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We will discuss velocities
and acceleration.

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I'll start
with something simple.

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I have a motion of an object
along a straight line--

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we'll call that
one-dimensional motion.

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And I'll tell you that
the object is here at time t1.

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At time t2, it's here.

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At time t3, it's there.

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At time t4, it's here

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and at time t5, it's back
where it was at t1.

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And here you see
the positions in x
 

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where it is located
at that moment in time.

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I will define this to be
the increasing value of x.

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It's my free choice,
but I've chosen this now.

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Now we will introduce what
we call the average velocity.

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I put a bar over it.

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That stands for average
between time t1 and time t2.

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That we define in physics

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as x at time t2
minus x at time t1

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divided by t2 minus t1.

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That is our definition.

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In our case, because
of the way that I define

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the increasing value of x,
this is larger than 0.

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However, if I take the average
velocity between t1 and t5

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that would be 0, because
they are at the same position

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so the upstairs is 0.

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If I had chosen t4 and t2--

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average velocity
between time t2 and t4--

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you would have seen
that that is negative

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because the upstairs
is negative.

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Notice that I haven't told you

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where I choose my zero
on my x axis.

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It's completely unimportant
for the average velocity.

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It makes no difference.

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However, if I had chosen this

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to be the direction
of increasing x

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then, of course,
the signs would flip.

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Then this would
have been negative

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and this would
have been positive.

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So the direction,
that you are free to choose

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determines the signs.

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The location where you put
your zero is not important

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but signs in physics do matter.

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Signs are important.

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Whether you owe me money
or I owe you money

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the difference
is only a minus sign

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but I think it's important
for you.

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Now I will give you
not only the positions--

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as I did here on the x axis
at discrete moments in time--

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but I'm going to tell you

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exactly where the object is
at any moment in time.

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Here you see an xt diagram

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so you see that at t1,
the object is at position xt1.

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This is the road of the object.

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This is the straight line,
where it's moving.

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It starts here and
it goes to this position.

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It goes to this one,
it comes back to t4

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and it comes back here.

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I will tell you now
every moment in time in between.

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And there it goes.

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Voilà.

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This is now information
that is way more.

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You have the information
at any moment in time.

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Notice that I now did choose
x = 0.

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I chose it somewhere here

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but I could have chosen it
at any other point--

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for whatever follows

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you will see that it makes
no difference--

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so I have chosen a zero point
so that I can make a graph.

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And now we will look
at the average velocity

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in a somewhat different way.

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Say I choose my time t2 and t3.

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I draw here now this line.

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And this angle I call alpha

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and this part here
I call delta x

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and this here is delta t.

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And so you could right now--

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if you're careful
about your sign convention--

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you could write down now
that the average velocity

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equals delta x
divided by delta t.

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But be careful.

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If the angle is positive--
I call this a positive angle--

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then the average velocity
is positive

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but if I have a negative angle

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then the average velocity
would be negative.

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For instance, between t4 and t5,
if I draw this line

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then this angle here is negative

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and so the average velocity
between t4 and t5

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is now negative.

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Again, if I had changed
the zero points

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you would have found
the same values

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for the average velocity.

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The only difference
would have been

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the position of the curve
in that plot.

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There is a very
big difference in physics

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between speed and velocity.

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The average velocity
between time t1 and t5 is zero

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but the average speed is not.

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The average speed is defined
as the distance traveled

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divided by the time that it
takes to travel that distance.

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Now, what is the distance
that the object traveled

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between time t1 and time t5?

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Well, the object started at
a position here on this x axis

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and then it went up,
reached the highest point here

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so I'll make a drawing
for you here.

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It reached the highest point
here, then it went down.

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And then when it went here

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it went up again and comes
down again and it's back.

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And in order to find
the average speed

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you would now have to know
exactly what this distance is

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add up this distance

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add up this distance
and this distance.

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And if that distance altogether
were, for instance, 300 meters

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and if the time between t1
and t5 were three seconds

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then the average speed

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would be 300 meters divided
by three seconds.

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That would be
100 meters per second

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so the average speed
would be 100 meters per second

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yet the average velocity
would be zero.

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If you look
at the location t3 and t2

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and I bring t3 closer
and closer to t2

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then this angle
of alpha will increase

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and I can go to the extreme

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that I bring t3
almost right at t2.

