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The bad news today is that there
will be quite a bit of math.
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But the good news
is that we will only do it once
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and it will only take
something like half-hour.
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There are quantities in physics
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which are determined uniquely
by one number.
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Mass is one of them.
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Temperature is one of them.
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Speed is one of them.
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We call those scalars.
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There are others where
you need more than one number
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for instance, on a one-
dimensional motion, velocity
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it has a certain magnitude--
that's the speed--
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but you also have to know
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whether it goes this way
or that way.
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So there has to be a direction.
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Velocity is a vector and
acceleration is a vector
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and today we're going to learn
how to work with these vectors.
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A vector has a length
and a vector has a direction
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and that's why we actually
represent it by an arrow.
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We all have seen...
this is a vector.
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Remember this--
this is a vector.
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If you look at the vector
head-on, you see a dot.
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If you look at the vector
from behind, you see a cross.
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This is a vector
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and that will be
our representation of vectors.
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Imagine that I am standing
on the table in 26.100.
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This is the table and I am
standing, say, at point O
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and I move along a straight line
from O to point P
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so I move like so.
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That's why I am on the table
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and that's where you will see me
when you look from 26.100.
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It just so happens
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that someone is also going
to move the table--
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in that same amount of time--
from here to there.
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So that means that the table
will have moved down
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and so my point P will have
moved down exactly the same way
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and so you will see me
now at point S.
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You will see me
at point S in 26.100
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although I am still standing
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at the same location
on the table.
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The table has moved.
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This is now the position
of the table.
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See, the whole table
has shifted.
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Now, if these two motions
take place simultaneously
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then what you will see
from where you are sitting...
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you will see me move in 26.100
from O straight line to S
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and this holds the secret
behind the adding of vectors.
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We say here that the vector OS--
we'll put an arrow over it--
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is the vector OP, with an arrow
over it, plus PS.
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This defines how we add vectors.
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There are various ways
that you can add vectors.
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Suppose I have here vector A
and I have here vector B.
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Then you can do it this way
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which I call
the "head-tail" technique.
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I take B and I bring it
to the head of A.
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So this is B, this is a vector
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and then the net result
is A plus B.
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This vector C equals A plus B.
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That's one way of doing it.
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It doesn't matter
whether you take B...
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the tail of B to the head of A
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or whether you take the tail of
A and bring it to the head of B.
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You will get the same result.
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There's another way
you can do it
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and I call that
"the parallelogram method."
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Here you have A.
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You bring the two tails
together, so here is B now
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so the tails are touching
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and now you complete
this parallelogram.
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And now this vector C
is the same sum vector
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that you have here,
whichever way you prefer.
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You see immediately
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that A plus B is the same
as B plus A.
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There is no difference.
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What is the meaning
of a negative vector?
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Well, A minus A equals zero--
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vector A subtract
from vector A equals zero.
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So here is vector A.
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So which vector do I have
to add to get zero?
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I have to add minus A.
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Well, if you use
the head-tail technique...
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This is A.
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You have to add this vector
to have zero
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so this is minus A
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and so minus A is nothing
but the same as A
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but flipped over 180 degrees.
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We'll use that very often.
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And that brings us to the point
of subtraction of vectors.
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How do we subtract vectors?
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So A minus B equals C.
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Here we have vector A
and here we have--
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let me write this down here--
and here we have vector B.
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One way to look at this
is the following.
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You can say A minus B
is A plus minus B
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and we know how to add vectors
and we know what minus B is.
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Minus B is the same vector
but flipped over
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so we put here minus B
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and so this vector now
here equals A minus B.
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Here's vector C,
here's A minus B.
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And, of course, you can do it
in different ways.
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You can also think of it
as A plus... as C plus B is A.
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Right? You can say you can
bring this to the other side.
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You can say C plus B is A,
C plus B is A.
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In other words, which vector
do I have to add to B to get A?
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And then you have the
parallelogram technique again.
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There are many ways
you can do it.
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The head-tail technique
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is perhaps the easiest
and the safest.
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So you can add
a countless number of vectors
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one plus the other,
and the next one
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and you finally have the sum
of five or six or seven vectors
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which, then, can be
represented by only one.
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When you add scalars,
for instance, five and four
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then there is only one answer,
that is nine.
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Five plus four is nine.
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Suppose you have two vectors.
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You have no information
on their direction
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but you do know that the
magnitude of one is four
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and the magnitude
of the other is five.
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That's all you know.
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Then the magnitude of
the sum vector could be nine
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if they are both in the same
direction-- that's the maximum--
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or it could be one, if they
are in opposite directions.
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So then you have
a whole range of possibilities
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because you do not know
the direction.
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So the adding and the
subtraction of vectors
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is way more complicated
than just scalars.
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As we have seen,
that the sum of vectors
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can be represented by one vector
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equally can we take one vector
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and we can replace it
by the sum of others.
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And we call that
"decomposition" of a vector.
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And that's going to be
very important in 801
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and I want you to follow this,
therefore, quite closely.
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I have a vector which is
in three-dimensional space.
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This is my z axis...
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this is my x axis,
y axis and z axis.
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This is the origin O
and here is a point P
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and I have a vector OP--
that's the vector.
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And what I do now,
I project this vector
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onto the three axes,
x, y and z.
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So there we go.
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Each one has her or his
own method of doing this.
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There we are.
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I call this vector
vector A.
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Now, this angle will be theta,
and this angle will be phi.
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Notice that the projection of A
on the y axis has here
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a number which I call A of y.
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This number is A of x and
this number here is A of z--
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simply a projection of
that vector onto the three axes.
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We now introduce
what we call "unit vectors."
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Unit vectors are always pointing
in the direction
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of the positive axis
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and the unit vector in
the x direction is this one.
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It has a length one,
and we write for it "x roof."
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"Roof" always means unit vector.
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And this is the unit vector
in the y direction
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and this is the unit vector
in the z direction.
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And now I'm going
to rewrite vector A
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in terms of the three components
that we have here.
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So the vector A,
I'm going to write
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as "A of x times x roof,
plus A of y times y roof
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plus A of z times z roof."
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And this A of x times x
is really a vector
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that runs from the origin
to this point.
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So we could put in that
as a vector, if you want to.
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This makes it a vector.
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This is that vector.
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A of y times... oh, sorry,
it is A of x, this one.
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A of y times y roof is this one
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and A of z times z roof
is this one.
