133.
626
00:38:33 --> 00:38:39
You see, this is sign-sensitive.
627
00:38:36 --> 00:38:42
This is wonderful.
628
00:38:37 --> 00:38:43
That's the great thing
about treating it that way.
629
00:38:39 --> 00:38:45
So you now know that it comes
in with a velocity
630
00:38:43 --> 00:38:49
of 133 in the x direction--
positive--
631
00:38:50 --> 00:38:56
133 in the minus y direction,
632
00:38:53 --> 00:38:59
and so what is the net,
the sum of the two vectors?
633
00:39:03 --> 00:39:09
That, of course, is this vector
634
00:39:05 --> 00:39:11
and no surprise,
this angle is 45 degrees
635
00:39:10 --> 00:39:16
and this one is
the square root of 2 times 133
636
00:39:13 --> 00:39:19
and that, of course, gives you
back your 189 meters per second.
637
00:39:18 --> 00:39:24
189 meters per second, and
that is 425 miles per hour.
638
00:39:27 --> 00:39:33
I'm not recommending that
you would do this, of course.
639
00:39:30 --> 00:39:36
It is perfectly reasonable
640
00:39:32 --> 00:39:38
to immediately come
to that conclusion
641
00:39:35 --> 00:39:41
because of the symmetry
of the parabola.
642
00:39:40 --> 00:39:46
643
00:39:42 --> 00:39:48
Let's now turn
to uniform circular motion.
644
00:39:49 --> 00:39:55
Uniform circular motion occurs
645
00:39:53 --> 00:39:59
when an object goes around
in a circle
646
00:39:58 --> 00:40:04
and when the speed
never changes.
647
00:40:02 --> 00:40:08
If the speed doesn't change,
648
00:40:05 --> 00:40:11
then the velocity,
of course, does change
649
00:40:08 --> 00:40:14
because the direction changes
all the time,
650
00:40:11 --> 00:40:17
but the speed does not.
651
00:40:17 --> 00:40:23
So here we have our circle.
652
00:40:23 --> 00:40:29
Let this be radius r,
653
00:40:25 --> 00:40:31
and at this moment in time,
the object is here.
654
00:40:30 --> 00:40:36
It has a certain velocity.
655
00:40:33 --> 00:40:39
This is 90 degrees.
656
00:40:35 --> 00:40:41
And later in time,
657
00:40:37 --> 00:40:43
the object is here,
the speed is the same,
658
00:40:40 --> 00:40:46
but the direction
has changed, 90 degrees.
659
00:40:45 --> 00:40:51
So these vectors,
they have the same length.
660
00:40:48 --> 00:40:54
661
00:40:51 --> 00:40:57
In a situation like this
662
00:40:52 --> 00:40:58
that we have
uniform circular motion--
663
00:40:55 --> 00:41:01
so it's uniform...
664
00:41:01 --> 00:41:07
circular motion--
665
00:41:06 --> 00:41:12
we first identify what we call
the period T in seconds.
666
00:41:13 --> 00:41:19
That's the time to go around.
667
00:41:17 --> 00:41:23
Then we identify what we call
the frequency,
668
00:41:21 --> 00:41:27
that is, how many times
it goes around per second.
669
00:41:25 --> 00:41:31
I prefer the letter f, but our
book uses the Greek letter nu.
670
00:41:31 --> 00:41:37
I find the nu
often very confusing
671
00:41:33 --> 00:41:39
with the v of velocity.
672
00:41:35 --> 00:41:41
That's why I prefer the f.
673
00:41:37 --> 00:41:43
It is one over T, and so
the units are seconds minus one
674
00:41:42 --> 00:41:48
but most physicists
would call that "hertz."
675
00:41:46 --> 00:41:52
Ten hertz means to go
ten times around per second.
676
00:41:50 --> 00:41:56
And then we identify omega,
the angular velocity.
677
00:41:59 --> 00:42:05
Omega, which is
in radians per second.
678
00:42:05 --> 00:42:11
679
00:42:06 --> 00:42:12
Since it takes T seconds
to go around two pi radians,
680
00:42:12 --> 00:42:18
omega is two pi divided by T.
681
00:42:16 --> 00:42:22
Now, then we have the speed,
682
00:42:19 --> 00:42:25
which we can also think
of as a linear velocity.
683
00:42:22 --> 00:42:28
How many meters per second
is linear,
684
00:42:25 --> 00:42:31
as opposed to
how many radians per second,
685
00:42:28 --> 00:42:34
which is angular velocity.
686
00:42:29 --> 00:42:35
So this is a linear velocity,
this is an angular velocity.
687
00:42:33 --> 00:42:39
And that linear velocity,
688
00:42:35 --> 00:42:41
which, in this case,
is really your speed,
689
00:42:38 --> 00:42:44
is of course
the circumference of the circle
690
00:42:40 --> 00:42:46
divided by how many seconds
it takes to go around.