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The angle of alpha will then
be tangential to this point.

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This will then be
my angle of alpha.

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And now you will understand
how we define

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the instantaneous velocity
at time t

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which is different
from an average velocity

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between two time intervals.

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The instantaneous velocity, v--
and I pick a random time, t--

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equals the limiting case

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for x measured at time t
plus delta t

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minus x measured at time t
divided by delta t

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and I do that for delta t--
goes to zero.

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So think of this
as being t3 and this as t2.

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I bring t3 closer and closer
and closer to t2

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and the time between them
then goes to zero.

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And this is something
that you undoubtedly recognize.

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That's the first derivative
of the position versus time.

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And now comes an equation

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which is one of the very few

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that I want you to remember
in x... in 801:

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v equals dx/dt.

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This is one that you must
remember, not only in 801

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but for the rest
of your time at MIT.

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And this could be larger than 0,
this could be 0

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and this could be
smaller than 0.

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If the angle of alpha,
the tangential, is positive

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then it is a positive value.

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If it is negative, however,
when you're here

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then it is a negative velocity.

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And if the angle
of alpha is zero

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then the velocity is zero.

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So if we now look at this plot

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we can search for the times
that the velocity is zero

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so you have to look
for the derivative being zero.

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That means
the angle alpha being zero.

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Clearly, here the velocity
is zero.

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Right here,
at this turning point--

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that means when the object
is here-- it is zero.

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When the object is here

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it is again zero
at this moment in time.

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Again, the angle is zero,
and it is again zero here.

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So those are the times
that the velocity is zero.

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What are the times
that the velocity is positive?

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Well, it's positive here.

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The velocity's positive here

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still positive, positive,
becomes negative

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negative, positive,
zero, negative.

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So that's the definition
of v, instantaneous velocity.

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What is the instantaneous speed?

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Well, speed is
not sign-sensitive.

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Suppose that the velocity here--
just... I call that v1--

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suppose that was
plus 30 meters per second.

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I just grabbed this number
out of the blue.

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And suppose here, somewhere,
it was... I call that v2--

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suppose that was
minus 100 meters per second.

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This is negative
and this is positive.

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Then we would have
to say, in physics--

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whether you like it or not,
it's not very pleasing--

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but you would have to say

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that this velocity
is lower than that one

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because minus 100
is lower than plus 30.

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But the speed, of course,
is higher here

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because the speed is
the magnitude of the velocity

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and is not sign-sensitive.

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So this has the highest speed,
of 100 meters per second

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and this has a lower speed

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but this has
the lowest velocity.

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It's just an algebraic game

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but very important
when you make your calculations.

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I have always wondered
what the average speed

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or the average velocity
is of a bullet.

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Now I want you to realize
I am not a fan of guns at all

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but it always intrigued me.

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How can I measure the average
speed of a bullet--

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and I have discussed it
with some people here--

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and we came up
with an easy way to do that.

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We have a wire, which goes
into the blackboards, wire I

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and we have
another wire that goes

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into the blackboards, wire II,
and the separation is D meters.

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We have to measure that.

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The set-up is here

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so this is wire number I
and this is wire number II.

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So you will see D
coming in like this,
 

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so I'll make this a I
and I'll make this a II.

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That's the way it's set up.

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And we fire the bullet,
which breaks this wire.

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At that moment, the timer starts

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and then it breaks this wire
and that's when the timer stops.

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Now, I told you
a measurement is meaningless

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without knowledge of the
uncertainty in your measurement.

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So there are
two uncertainties involved--

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the distance
and the timing uncertainty.

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This distance
I will measure for you, D.

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I have here a large ruler.

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Here's one wire,
here's the other wire.

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I cannot do that
any better, really

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than maybe even half
a centimeter

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because the situation
is not all that stable--

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I don't know what happens when
the bullet will hit the wire--

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so I would say
it is 148½ centimeters

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but I cannot guarantee it to
better than half a centimeter--

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148½ plus or minus
0.5 centimeters.

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I want you to appreciate

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that this is a very small
percentage error.

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This is only five parts
out of 1,500.

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That is one out of 300,

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so that is only
a one-third percent error.

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That's very small-- that's
what we call the relative error.

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Then I ask myself the question--

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I want to measure the accuracy
of the speed of the bullet

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to about two percent.

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That was my goal.

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How accurate
should I do the timing?