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And so these three green vectors
added together
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are exactly identical
to the vector OP
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so we have decomposed one vector
into three directions.
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And we will see that very often,
this is of great use in 801.
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The magnitude of the vector is
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the square root of Ax squared
plus Ay squared plus Az squared
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and so we can take
a simple example.
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For instance,
I take a vector A--
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this is just an example,
to see this in action--
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and we call A three X roof,
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so A of axis is three
minus five y roof plus 6 Z roof
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so that means that it's
three units in this direction
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it is five units
in this direction--
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in the minus y direction--
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and six in the plus z direction.
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That makes up a vector
and I call that vector A.
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What is the magnitude
of that vector--
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which I always write down
with vertical bars--
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if I put two bars on one side,
that's always the magnitude
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or sometimes
I simply leave the arrow off,
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but to be always on
the safe side, I like this idea
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that you know it's
really the magnitude
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becomes the scalar
when you do that.
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So that would be the square root
of three squared is nine
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five squared is 25,
six squared is 36
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so that's the square root of 70.
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And suppose I asked you,
"What is theta?"
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It's uniquely determined,
of course.
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This vector
is uniquely determined
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in three-dimensional space
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so you should be able to find
phi and theta.
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Well, the cosine of theta...
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See, this angle here...
90 degrees projection.
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So the cosine of theta
is A of z divided by A itself.
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So the cosine of theta equals
A of z divided by A itself
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which in our case
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would be six divided
by the square root of 70.
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And you can do fine.
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It's just simply a matter
of manipulating some numbers.
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We now come to a much more
difficult part of vectors
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and that is multiplication
of vectors.
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We're not going to need this
until October, but I decided
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we might as well
get it over with now.
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Now that we introduced vectors,
you can add and subtract
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you might as well learn
about multiplication.
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It's sort of, the job is done,
it's like going to the dentist.
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It's a little painful,
but it's good for you
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and when it's behind you,
the pain disappears.
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So we're going to talk about
multiplication of vectors
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something that will not
come back until October
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and later in the course.
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There are two ways
that we multiply vectors
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and one is called
the "dot product"
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often also called
the scalar product.
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A dot B, a fat dot, and that
is defined as it is a scalar.
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A of x times B of x,
just a number
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plus A of y times B of y--
that's another number--
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plus A of z times B of z--
that's another number.
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It is a scalar.
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It has no longer a direction.
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That is the dot product.
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So that's method number one.
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That's completely legitimate
and you can always use that.
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There is another way
to find the dot product
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depending upon
what you're being given--
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how the problem is
presented to you.
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If someone gives you the vector
A and you have the vector B
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and you happen to know
this angle between them,
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this angle theta--
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which has nothing to do
with that angle theta;
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it's the angle between the two--
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then the dot product
is also the following
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and you may make an attempt
to prove that.
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You project the vector B on A.
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This is that projection.
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The length of this vector
is B cosine theta.
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And then the dot product
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is the magnitude of A
times the magnitude of B
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times the cosine
of the angle theta.
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The two are
completely identical.
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Now, you may ask me,
you may say,
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"Gee, how do I know
what theta is?
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"How do I know
I should take theta this angle
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"or maybe I should take
theta this angle?
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I mean, what angle
is A making with B?"
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It makes no difference
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because the cosine
of this angle here
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is the same as the cosine
of 360 degrees minus theta
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so that makes no difference.
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Sometimes this is faster
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depending upon how the problem
is presented to you;
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sometimes the other is faster.
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You can immediately see
by looking at this--
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it's easier to see
than looking here--
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that the dot product can be
larger than zero
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it can be equal to zero
and it can be smaller than zero.
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A and B are, by definition,
always positive.
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They are a magnitude.
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That's always determined
by the cosine of theta.
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If the cosine of theta
is larger than zero,
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well, then
it's larger than zero.
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The cosine of theta can be zero.
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If the angle for theta
is pi over two--
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in other words,
if the two vectors
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are perpendicular
to each other--
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then the dot product is zero,
and if this angle theta
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00:17:20 --> 00:17:26
is between 90 degrees
and 180 degrees
284
00:17:23 --> 00:17:29
then the cosine is negative.
285
00:17:26 --> 00:17:32
We will see that at work,
no pun implied
286
00:17:29 --> 00:17:35
when we're going to deal
with work in physics.
287
00:17:32 --> 00:17:38
You will see
that we can do positive work
288
00:17:34 --> 00:17:40
and we can do negative work
289
00:17:36 --> 00:17:42
and that has to do
with this dot product.
290
00:17:38 --> 00:17:44
Work and energy are
dot products.
291
00:17:43 --> 00:17:49
I could do an extremely
simple example with you;
292
00:17:48 --> 00:17:54
the simplest
that I can think of.
293
00:17:51 --> 00:17:57
Perhaps it's almost an insult--
it's not meant that way.
294
00:17:56 --> 00:18:02
Suppose we have A dot B
295
00:18:01 --> 00:18:07
and A is the one that you really
have on the blackboard there.
296
00:18:06 --> 00:18:12
Right here, that's A.
297
00:18:08 --> 00:18:14
But B is just two y roof.
298
00:18:14 --> 00:18:20
Two y roof, that's all it is.
299
00:18:19 --> 00:18:25
Well, what is A dot B?
300
00:18:21 --> 00:18:27
A dot B... there's
no x component of B
301
00:18:26 --> 00:18:32
so that becomes zero,
this term becomes zero.
302
00:18:29 --> 00:18:35
There is only
a y component of B
303
00:18:31 --> 00:18:37
so it is minus five
times plus two
304
00:18:35 --> 00:18:41
so I get minus ten, because
there was no z component.
305
00:18:39 --> 00:18:45
Simple as that,
so it's minus ten.
306
00:18:43 --> 00:18:49
I can give you another example,
example two.
307
00:18:48 --> 00:18:54
Suppose A itself is the unit
vector in the y direction
308
00:18:53 --> 00:18:59
and B is the unit vector
in the z direction.
309
00:18:59 --> 00:19:05
Then A dot B is what?
310
00:19:05 --> 00:19:11
I want to hear it
loud and clear.
311
00:19:07 --> 00:19:13
CLASS:
Zero.
312
00:19:08 --> 00:19:14
LEWIN:
Yeah! Zero.
313
00:19:10 --> 00:19:16
It is zero-- you don't even
have to think about anything.
314
00:19:13 --> 00:19:19
You know that these two
are at 90 degrees.