691
00:42:44 --> 00:42:50
And that is also omega r
692
00:42:47 --> 00:42:53
and that is now
in meters per second.
693
00:42:51 --> 00:42:57
All this is only possible
if there is an acceleration,
694
00:42:59 --> 00:43:05
and the acceleration is called
the centripetal acceleration.
695
00:43:03 --> 00:43:09
It is always pointed
towards the center:
696
00:43:11 --> 00:43:17
"a" centripetal,
697
00:43:13 --> 00:43:19
"a" centripetal.
698
00:43:16 --> 00:43:22
And the centripetal
acceleration-- the magnitude--
699
00:43:21 --> 00:43:27
is v squared divided by r,
700
00:43:25 --> 00:43:31
which is therefore
also omega squared r,
701
00:43:28 --> 00:43:34
and that, of course,
is in meters per second squared.
702
00:43:32 --> 00:43:38
703
00:43:37 --> 00:43:43
I want to work out
a specific example,
704
00:43:41 --> 00:43:47
and the example that I have
chosen is the human centrifuge
705
00:43:49 --> 00:43:55
that is used by NASA in Houston
for experiments on humans
706
00:43:58 --> 00:44:04
to see how they deal
with strong accelerations.
707
00:44:05 --> 00:44:11
This is that centrifuge.
708
00:44:09 --> 00:44:15
709
00:44:13 --> 00:44:19
The radius
from the axis of rotation--
710
00:44:18 --> 00:44:24
the axis of rotation is here--
711
00:44:22 --> 00:44:28
and the distance
from here to here,
712
00:44:23 --> 00:44:29
though you may not think so,
is about 15 meters.
713
00:44:26 --> 00:44:32
So the astronauts go in here
and then the thing goes around.
714
00:44:30 --> 00:44:36
And so I would like to work out
this with some numbers.
715
00:44:36 --> 00:44:42
716
00:44:44 --> 00:44:50
The radius r--
I'll give your light back
717
00:44:48 --> 00:44:54
because it may
be nicer for you...
718
00:44:51 --> 00:44:57
The radius is 15 meters.
719
00:44:53 --> 00:44:59
It depends, of course,
a little bit
720
00:44:55 --> 00:45:01
on where the person
is located in that sphere.
721
00:44:58 --> 00:45:04
It goes around
24 revolutions per minute
722
00:45:05 --> 00:45:11
and that translates
into 0.4 hertz.
723
00:45:09 --> 00:45:15
724
00:45:11 --> 00:45:17
So the period to go around
for one rotation
725
00:45:16 --> 00:45:22
is 2.5 seconds.
726
00:45:17 --> 00:45:23
The thing goes around
once in 2.5 seconds.
727
00:45:21 --> 00:45:27
So the angular velocity omega,
which is two pi divided by T...
728
00:45:28 --> 00:45:34
If you take two pi
and divide it by 2.5,
729
00:45:30 --> 00:45:36
it just comes out
to be roughly 2.5.
730
00:45:33 --> 00:45:39
(chuckling ):
It's a purely accident,
that's the way it is.
731
00:45:36 --> 00:45:42
Don't ever think that that
has to be the same, of course.
732
00:45:40 --> 00:45:46
It just happens to come out
that way for these dimensions.
733
00:45:43 --> 00:45:49
So omega is about 2.5 radians
per second.
734
00:45:47 --> 00:45:53
And the speed, linear speed--
735
00:45:53 --> 00:45:59
tangential speed, if you
want to call it-- is omega r.
736
00:45:59 --> 00:46:05
That comes out to be about 35...
37.7 meters per second,
737
00:46:08 --> 00:46:14
and that translates into
about 85 miles per hour,
738
00:46:11 --> 00:46:17
so it's a sizable speed.
739
00:46:15 --> 00:46:21
What, of course,
the goal is for NASA:
740
00:46:18 --> 00:46:24
What is the centripetal
acceleration--
741
00:46:20 --> 00:46:26
that is omega squared r--
742
00:46:22 --> 00:46:28
or, if you prefer to take
v squared divided by r,
743
00:46:26 --> 00:46:32
you'll find, of course,
exactly the same answer
744
00:46:28 --> 00:46:34
if you haven't made a slip,
745
00:46:31 --> 00:46:37
and that is 95 meters
per second squared.
746
00:46:37 --> 00:46:43
And that is about ten times
747
00:46:40 --> 00:46:46
the gravitational acceleration
on Earth,
748
00:46:41 --> 00:46:47
which is really phenomenal,
if you add, too, the fact
749
00:46:46 --> 00:46:52
that the direction is changing
all the time when you go around,
750
00:46:49 --> 00:46:55
so you feel the 10 g
in this direction
751
00:46:52 --> 00:46:58
and then you feel it
in a different direction.
752
00:46:55 --> 00:47:01
I can't imagine how people
can actually survive that--
753
00:46:58 --> 00:47:04
I mean, not faint.