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Well, I had to make
an estimate very roughly

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how fast the speed
of the bullet is

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and I would think it is probably
lower than the speed of sound.

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The speed of sound
is 340 meters per second.

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I don't know
whether it's 200 or 300

253
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but it's got to be somewhere
in that ballpark

254
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of the kind of bullets
that we have--

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200 or 300 meters per second.

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Let us assume that the speed
is 300 meters per second--

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just a wild guess.

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Then it would take
5 milliseconds

259
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for this bullet to cross
from here to here.

260
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And if I want
to make a measurement

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to two percent accuracy

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I have to know this timing

263
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to about one-tenth
of a millisecond

264
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because one-tenth
of a millisecond

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is about two percent of five.

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So that sets the accuracy

267
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that I need to make
the time measurements.

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And so we do have a timer.

269
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It is about accurate to
about a tenth of a millisecond

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and so now I can measure
that time.

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So I am going to have here

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some time that we measure
plus or minus 0.1

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and we'll do the whole thing
in milliseconds.

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But our final answer will be
in meters per second.

276
00:15:05 --> 00:15:11

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All right, I always have
to think hard when I do this

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because when we deal with
bullets, that is no kid stuff

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and I... as I said

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I have really no experience
firing guns.

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This is the bolt.

282
00:15:24 --> 00:15:30

283
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There we go.

284
00:15:29 --> 00:15:35

285
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Here's the bolt.

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287
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There we go.

288
00:15:37 --> 00:15:43

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It's in place.

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Before I do that, I want
to check... check the circuits.

291
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I want to make sure
that the electronic circuit

292
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is properly working.

293
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You see the timing here, right?

294
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So I do a small test

295
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just to see whether
the circuit is working.

296
00:15:57 --> 00:16:03

297
00:15:59 --> 00:16:05
Yep, should be working.

298
00:16:01 --> 00:16:07
Here comes the bullet.

299
00:16:04 --> 00:16:10

300
00:16:11 --> 00:16:17
You ready?

301
00:16:12 --> 00:16:18
I'm ready.

302
00:16:13 --> 00:16:19
Three, two, one, zero.

303
00:16:15 --> 00:16:21
(bullet whacks metal )

304
00:16:17 --> 00:16:23
What do we see?

305
00:16:18 --> 00:16:24

306
00:16:20 --> 00:16:26
5.8 milliseconds.

307
00:16:23 --> 00:16:29

308
00:16:25 --> 00:16:31
5.8.

309
00:16:26 --> 00:16:32
Is that what you see?

310
00:16:28 --> 00:16:34
Yeah?

311
00:16:30 --> 00:16:36
5.8 milliseconds.

312
00:16:32 --> 00:16:38

313
00:16:34 --> 00:16:40
5.8 plus or minus 0.1.

314
00:16:37 --> 00:16:43
So out comes the average.

315
00:16:39 --> 00:16:45
Call it speed
or call it velocity

316
00:16:41 --> 00:16:47
it's the same thing
in this case.

317
00:16:44 --> 00:16:50
148.5, 5.8, and I have to
convert it to meters per second.

318
00:16:52 --> 00:16:58
That brings it at 256,
plus or minus.

319
00:16:59 --> 00:17:05
Now you come in here,
with your plus or minuses.

320
00:17:02 --> 00:17:08
This is a one point...
one-third percent error.

321
00:17:05 --> 00:17:11
It's negligible to this one.

322
00:17:07 --> 00:17:13
One out of 58 is about 1.7%

323
00:17:10 --> 00:17:16
so this is the only one
we have to worry about

324
00:17:13 --> 00:17:19
so the uncertainty in there
is about 1.7%.

325
00:17:16 --> 00:17:22
It's less than two--

326
00:17:17 --> 00:17:23
that's what I wanted
and it gives me an error

327
00:17:20 --> 00:17:26
of about four meters per second.

328
00:17:22 --> 00:17:28
And so this is the result.

329
00:17:24 --> 00:17:30
And you see,
it's only meaningful

330
00:17:27 --> 00:17:33
because we have a good idea
about the uncertainties

331
00:17:31 --> 00:17:37
in the measurement.

332
00:17:33 --> 00:17:39

333
00:17:35 --> 00:17:41
Just as we introduced
average velocity

334
00:17:40 --> 00:17:46
now I am going to introduce
average acceleration.