315
00:19:15 --> 00:19:21
If you want to waste your time
316
00:19:17 --> 00:19:23
and want to substitute it
in here
317
00:19:19 --> 00:19:25
you will see
that it comes out to be zero.
318
00:19:21 --> 00:19:27
It should work, because clearly
A of y means
319
00:19:24 --> 00:19:30
that this... this is one.
320
00:19:28 --> 00:19:34
That's what it means.
321
00:19:29 --> 00:19:35
And B is z, that means
that B of z... this is one
322
00:19:33 --> 00:19:39
and all the others do not exist.
323
00:19:36 --> 00:19:42
Well, I wish you luck
with that and we now go
324
00:19:39 --> 00:19:45
to a way more difficult
part of multiplication
325
00:19:43 --> 00:19:49
and that is
vector multiplication
326
00:19:46 --> 00:19:52
which is called
"the vector product."
327
00:19:52 --> 00:19:58
Or also called...
most of the time
328
00:19:54 --> 00:20:00
I refer to it
as "the cross product."
329
00:19:58 --> 00:20:04
The cross product is written
like so: A cross B equals C.
330
00:20:06 --> 00:20:12
It's a cross, very clear cross.
331
00:20:10 --> 00:20:16
And I will tell you
how I remember...
332
00:20:12 --> 00:20:18
that is, method number one.
333
00:20:13 --> 00:20:19
I'm going to teach you--
just like with the dot product--
334
00:20:15 --> 00:20:21
two methods.
335
00:20:16 --> 00:20:22
I will tell you
method number one
336
00:20:18 --> 00:20:24
which is the one
that always works.
337
00:20:20 --> 00:20:26
It's time-consuming,
but it always works.
338
00:20:23 --> 00:20:29
You write down here
a matrix with three rows.
339
00:20:27 --> 00:20:33
The first row is x roof,
y roof, z roof.
340
00:20:33 --> 00:20:39
The second one is
A of x, A of y, A of z.
341
00:20:38 --> 00:20:44
It's important,
if A is here first
342
00:20:40 --> 00:20:46
that that second row must be A
and the third row is then B.
343
00:20:45 --> 00:20:51
B of x, B of y, B of z.
344
00:20:49 --> 00:20:55
So these six are numbers
and these are the unit vectors.
345
00:20:54 --> 00:21:00
I repeat this here verbatim--
346
00:21:00 --> 00:21:06
you will see in a minute
why I need that--
347
00:21:07 --> 00:21:13
and I will do the same here.
348
00:21:09 --> 00:21:15
349
00:21:17 --> 00:21:23
Okay, and now comes the recipe.
350
00:21:19 --> 00:21:25
You take... you go from
the upper left-hand corner
351
00:21:24 --> 00:21:30
to the one in this direction.
352
00:21:28 --> 00:21:34
You multiply them, all three,
and that's a plus sign.
353
00:21:32 --> 00:21:38
So you get Ay... so C
354
00:21:35 --> 00:21:41
which is A cross B equals Ay,
times Bz, times the x roof--
355
00:21:45 --> 00:21:51
but I'm not going to put
the x roof in yet--
356
00:21:48 --> 00:21:54
because I have to subtract
this one... minus sign
357
00:21:55 --> 00:22:01
which has Az By
358
00:21:58 --> 00:22:04
so it is minus Az By, and
that is in the direction x.
359
00:22:07 --> 00:22:13
The next one is this one.
360
00:22:11 --> 00:22:17
Az Bx...
361
00:22:17 --> 00:22:23
minus this one
362
00:22:23 --> 00:22:29
Ax Bz
363
00:22:29 --> 00:22:35
in the direction y.
364
00:22:32 --> 00:22:38
And last but not least
365
00:22:35 --> 00:22:41
Ax By...
366
00:22:42 --> 00:22:48
minus Ay Bx...
367
00:22:52 --> 00:22:58
in the direction
of the unit vector z.
368
00:22:59 --> 00:23:05
So this part here is
what we call "C of x".
369
00:23:04 --> 00:23:10
It's the x component
of this vector
370
00:23:07 --> 00:23:13
and this we can call "C of y"
and this we can call "C of z."
371
00:23:13 --> 00:23:19
So we can also write
that vector, then,
372
00:23:16 --> 00:23:22
that C equals C of x, x roof,
plus C of y, y roof
373
00:23:23 --> 00:23:29
plus C of z, z roof.
374
00:23:27 --> 00:23:33
Cross product of A and B.
375
00:23:31 --> 00:23:37
We will have lots of exercises,
376
00:23:33 --> 00:23:39
lots of chances you will have
on assignment, too
377
00:23:36 --> 00:23:42
to play with this a little bit.
378
00:23:37 --> 00:23:43
Now comes my method number two
and method number two is, again
379
00:23:42 --> 00:23:48
as we had with the dot product,
is a geometrical method.
380
00:23:51 --> 00:23:57
Let me try to work
on this board in between.
381
00:23:56 --> 00:24:02
If you know vector A
and you know vector B
382
00:24:05 --> 00:24:11
and you know
that the angle is theta
383
00:24:08 --> 00:24:14
then the cross product, C,
equals A cross B
384
00:24:14 --> 00:24:20
is the magnitude of A
times the magnitude of B
385
00:24:19 --> 00:24:25
times the sine of theta
386
00:24:21 --> 00:24:27
not the cosine of theta as we
had before with the dot product.
387
00:24:25 --> 00:24:31
It is the sine of theta.
388
00:24:29 --> 00:24:35
So you can really immediately
see that this will be zero
389
00:24:32 --> 00:24:38
if theta is either zero degrees
or 180 degrees
390
00:24:36 --> 00:24:42
whereas the dot product was zero
391
00:24:37 --> 00:24:43
when the angle between them
was 90 degrees.
392
00:24:43 --> 00:24:49
This number can be larger
than zero
393
00:24:45 --> 00:24:51
if the sine theta
is larger than zero.
394
00:24:46 --> 00:24:52
It can also be smaller
than zero.
395
00:24:48 --> 00:24:54
Now we only have
the magnitude of the vector
396
00:24:51 --> 00:24:57
and now comes the hardest part.
397
00:24:53 --> 00:24:59
What is the direction
of the vector?
398
00:24:55 --> 00:25:01
And that is something
399
00:24:57 --> 00:25:03
that you have to engrave
in your mind and not forget.