754
00:47:01 --> 00:47:07
Most people, like you and me,
755
00:47:03 --> 00:47:09
if we were to be accelerated
along a straight line,
756
00:47:06 --> 00:47:12
not even a circle,
where the direction changed,
757
00:47:09 --> 00:47:15
but along a straight line,
758
00:47:10 --> 00:47:16
most of us faint
when we get close to 6 g.
759
00:47:15 --> 00:47:21
And there is a reason for that.
760
00:47:16 --> 00:47:22
You get problems
with your blood circulation
761
00:47:19 --> 00:47:25
and not enough oxygen goes
to your brains,
762
00:47:22 --> 00:47:28
and that's why you faint.
763
00:47:24 --> 00:47:30
How these astronauts
can do it at 10 g
764
00:47:27 --> 00:47:33
and the direction changing
all the time, it beats me.
765
00:47:30 --> 00:47:36
If you take a Boeing 747,
it takes 30 seconds
766
00:47:35 --> 00:47:41
from the moment
that it starts on the runway
767
00:47:37 --> 00:47:43
until it takes off.
768
00:47:38 --> 00:47:44
You should time that,
when you get a chance.
769
00:47:40 --> 00:47:46
It's very close to 30 seconds,
770
00:47:42 --> 00:47:48
and by that time
the plane has reached a speed
771
00:47:44 --> 00:47:50
of about 150 miles per hour.
772
00:47:48 --> 00:47:54
And if you calculate,
773
00:47:49 --> 00:47:55
if you assume that the
acceleration is constant--
774
00:47:51 --> 00:47:57
it's an easy calculation--
it turns out
775
00:47:53 --> 00:47:59
that the acceleration is only
two meters per second squared.
776
00:47:56 --> 00:48:02
That is only one-fifth of
the gravitational acceleration.
777
00:47:59 --> 00:48:05
Feels sort of good, right?
778
00:48:01 --> 00:48:07
It's very comfortable,
when you're taking off.
779
00:48:03 --> 00:48:09
It's only 2 meters
per second squared.
780
00:48:06 --> 00:48:12
These poor people,
men and women,
781
00:48:08 --> 00:48:14
95 meters per second squared.
782
00:48:11 --> 00:48:17
783
00:48:16 --> 00:48:22
I would like
to address something
784
00:48:18 --> 00:48:24
that is not part of the exam,
785
00:48:20 --> 00:48:26
but that is something
that I want you to think about,
786
00:48:24 --> 00:48:30
something that is fun,
787
00:48:25 --> 00:48:31
and it's always nice
to do something that is fun.
788
00:48:30 --> 00:48:36
It has to do
with my last lecture.
789
00:48:32 --> 00:48:38
I have to clean my hands first
for it to work quite well.
790
00:48:40 --> 00:48:46
I have a yardstick here, and I
am going to put the yardstick
791
00:48:44 --> 00:48:50
on my hands, on my two fingers,
which I hold in front of me.
792
00:48:51 --> 00:48:57
Here it is.
793
00:48:54 --> 00:49:00
It's resting on my two fingers,
794
00:48:57 --> 00:49:03
and I'm going to move my
two fingers towards each other.
795
00:49:03 --> 00:49:09
One of them begins
to slide first, of course.
796
00:49:06 --> 00:49:12
I can't tell you which one.
797
00:49:08 --> 00:49:14
But something
very strange will happen.
798
00:49:12 --> 00:49:18
If this one starts to slide
first, it comes to a stop
799
00:49:15 --> 00:49:21
and then the other one
starts to slide
800
00:49:17 --> 00:49:23
and it comes to a stop.
801
00:49:18 --> 00:49:24
And then this one starts
to slide and so on.
802
00:49:21 --> 00:49:27
And that is very strange.
803
00:49:23 --> 00:49:29
This is something
you should be able to explain,
804
00:49:27 --> 00:49:33
certainly after the lecture
we had last time.
805
00:49:30 --> 00:49:36
Look at this.
806
00:49:35 --> 00:49:41
Did you see the alternation?
I'll do it a little faster.
807
00:49:38 --> 00:49:44
Left is going, right is going,
left is going, right is going,
808
00:49:41 --> 00:49:47
left is going.
809
00:49:42 --> 00:49:48
Once more... look at it.
810
00:49:43 --> 00:49:49
Left is going, right is going,
left is going, right is going.
811
00:49:47 --> 00:49:53
They alternate.
812
00:49:49 --> 00:49:55
Give this some thought,
813
00:49:51 --> 00:49:57
and you know,
PIVoT has an option
814
00:49:54 --> 00:50:00
that you can discuss
problems with other students,
815
00:49:57 --> 00:50:03
so make use
of this discussion button
816
00:49:59 --> 00:50:05
and see whether you can
come to an explanation.
817
00:50:02 --> 00:50:08
Good luck on your exam.
818
00:50:04 --> 00:50:10
See you next Friday.
819
00:50:07 --> 00:50:13
820
00:50:11 --> 00:50:17.000