335
00:17:45 --> 00:17:51
Notice that the velocity changes
here throughout time.

336
00:17:52 --> 00:17:58
And that brings me
to the next part

337
00:17:56 --> 00:18:02
the logical part, namely,
that we are going to introduce

338
00:18:02 --> 00:18:08
an average acceleration
 

339
00:18:05 --> 00:18:11
and with a little bit
of imagination

340
00:18:08 --> 00:18:14
you can probably guess
what that looks like.

341
00:18:11 --> 00:18:17
The average acceleration
between time t1 and time t2

342
00:18:16 --> 00:18:22
would then be the velocity
at time t2

343
00:18:20 --> 00:18:26
minus the velocity at time t1,
divided by t2 minus t1.

344
00:18:26 --> 00:18:32
And the dimension is lengths
per seconds per time squared

345
00:18:30 --> 00:18:36
so it's meters
per second squared.

346
00:18:32 --> 00:18:38
This is done for
a one-dimensional situation.

347
00:18:35 --> 00:18:41
This number can be larger than
zero, it can be equal to zero

348
00:18:39 --> 00:18:45
and it can be smaller than zero.

349
00:18:41 --> 00:18:47
In our case, t1 to t2 here

350
00:18:45 --> 00:18:51
notice the velocity is zero
as a start.

351
00:18:51 --> 00:18:57
And it begins to increase

352
00:18:52 --> 00:18:58
because this angle of alpha
increases.

353
00:18:54 --> 00:19:00
It's the angle that matters.

354
00:18:56 --> 00:19:02
The angle increases,
so in our case from t1 to t2

355
00:19:00 --> 00:19:06
the average acceleration
is larger than zero.

356
00:19:04 --> 00:19:10
Look at the angle.

357
00:19:05 --> 00:19:11
However, if you take the average
acceleration between t1 and t5

358
00:19:12 --> 00:19:18
that is smaller than zero

359
00:19:15 --> 00:19:21
because here
the velocity is zero

360
00:19:19 --> 00:19:25
but here
the velocity is negative.

361
00:19:23 --> 00:19:29
So if you substitute
that in there

362
00:19:25 --> 00:19:31
you get an average acceleration
which is smaller than zero.

363
00:19:29 --> 00:19:35
So the signs in the velocity

364
00:19:31 --> 00:19:37
and the signs in average
acceleration depend crucially

365
00:19:34 --> 00:19:40
on how I have defined
my increasing value of x

366
00:19:37 --> 00:19:43
not where I choose
my zero points.

367
00:19:40 --> 00:19:46
If I reverse the direction
of increasing x

368
00:19:44 --> 00:19:50
then all my signs will change.

369
00:19:46 --> 00:19:52
So you can also write down then

370
00:19:48 --> 00:19:54
that average acceleration,
if you like that

371
00:19:51 --> 00:19:57
is delta v divided by delta t

372
00:19:53 --> 00:19:59
but you must be careful because
the delta t is sign-sensitive.

373
00:19:57 --> 00:20:03
You must obey
your sign convention.

374
00:20:00 --> 00:20:06

375
00:20:02 --> 00:20:08
I have here a tennis ball

376
00:20:04 --> 00:20:10
and I can bounce this tennis
ball, I can throw it down.

377
00:20:09 --> 00:20:15
And let us assume,
just for simplicity

378
00:20:12 --> 00:20:18
that it hits the floor
at about five meters per second

379
00:20:16 --> 00:20:22
and that it's a very,
very good tennis ball

380
00:20:20 --> 00:20:26
and that it also bounces back

381
00:20:22 --> 00:20:28
with a velocity of
about five meters per second.

382
00:20:26 --> 00:20:32
I will choose this to be
my increasing value of x

383
00:20:31 --> 00:20:37
and so it hits the floor
like this.

384
00:20:35 --> 00:20:41
That means the velocity
at which it hits the floor

385
00:20:39 --> 00:20:45
is minus five meters per second.

386
00:20:42 --> 00:20:48
It bounces off, there it comes

387
00:20:45 --> 00:20:51
and it goes up with
plus five meters per second.

388
00:20:50 --> 00:20:56
I call this v1
and I call this v2.

389
00:20:53 --> 00:20:59
So what, now, is
the average acceleration?