400
00:25:01 --> 00:25:07
The direction is found
as follows.
401
00:25:04 --> 00:25:10
You take A,
because it's first mentioned
402
00:25:08 --> 00:25:14
and you rotate A over the
shortest possible angle to B.
403
00:25:13 --> 00:25:19
If you had in your hand
a corkscrew--
404
00:25:15 --> 00:25:21
and I will show that
in a minute--
405
00:25:16 --> 00:25:22
then you turn the corkscrew as
seen from your seats clockwise
406
00:25:20 --> 00:25:26
and the corkscrew would go
into the blackboard.
407
00:25:23 --> 00:25:29
And if the corkscrew goes
into the blackboard
408
00:25:25 --> 00:25:31
you will see the tail
of the vector
409
00:25:27 --> 00:25:33
and you will see a cross,
little plus sign
410
00:25:30 --> 00:25:36
and therefore we put that
like so.
411
00:25:34 --> 00:25:40
A cross product is always
perpendicular to both A and B
412
00:25:40 --> 00:25:46
but it leaves you
with two choices:
413
00:25:41 --> 00:25:47
It can either come
out of the blackboard
414
00:25:43 --> 00:25:49
or it can go
in the blackboard
415
00:25:46 --> 00:25:52
and I just told you
which convention to use.
416
00:25:50 --> 00:25:56
And I want to show that to you
417
00:25:52 --> 00:25:58
in a way that may appeal
to you more.
418
00:25:56 --> 00:26:02
This is what I have used before
419
00:25:59 --> 00:26:05
on my television help sessions
420
00:26:05 --> 00:26:11
that I have given at MIT.
421
00:26:07 --> 00:26:13
I have an apple--
not an apple...
422
00:26:08 --> 00:26:14
This is a tomato--
not a tomato...
423
00:26:10 --> 00:26:16
It's a potato.
424
00:26:11 --> 00:26:17
(class laughs )
425
00:26:12 --> 00:26:18
I have a potato here
and here is a corkscrew.
426
00:26:15 --> 00:26:21
Here is a corkscrew.
427
00:26:17 --> 00:26:23
I'm going to turn the corkscrew
428
00:26:19 --> 00:26:25
as seen from your side,
clockwise.
429
00:26:22 --> 00:26:28
And you'll see that the
corkscrew goes into the potato
430
00:26:29 --> 00:26:35
in -- that's the direction,
then, of the vector.
431
00:26:33 --> 00:26:39
If we had B cross A,
then you take B in your hands
432
00:26:38 --> 00:26:44
and you rotate it over
the shortest angle to A.
433
00:26:40 --> 00:26:46
Now you have to rotate
counterclockwise
434
00:26:43 --> 00:26:49
and when you rotate
counterclockwise
435
00:26:45 --> 00:26:51
the corkscrew comes to you--
there you go--
436
00:26:48 --> 00:26:54
and so the vector is now
pointing in this direction.
437
00:26:51 --> 00:26:57
And if the vector
is pointing towards you
438
00:26:54 --> 00:27:00
then we would indicate that
with a circle and a dot.
439
00:26:58 --> 00:27:04
In other words, for this vector
440
00:27:00 --> 00:27:06
B cross A would have
exactly the same magnitude--
441
00:27:04 --> 00:27:10
no difference-- but it would
be coming out of the blackboard.
442
00:27:09 --> 00:27:15
In other words, A cross B
equals minus B cross A
443
00:27:21 --> 00:27:27
whereas A dot B
is the same as B dot A.
444
00:27:25 --> 00:27:31
We will encounter cross products
when we deal with torques
445
00:27:29 --> 00:27:35
and when we deal
with angular momentum
446
00:27:31 --> 00:27:37
which is not
the easiest part of 801.
447
00:27:35 --> 00:27:41
Let's take
an extremely simple example.
448
00:27:39 --> 00:27:45
Again, I don't mean to insult
you with such a simple example
449
00:27:44 --> 00:27:50
but you will get chances,
450
00:27:46 --> 00:27:52
more advanced chances
on your assignment.
451
00:27:49 --> 00:27:55
Suppose I gave
to vector A this x roof.
452
00:27:54 --> 00:28:00
It's a unit vector
in the x direction.
453
00:27:57 --> 00:28:03
That means A of x is one
454
00:28:00 --> 00:28:06
and A of y is zero
and A of z is zero.
455
00:28:04 --> 00:28:10
And suppose B is y roof.
456
00:28:09 --> 00:28:15
That means B of y is one
457
00:28:13 --> 00:28:19
and B of x is zero
and B of z is zero.
458
00:28:17 --> 00:28:23
What, now, is the dot product,
the cross product, A cross B?
459
00:28:23 --> 00:28:29
460
00:28:26 --> 00:28:32
Well, you can apply that recipe
461
00:28:31 --> 00:28:37
but it's much easier to go
462
00:28:34 --> 00:28:40
to the x, y, z axes
that we have here.
463
00:28:39 --> 00:28:45
A was in the x direction,
the unit vector
464
00:28:41 --> 00:28:47
and B in the y direction.
465
00:28:44 --> 00:28:50
I take A in my hand, I rotate
over the smallest angle
466
00:28:47 --> 00:28:53
which is 90 degrees to y,
and my corkscrew will go up.
467
00:28:51 --> 00:28:57
So I know
the whole thing already.
468
00:28:53 --> 00:28:59
I know that this cross product
must be z roof.
469
00:28:58 --> 00:29:04
The magnitude must be one.
470
00:29:00 --> 00:29:06
That's immediately clear.
471
00:29:01 --> 00:29:07
But I immediately have
the direction
472
00:29:04 --> 00:29:10
by using the corkscrew rule.
473
00:29:06 --> 00:29:12
Now if you're very smart
474
00:29:09 --> 00:29:15
you may say,
"Aha! You find plus z
475
00:29:14 --> 00:29:20
"only because you have used
this coordinate system.
476
00:29:16 --> 00:29:22
"If this axis had been x,
and this one had been y
477
00:29:22 --> 00:29:28
"then the cross product
of x and y would be
478
00:29:24 --> 00:29:30
in the minus z direction."
479
00:29:26 --> 00:29:32
Yeah, you're right.
480
00:29:28 --> 00:29:34
But if you ever do that,
I willkill you!
481
00:29:31 --> 00:29:37
(class laughs )
482
00:29:32 --> 00:29:38
You will always,
always have to work
483
00:29:34 --> 00:29:40
with what we call "a right-
handed coordinate system."