390
00:20:57 --> 00:21:03
Well, I would have to know
the time that it takes

391
00:21:02 --> 00:21:08
for this change in direction.

392
00:21:05 --> 00:21:11
In other words, we call that
the impact time.

393
00:21:08 --> 00:21:14
I would say, in this case,
the impact time delta t

394
00:21:11 --> 00:21:17
is probably about
a hundredth of a second

395
00:21:14 --> 00:21:20
and so my average acceleration
would be v2 minus v1--

396
00:21:19 --> 00:21:25
that is
plus five minus minus five--

397
00:21:22 --> 00:21:28
that is ten divided
by ten to the minus two

398
00:21:26 --> 00:21:32
and that is plus 1,000 meters
per second squared.

399
00:21:31 --> 00:21:37
I have observed carefully
the signs.

400
00:21:34 --> 00:21:40
If now I say,
"Aha, I don't like this

401
00:21:37 --> 00:21:43
I want to go this-- the value
of increasing x."

402
00:21:41 --> 00:21:47
No big deal.

403
00:21:42 --> 00:21:48
This will become a plus,
this will become a minus

404
00:21:46 --> 00:21:52
and then this would
become a minus.

405
00:21:48 --> 00:21:54
So then the acceleration

406
00:21:49 --> 00:21:55
is minus 1,000 meters
per second squared.

407
00:21:52 --> 00:21:58

408
00:21:57 --> 00:22:03
I have also here a tomato
and I have here some eggs.

409
00:22:04 --> 00:22:10

410
00:22:05 --> 00:22:11
Now, imagine now
that I throw the tomato down

411
00:22:09 --> 00:22:15
or, for that matter, the egg

412
00:22:12 --> 00:22:18
and that they hit the floor
at five meters per second.

413
00:22:16 --> 00:22:22
I could do that.

414
00:22:18 --> 00:22:24
They would not come back up.

415
00:22:21 --> 00:22:27
They would go...
(blows raspberry )

416
00:22:24 --> 00:22:30
So therefore the change
in velocity would not be ten--

417
00:22:27 --> 00:22:33
apart from the sign
that you have to think about--

418
00:22:30 --> 00:22:36
but it would only be
five meters per second.

419
00:22:34 --> 00:22:40
The impact time
would probably be much longer

420
00:22:39 --> 00:22:45
maybe a quarter of a second.

421
00:22:41 --> 00:22:47
So therefore the average
acceleration during the impact

422
00:22:46 --> 00:22:52
would then be only five
divided by one quarter...

423
00:22:50 --> 00:22:56
would be something like
20 meters per second squared.

424
00:22:53 --> 00:22:59
Now, whether you call it plus

425
00:22:55 --> 00:23:01
or whether you call it minus
20 meters per second squared

426
00:22:59 --> 00:23:05
depends on your convention
of what you call increasing x.

427
00:23:03 --> 00:23:09
But the eggs and the tomatoes
don't care

428
00:23:05 --> 00:23:11
what you call minus
and what you call plus.

429
00:23:08 --> 00:23:14
Whether the acceleration is

430
00:23:09 --> 00:23:15
minus 20 meters
per second squared

431
00:23:11 --> 00:23:17
or plus 20 meters
per second squared

432
00:23:14 --> 00:23:20
you'd better believe it,
the egg will break.

433
00:23:16 --> 00:23:22
So it's only in your convention
that it matters

434
00:23:19 --> 00:23:25
but, of course,
the physics will not change.

435
00:23:22 --> 00:23:28
The eggs couldn't care less

436
00:23:24 --> 00:23:30
what you have chosen
for your sign convention.

437
00:23:28 --> 00:23:34
Something breaks

438
00:23:29 --> 00:23:35
because the magnitude of
acceleration becomes too high.

439
00:23:33 --> 00:23:39
That's why something breaks.

440
00:23:36 --> 00:23:42
A few days ago, I saw
a Sherlock Holmes movie

441
00:23:40 --> 00:23:46
and there was a guy
who fell on the floor--

442
00:23:44 --> 00:23:50
marble floor-- hit his head,
was lying there motionless.

443
00:23:49 --> 00:23:55
And here was Watson, and
Watson said to Sherlock Holmes

444
00:23:54 --> 00:24:00
"What happened?"

445
00:23:56 --> 00:24:02
Sherlock Holmes
walks over to the guy

446
00:23:59 --> 00:24:05
touches him and he says,
"He crushed his skull."

447
00:24:03 --> 00:24:09
He looked very intelligent,
I must say, when he said that.

448
00:24:06 --> 00:24:12
"He crushed his skull."

449
00:24:08 --> 00:24:14
And I said, "Gee, that's
really physics in action--

450
00:24:10 --> 00:24:16
It's 801 all the way."

451
00:24:11 --> 00:24:17
(class laughs )

452
00:24:13 --> 00:24:19
A modest... a really modest
velocity when he hits the floor

453
00:24:16 --> 00:24:22
but he hit the floor
like a billiard ball.

454
00:24:19 --> 00:24:25
The guy was bald, for one thing

455
00:24:21 --> 00:24:27
and so the impact time
was very short.

456
00:24:23 --> 00:24:29
And when the impact time
is short

457
00:24:25 --> 00:24:31
even if you hit the floor
with a modest speed

458
00:24:27 --> 00:24:33
the acceleration is high...
(blows raspberry )

459
00:24:31 --> 00:24:37
And that was too much

460
00:24:32 --> 00:24:38
and so that's
why his skull was crushed.

461
00:24:36 --> 00:24:42
So what matters is this changing
velocity and the impact time.

462
00:24:43 --> 00:24:49
We now want to make one last
step from average acceleration.

463
00:24:49 --> 00:24:55
We want to go
to the acceleration

464
00:24:53 --> 00:24:59
at any moment in time

465
00:24:55 --> 00:25:01
just the way we did that
with velocity.

466
00:24:59 --> 00:25:05
And that now is a natural step.

467
00:25:01 --> 00:25:07
The acceleration
at any moment in time

468
00:25:04 --> 00:25:10
will be the limit for delta t
goes to zero for v

469
00:25:11 --> 00:25:17
measured at t plus delta t
minus vt divided by delta t.

470
00:25:19 --> 00:25:25
That is
the instantaneous acceleration.

471
00:25:22 --> 00:25:28
And this, you will recognize

472
00:25:25 --> 00:25:31
is the first derivative
of velocity versus time

473
00:25:29 --> 00:25:35
which is also
the second derivative

474
00:25:31 --> 00:25:37
of position versus time.

475
00:25:34 --> 00:25:40
And so here comes
the second equation

476
00:25:36 --> 00:25:42
that I really want
you to remember

477
00:25:38 --> 00:25:44
forever and ever and ever

478
00:25:40 --> 00:25:46
that the acceleration
is dv/dt

479
00:25:45 --> 00:25:51
which is also d2x/dt squared.

480
00:25:50 --> 00:25:56

481
00:25:54 --> 00:26:00
We can go to our plot and we can
ask ourselves the question now:

482
00:25:58 --> 00:26:04
where is the acceleration zero,
where is it larger than zero

483
00:26:02 --> 00:26:08
and where is it
smaller than zero?

484
00:26:05 --> 00:26:11
Because this value can be
larger than zero, equal to zero

485
00:26:09 --> 00:26:15
and smaller than zero.

486
00:26:10 --> 00:26:16
And now you have
to be very careful
 

487
00:26:13 --> 00:26:19
when you try to derive that
from this plot.

488
00:26:16 --> 00:26:22
You have to be very careful

489
00:26:18 --> 00:26:24
because you and I have no good
feeling for second derivatives.

490
00:26:22 --> 00:26:28
Velocity is easy--

491
00:26:23 --> 00:26:29
all you have to do
is looking at alpha.

492
00:26:25 --> 00:26:31
But when it comes
to the second derivative

493
00:26:27 --> 00:26:33
you have to see
how alpha is changing.

494
00:26:30 --> 00:26:36
Well, right here,
the velocity is not changing

495
00:26:34 --> 00:26:40
so the acceleration
everywhere here must be zero.

496
00:26:39 --> 00:26:45
Here the velocity is increasing

497
00:26:43 --> 00:26:49
so the acceleration must be
larger than zero here.

498
00:26:48 --> 00:26:54
Here, the velocity
is almost constant--

499
00:26:52 --> 00:26:58
it's almost a straight line.

500
00:26:55 --> 00:27:01
What does that mean
for the acceleration?

501
00:26:59 --> 00:27:05
Zero, exactly.