484
00:29:38 --> 00:29:44
And a right-handed coordinate
system, by definition
485
00:29:41 --> 00:29:47
is one whereby the cross product
486
00:29:44 --> 00:29:50
of x with y is z
and not y minus z.
487
00:29:48 --> 00:29:54
So whenever you get,
in the future, involved
488
00:29:50 --> 00:29:56
with cross products and torques
and angular momentum
489
00:29:52 --> 00:29:58
always make yourself
an xyz diagram
490
00:29:56 --> 00:30:02
for which x cross y is z.
491
00:29:59 --> 00:30:05
Never, ever make it such
that x cross y is minus z.
492
00:30:02 --> 00:30:08
You're going to hang yourself.
493
00:30:04 --> 00:30:10
For one thing,
that wouldn't work anymore.
494
00:30:08 --> 00:30:14
So be very, very careful.
495
00:30:10 --> 00:30:16
You must work... if you use
the right-hand corkscrew rule
496
00:30:12 --> 00:30:18
make sure you work with the
right-handed coordinate system.
497
00:30:19 --> 00:30:25
All right,
now the worst part is over.
498
00:30:24 --> 00:30:30
And now I would like
to write down for you...
499
00:30:30 --> 00:30:36
We pick up some
of the fruits now
500
00:30:33 --> 00:30:39
although it will
penetrate slowly.
501
00:30:36 --> 00:30:42
I want to write down for you
equations for a moving particle
502
00:30:42 --> 00:30:48
a moving object
in three-dimensional space--
503
00:30:48 --> 00:30:54
very complicated motion
504
00:30:49 --> 00:30:55
which I can hardly imagine
what it's like.
505
00:30:55 --> 00:31:01
It is a point that is going
to move around in space
506
00:30:59 --> 00:31:05
and it is this point P
507
00:31:02 --> 00:31:08
this point P is going
to move around in space
508
00:31:05 --> 00:31:11
and I call this vector OP,
I call that now vector r
509
00:31:11 --> 00:31:17
and I give it a sub-index t
510
00:31:13 --> 00:31:19
which indicates
it's changing with time.
511
00:31:16 --> 00:31:22
I call this location A of y,
I am going to call that y of t.
512
00:31:21 --> 00:31:27
It's changing with time.
513
00:31:23 --> 00:31:29
I call this x of t--
it's going to change with time--
514
00:31:26 --> 00:31:32
and I call this point z of t
515
00:31:29 --> 00:31:35
which is going
to change with time
516
00:31:31 --> 00:31:37
because point P
is going to move.
517
00:31:34 --> 00:31:40
And so I'm going to write down
the vector r
518
00:31:38 --> 00:31:44
in its most general form
that I can do that.
519
00:31:41 --> 00:31:47
R, which changes with time
520
00:31:43 --> 00:31:49
is now x of t-- which is the
same as a over x there, before--
521
00:31:49 --> 00:31:55
times x roof plus y of t,
522
00:31:55 --> 00:32:01
y roof plus z of t, z roof.
523
00:32:00 --> 00:32:06
I have decomposed my vector r
into three independent vectors.
524
00:32:06 --> 00:32:12
Each one of those change
with time.
525
00:32:10 --> 00:32:16
What is the velocity
of this particle?
526
00:32:12 --> 00:32:18
Well, the velocity is the first
derivative of the position
527
00:32:18 --> 00:32:24
so that it is dr dt.
528
00:32:22 --> 00:32:28
So there we go-- first
the derivative of this one
529
00:32:26 --> 00:32:32
which is dx dt, x roof.
530
00:32:30 --> 00:32:36
I am going to write for dx dt
"x dot," because I am lazy
531
00:32:36 --> 00:32:42
and I am going to write for
d2x dt squared, "x double dots."
532
00:32:41 --> 00:32:47
It's often done,
but not in your book.
533
00:32:43 --> 00:32:49
But it is a notation
that I will often use
534
00:32:45 --> 00:32:51
because otherwise the equations
look so clumsy.
535
00:32:48 --> 00:32:54
Plus y dot times y roof
plus z dot times z roof.
536
00:32:56 --> 00:33:02
So z dot is the dz/dt.
537
00:32:59 --> 00:33:05
What is the acceleration
as a function of time?
538
00:33:02 --> 00:33:08
Well, the acceleration
as a function of time
539
00:33:05 --> 00:33:11
equals dv/dt.
540
00:33:08 --> 00:33:14
So that's the section derivative
of x versus time
541
00:33:12 --> 00:33:18
and so that becomes
x double dot times x roof
542
00:33:17 --> 00:33:23
plus y double dot times y roof
plus z double dot times z roof.
543
00:33:26 --> 00:33:32
And look
what we have now accomplished.
544
00:33:30 --> 00:33:36
It looks like minor, but
it's going to be big later on.
545
00:33:33 --> 00:33:39
We have a point P going
in three-dimensional space
546
00:33:38 --> 00:33:44
and here we have the entire
behavior of the object
547
00:33:44 --> 00:33:50
as it moves its projection
along the x axis.
548
00:33:49 --> 00:33:55
This is the position,
this is its velocity
549
00:33:53 --> 00:33:59
and this is its acceleration.
550
00:33:55 --> 00:34:01
And here you can see the
entire behavior on the z axis.
551
00:34:01 --> 00:34:07
This is the position
on the z axis
552
00:34:03 --> 00:34:09
this is the velocity component
in the z direction
553
00:34:05 --> 00:34:11
and this is the acceleration
on the z axis.
554
00:34:08 --> 00:34:14
And here you have the y.
555
00:34:09 --> 00:34:15
In other words, we have now...
the three-dimensional motion
556
00:34:13 --> 00:34:19
we have cut into three
one-dimensional motions.
557
00:34:19 --> 00:34:25
This is
a one-dimensional motion.
558
00:34:21 --> 00:34:27
This is behavior
only along the x axis
559
00:34:24 --> 00:34:30
and this is a behavior
only along the y axis
560
00:34:26 --> 00:34:32
and this is a behavior
only along the z axis
561
00:34:29 --> 00:34:35
and the three together make up
562
00:34:33 --> 00:34:39
the actual motion
of that particle.
563
00:34:37 --> 00:34:43
What have we gained now?
564
00:34:38 --> 00:34:44
It looks like... this looks
like a mathematical zoo.
565
00:34:41 --> 00:34:47
You would say, "Well, if this
is what it is going to be like
566
00:34:44 --> 00:34:50
it's going to be hell."
567
00:34:45 --> 00:34:51
Well, not quite--
568
00:34:49 --> 00:34:55
in fact, it's going
to help you a great deal.
569
00:34:53 --> 00:34:59
First of all, if I throw up
a tennis ball in class
570
00:34:56 --> 00:35:02
like this, then
the whole trajectory is...
571
00:35:02 --> 00:35:08
the whole trajectory is
in one plane
572
00:35:04 --> 00:35:10
in the vertical plane.
573
00:35:06 --> 00:35:12
So even though it is
in three dimensions
574
00:35:08 --> 00:35:14
we can always represent it by
two axes, by two dimensionally
575
00:35:12 --> 00:35:18
a y axis and an x axis
576
00:35:14 --> 00:35:20
so already
the three-dimensional problem
577
00:35:16 --> 00:35:22
often becomes
a two-dimensional problem.
578
00:35:20 --> 00:35:26
We will, with great success,
analyze these trajectories
579
00:35:25 --> 00:35:31
by decomposing
this very complicated motion.
580
00:35:28 --> 00:35:34
Imagine what an incredibly
complicated arc that is
581
00:35:31 --> 00:35:37
and yet we are going
to decompose it
582
00:35:33 --> 00:35:39
into a motion
in the x direction
583
00:35:36 --> 00:35:42
which lives a life of its own
584
00:35:38 --> 00:35:44
independent of the motion
in the y direction
585
00:35:40 --> 00:35:46
which lives a life ofits own
586
00:35:42 --> 00:35:48
and, of course, you always have
to combine the two
587
00:35:44 --> 00:35:50
to know what the particle
is doing.
588
00:35:51 --> 00:35:57
We know the equations so well
from our last lecture
589
00:35:56 --> 00:36:02
from one-dimensional motion
with constant acceleration.
590
00:36:03 --> 00:36:09
The first line tells you
591
00:36:04 --> 00:36:10
what the x position is
as a function of time.
592
00:36:07 --> 00:36:13
The index t tells you
that it is changing with time.
593
00:36:11 --> 00:36:17
It is the position
at t equals zero
594
00:36:13 --> 00:36:19
plus the velocity
at t equals zero
595
00:36:16 --> 00:36:22
times t plus
one-half ax t squared
596
00:36:19 --> 00:36:25
if there is an acceleration
in the x direction.
597
00:36:21 --> 00:36:27
The velocity immediately comes
598
00:36:23 --> 00:36:29
from taking the derivative
of this function
599
00:36:25 --> 00:36:31
and the acceleration comes
600
00:36:26 --> 00:36:32
from taking the derivative
of this function.
601
00:36:29 --> 00:36:35
Now, if we have a motion
which is more complicated--
602
00:36:34 --> 00:36:40
which reaches out to two
or three dimensions--
603
00:36:37 --> 00:36:43
we can decompose the motion
in three perpendicular axes
604
00:36:41 --> 00:36:47
and you can replace
every x here by a y
605
00:36:44 --> 00:36:50
which gives you the entire
behavior in the y direction
606
00:36:48 --> 00:36:54
and if you want to know
the behavior in the z direction
607
00:36:50 --> 00:36:56
you replace every x here by z
608
00:36:53 --> 00:36:59
and then you have decomposed
the motion in three directions.
609
00:36:59 --> 00:37:05
Each of them are linear.
610
00:37:04 --> 00:37:10
And that's
what I want to do now.
611
00:37:06 --> 00:37:12
I'm going to throw up an object,
golf ball or an apple in 26.100
612
00:37:21 --> 00:37:27
and we know that it's in the
vertical plane, so we have...
613
00:37:24 --> 00:37:30
we only deal
with a two-dimensional problem
614
00:37:27 --> 00:37:33
this being...
615
00:37:29 --> 00:37:35
I call this my x axis and I'm
going to call this my y axis.
616
00:37:35 --> 00:37:41
I call this
increasing value of x
617
00:37:38 --> 00:37:44
and I call this
increasing value of y.
618
00:37:42 --> 00:37:48
I could have calledthis
increasing value of y.
619
00:37:45 --> 00:37:51
Today I have decided to call
this increasing value of y.
620
00:37:49 --> 00:37:55
I am free in that choice.
621
00:37:52 --> 00:37:58
I throw up an object
at a certain angle
622
00:37:56 --> 00:38:02
and I see a motion
like this-- boing!--
623
00:37:59 --> 00:38:05
and it comes back to the ground.
624
00:38:04 --> 00:38:10
My initial speed
when I threw it was v zero
625
00:38:12 --> 00:38:18
and the angle here is alpha.
626
00:38:16 --> 00:38:22
The x component
of that initial velocity
627
00:38:22 --> 00:38:28
is v zero cosine alpha
628
00:38:26 --> 00:38:32
and the y component
equals v zero sine alpha.
629
00:38:33 --> 00:38:39
So that's the "begins" velocity
of the x direction.
630
00:38:37 --> 00:38:43
This is the "begins" velocity
in the y direction.
631
00:38:41 --> 00:38:47
A little later in time,
that object is here at point P
632
00:38:51 --> 00:38:57
and this is now
the position vector
633
00:38:54 --> 00:39:00
which we have called r of t,
it's this vector.
634
00:39:02 --> 00:39:08
That's the vector
that is moving through space.
635
00:39:06 --> 00:39:12
At this moment in time,
x of t is here
636
00:39:12 --> 00:39:18
and at this moment in time,
y of t is here.
637
00:39:22 --> 00:39:28
And now you're going to see,
for the first time
638
00:39:26 --> 00:39:32
the big gain by the way that we
have divided the two axes
639
00:39:33 --> 00:39:39
which live an independent life.
640
00:39:35 --> 00:39:41
First x.
641
00:39:37 --> 00:39:43
I want to know everything about
x that there has to be known.
642
00:39:40 --> 00:39:46
I want to know where it is
at any moment in time
643
00:39:44 --> 00:39:50
velocity and the acceleration,
only in x.
644
00:39:48 --> 00:39:54
First I want to know
that at t = 0.
645
00:39:53 --> 00:39:59
Well, at t = 0, I look there
646
00:39:56 --> 00:40:02
X zero-- that's the,
I can choose that to be zero.
647
00:40:00 --> 00:40:06
So I can say x zero is zero,
that's my free choice.
648
00:40:04 --> 00:40:10
Now I need v zero x--
what is the velocity?
649
00:40:08 --> 00:40:14
The velocity at t = 0,
which we have called v zero x
650
00:40:13 --> 00:40:19
is this velocity--
v zero cosine alpha.
651
00:40:17 --> 00:40:23
And it's not going to change.
652
00:40:20 --> 00:40:26
Why is it not going to change?
653
00:40:22 --> 00:40:28
Because there is no a of x,
so this term here is zero
654
00:40:29 --> 00:40:35
we only have this one.
655
00:40:30 --> 00:40:36
So at all moments in time
656
00:40:32 --> 00:40:38
the velocity in the x direction
is v zero cosine alpha
657
00:40:36 --> 00:40:42
and the a of x equals zero.
658
00:40:40 --> 00:40:46
659
00:40:43 --> 00:40:49
Now I want to do the same
in the x direction for time t.
660
00:40:50 --> 00:40:56
Well, at time t, I look there
at the first equation.
661
00:40:55 --> 00:41:01
There it is-- x zero is zero.
662
00:40:58 --> 00:41:04
I know v zero x,
that is v zero cosine alpha
663
00:41:03 --> 00:41:09
so x of t is
v zero cosine alpha times t
664
00:41:09 --> 00:41:15
but there is no acceleration,
so that's it.
665
00:41:13 --> 00:41:19
What is vx of t?
666
00:41:16 --> 00:41:22
The velocity in the x direction
at any moment in time.
667
00:41:20 --> 00:41:26
That is that equation,
that is simply v zero x.
668
00:41:24 --> 00:41:30
It is not changing in time
669
00:41:26 --> 00:41:32
because there is
no acceleration.
670
00:41:29 --> 00:41:35
So the initial velocity
at t zero is the same
671
00:41:32 --> 00:41:38
as t seconds later
and the acceleration is zero.
672
00:41:36 --> 00:41:42
Now we're going to do this
for the y direction.
673
00:41:42 --> 00:41:48
And now you begin to see
the gain for the decomposition.
674
00:41:46 --> 00:41:52
In the y direction,
we change the x by y
675
00:41:51 --> 00:41:57
and so we do it first
at t = 0.
676
00:41:55 --> 00:42:01
So look there.
677
00:41:57 --> 00:42:03
This becomes y zero--
I call that zero.
678
00:42:00 --> 00:42:06
I can always call
my origin zero.
679
00:42:03 --> 00:42:09
I get v zero y times t.
680
00:42:06 --> 00:42:12
Well, v zero y is this quantity
681
00:42:09 --> 00:42:15
is v zero sine alpha,
v zero sine alpha.
682
00:42:19 --> 00:42:25
This is v zero sine alpha.
683
00:42:20 --> 00:42:26
That is the velocity at
time zero, and this is zero.
684
00:42:25 --> 00:42:31
At time zero...
this is zero at time zero.
685
00:42:30 --> 00:42:36
What is the acceleration
in the y direction at time zero?
686
00:42:37 --> 00:42:43
What is the acceleration?
That has to do with gravity.
687
00:42:40 --> 00:42:46
There is no acceleration
in the x direction
688
00:42:43 --> 00:42:49
but you better believe that
there is one in the y direction.
689
00:42:45 --> 00:42:51
So only when we deal
with the y equations
690
00:42:48 --> 00:42:54
does this acceleration come in--
691
00:42:50 --> 00:42:56
not at all when we deal
with the x direction.
692
00:42:53 --> 00:42:59
Well, if we call the
acceleration due to gravity
693
00:42:55 --> 00:43:01
g equals plus 9.80,
and I always call it g
694
00:43:02 --> 00:43:08
what would be the acceleration
in the y direction
695
00:43:04 --> 00:43:10
given the fact that I call this
increasing value of y?
696
00:43:08 --> 00:43:14
CLASS:
Minus 9.8.
697
00:43:12 --> 00:43:18
LEWIN:
Minus 9.8,
which I will also say
698
00:43:15 --> 00:43:21
always call minus g
because my g is always positive.
699
00:43:18 --> 00:43:24
So it is minus g.
700
00:43:22 --> 00:43:28
So that tells the story of t
equals zero in the y direction
701
00:43:26 --> 00:43:32
and now we have to complete it
at time t equals t.
702
00:43:31 --> 00:43:37
At time t equals t,
we have the first line there.
703
00:43:36 --> 00:43:42
Y zero is zero.
704
00:43:38 --> 00:43:44
So we have y as a function
of time, y zero is zero
705
00:43:42 --> 00:43:48
so we don't have to work
with that.
706
00:43:45 --> 00:43:51
Where is my... so this is zero,
so I get v zero y times t
707
00:43:50 --> 00:43:56
so I get
v zero sine alpha times t
708
00:43:58 --> 00:44:04
plus one-half, but it
is minus one-half g t squared
709
00:44:05 --> 00:44:11
and now I get the velocity
in the y direction at time t--
710
00:44:09 --> 00:44:15
that is my second line.
711
00:44:11 --> 00:44:17
That is going to be
v zero sine alpha minus g t
712
00:44:19 --> 00:44:25
and the acceleration in the y
direction at any moment in time
713
00:44:23 --> 00:44:29
equals minus g.
714
00:44:25 --> 00:44:31
And now I have done all I can
715
00:44:27 --> 00:44:33
to completely decompose
this complicated motion
716
00:44:31 --> 00:44:37
into two entirely independent
one-dimensional motions.
717
00:44:36 --> 00:44:42
And the next lecture
718
00:44:38 --> 00:44:44
we're going to use this again
and again and again and again.
719
00:44:41 --> 00:44:47
This lecture is not over yet
but I want you to know
720
00:44:44 --> 00:44:50
that this is
what we're going to apply
721
00:44:46 --> 00:44:52
for many lectures to come--
722
00:44:47 --> 00:44:53
the decomposition
of a complicated trajectory
723
00:44:51 --> 00:44:57
into two simple ones.
724
00:44:55 --> 00:45:01
Now, when you look at this
725
00:44:57 --> 00:45:03
there is something
quite remarkable
726
00:44:59 --> 00:45:05
and the remarkable thing
727
00:45:01 --> 00:45:07
is that the velocity
in the x direction
728
00:45:03 --> 00:45:09
throughout
this whole trajectory--
729
00:45:05 --> 00:45:11
if there is no air draft,
if there is no friction--
730
00:45:07 --> 00:45:13
is not changing.
731
00:45:09 --> 00:45:15
It's only the velocity in the
y direction that is changing.
732
00:45:13 --> 00:45:19
It means
if I throw up this golf ball--
733
00:45:15 --> 00:45:21
I throw it up like this--
734
00:45:17 --> 00:45:23
and it has a certain component
in x direction
735
00:45:19 --> 00:45:25
a certain velocity
736
00:45:21 --> 00:45:27
if I move myself with
exactly that same velocity--
737
00:45:24 --> 00:45:30
with exactly the same
horizontal velocity--
738
00:45:27 --> 00:45:33
I could catch the ball here.
739
00:45:28 --> 00:45:34
It would have to come back
exactly in my hands.
740
00:45:31 --> 00:45:37
That is because there is only an
acceleration in the y direction
741
00:45:36 --> 00:45:42
but the motion
in the y direction
742
00:45:38 --> 00:45:44
is completely independent
of the x direction.
743
00:45:41 --> 00:45:47
The x direction
doesn't even know
744
00:45:43 --> 00:45:49
what's going on
with the y direction.
745
00:45:45 --> 00:45:51
In the x direction,
if I throw an object like this
746
00:45:48 --> 00:45:54
the x direction simply,
very boringly,
747
00:45:51 --> 00:45:57
moves with a constant velocity.
748
00:45:54 --> 00:46:00
There is no time dependence.
749
00:45:57 --> 00:46:03
And the y direction,
on its own, does its own thing.
750
00:46:00 --> 00:46:06
It goes up, comes to a halt
and it stops.
751
00:46:04 --> 00:46:10
And, of course, the
actual motion is the sum
752
00:46:07 --> 00:46:13
the superposition of the two.
753
00:46:11 --> 00:46:17
We have tried to find a way
754
00:46:13 --> 00:46:19
to demonstrate
this quite bizarre behavior
755
00:46:18 --> 00:46:24
which is not so intuitive.
756
00:46:20 --> 00:46:26
That the x direction
really lives a life of its own.
757
00:46:24 --> 00:46:30
And the way we want
to do that is as follows.
758
00:46:31 --> 00:46:37
We have here a golf ball
759
00:46:34 --> 00:46:40
a gun we can shoot
up the golf ball
760
00:46:39 --> 00:46:45
and we do that in such a way
761
00:46:41 --> 00:46:47
that the golf ball,
if we do it correctly
762
00:46:45 --> 00:46:51
exactly comes back here.
763
00:46:48 --> 00:46:54
That's not easy-- that takes
hours and hours of adjustments.
764
00:46:52 --> 00:46:58
The golf ball goes up
and comes back here.
765
00:46:57 --> 00:47:03
Not here, not here,
not there-- that's easy.
766
00:47:00 --> 00:47:06
You can shoot it up
a little at an angle
767
00:47:02 --> 00:47:08
and the golf ball
will come back here.
768
00:47:05 --> 00:47:11
Once we have achieved that--
769
00:47:07 --> 00:47:13
that the golf ball
will come back there--
770
00:47:09 --> 00:47:15
then I'm going to give
this cart a push
771
00:47:13 --> 00:47:19
and the moment that it passes
through this switch
772
00:47:17 --> 00:47:23
the golf ball will fire
773
00:47:19 --> 00:47:25
so that the golf ball
will go straight up
774
00:47:22 --> 00:47:28
as seen from the cart
775
00:47:24 --> 00:47:30
but it has a horizontal velocity
776
00:47:26 --> 00:47:32
which is exactly the same
horizontal velocity as the cart
777
00:47:29 --> 00:47:35
so the cart are like my hands.
778
00:47:31 --> 00:47:37
As the golf ball goes like this
779
00:47:33 --> 00:47:39
the cart stays always exactly
under the golf ball
780
00:47:37 --> 00:47:43
always exactly
under the golf ball
781
00:47:39 --> 00:47:45
and if all works well
782
00:47:41 --> 00:47:47
the ball ends up exactly
on the cart again.
783
00:47:46 --> 00:47:52
Let me first show you--
784
00:47:47 --> 00:47:53
otherwise, if that doesn't work,
of course, it's all over--
785
00:47:51 --> 00:47:57
that if we shoot the ball
straight up
786
00:47:53 --> 00:47:59
that it comes back here.
787
00:47:54 --> 00:48:00
If it doesn't do that
788
00:47:56 --> 00:48:02
I don't even have to try this
more complicated experiment.
789
00:48:01 --> 00:48:07
So here's the golf ball.
790
00:48:03 --> 00:48:09
I'm going to fire the gun now.
791
00:48:04 --> 00:48:10
792
00:48:07 --> 00:48:13
Close... close.
793
00:48:12 --> 00:48:18
Reasonably close.
794
00:48:15 --> 00:48:21
Well, since it's only
reasonably close, perhaps...
795
00:48:19 --> 00:48:25
(class laughs )
796
00:48:24 --> 00:48:30
Perhaps it would help if we
give it a little bit of leeway.
797
00:48:28 --> 00:48:34
There goes the gun.
798
00:48:33 --> 00:48:39
Here comes the ball.
799
00:48:38 --> 00:48:44
And this is just in case.
800
00:48:46 --> 00:48:52
Tape it down.
801
00:48:48 --> 00:48:54
So as I'm going to push
this now, give it a push
802
00:48:54 --> 00:49:00
the gun will be triggered
803
00:48:57 --> 00:49:03
when the middle
of the car is here.
804
00:49:00 --> 00:49:06
You've seen how high
that ball goes
805
00:49:01 --> 00:49:07
so that ball will go...
(makes whooshing sound )
806
00:49:05 --> 00:49:11
And depending upon
how hard I push it
807
00:49:07 --> 00:49:13
they may meet here
or they may meet there.
808
00:49:14 --> 00:49:20
You ready for this?
809
00:49:16 --> 00:49:22
You ready?
810
00:49:17 --> 00:49:23
CLASS:
Ready.
811
00:49:18 --> 00:49:24
LEWIN:
I'm ready.
812
00:49:19 --> 00:49:25
813
00:49:23 --> 00:49:29
Physics works.
814
00:49:24 --> 00:49:30
(class applauds )
815
00:49:28 --> 00:49:34
LEWIN:
See you Wednesday.
816
00:49:29 --> 00:49:35.000