502
00:27:00 --> 00:27:06
Here, when it makes
this rounding curve

503
00:27:03 --> 00:27:09
the velocity is positive here,
but on this side it's negative

504
00:27:06 --> 00:27:12
so what does that mean
for the acceleration?

505
00:27:09 --> 00:27:15
Negative, you got it.

506
00:27:10 --> 00:27:16
And so you can now roughly find

507
00:27:13 --> 00:27:19
where the acceleration
is positive

508
00:27:16 --> 00:27:22
where it's negative
and where it is zero.

509
00:27:20 --> 00:27:26

510
00:27:24 --> 00:27:30
Let's do
a straightforward example

511
00:27:26 --> 00:27:32
the way that you could expect
it on an assignment

512
00:27:30 --> 00:27:36
or, if you were
extraordinarily lucky

513
00:27:33 --> 00:27:39
you might even get something
like that on an exam.

514
00:27:38 --> 00:27:44
Very straightforward.

515
00:27:39 --> 00:27:45
I'm going to give you the
position x as a function of time

516
00:27:44 --> 00:27:50
and then ask you lots
of questions about it.

517
00:27:50 --> 00:27:56
So this example
is a working example--

518
00:27:56 --> 00:28:02
x equals eight minus 60
plus t-squared.

519
00:28:01 --> 00:28:07
So this tells you where the
object is at any moment in time

520
00:28:07 --> 00:28:13
and let this be in meters.

521
00:28:09 --> 00:28:15
What now is the velocity
at any moment in time?

522
00:28:15 --> 00:28:21
Well, that's
the derivative dx/dt

523
00:28:19 --> 00:28:25
and I use the following--
x equals t to the power n.

524
00:28:27 --> 00:28:33
Then, as most
of you should know

525
00:28:29 --> 00:28:35
dx/dt is then n times t
to the power n minus one.

526
00:28:33 --> 00:28:39
That's all I'm using.

527
00:28:35 --> 00:28:41
So the derivative of eight
is zero.

528
00:28:38 --> 00:28:44
I get here minus six,
I get here plus 2t--

529
00:28:43 --> 00:28:49
this would be
in meters per second---

530
00:28:46 --> 00:28:52
and the acceleration...

531
00:28:48 --> 00:28:54
I have to take the derivative
of the velocity, I get plus two.

532
00:28:53 --> 00:28:59
So notice that the acceleration

533
00:28:56 --> 00:29:02
is constant in time,
is not changing

534
00:28:59 --> 00:29:05
but the velocity is changing.

535
00:29:01 --> 00:29:07
Well, at time t equals zero...

536
00:29:04 --> 00:29:10
just, I will start
to probe a little bit.

537
00:29:08 --> 00:29:14
I want to get a feeling
for what this object is doing.

538
00:29:13 --> 00:29:19
At time t equals zero,
x is plus eight

539
00:29:17 --> 00:29:23
The velocity is minus six
meters per second

540
00:29:21 --> 00:29:27
and the acceleration
equals plus two.

541
00:29:26 --> 00:29:32
I can also ask myself
at what time does x = 0?

542
00:29:32 --> 00:29:38
What are the times
that x is zero?

543
00:29:34 --> 00:29:40
Well, I have to solve
this second-order equation

544
00:29:37 --> 00:29:43
which is something that
you've all done in high school

545
00:29:41 --> 00:29:47
and you will find that
that's the case

546
00:29:44 --> 00:29:50
when the time is plus two
and when the time is plus four.

547
00:29:48 --> 00:29:54
Take the plus two...
that makes this four.

548
00:29:55 --> 00:30:01
4 + 8 = 12, minus 6 x 2,
that's 0.

549
00:29:58 --> 00:30:04
So you see the 2 works and
you check that the 4 also works.

550
00:30:03 --> 00:30:09

551
00:30:05 --> 00:30:11
Just for my curiosity,
when is the velocity zero?

552
00:30:08 --> 00:30:14
Oh, that's easy--

553
00:30:09 --> 00:30:15
that's when this equation
is zero

554
00:30:11 --> 00:30:17
so that's when t equals three.

555
00:30:13 --> 00:30:19

556
00:30:15 --> 00:30:21
What is, at that moment,
the position?

557
00:30:18 --> 00:30:24
Well, I substitute
t equals three in here

558
00:30:20 --> 00:30:26
and that gives me minus one.

559
00:30:22 --> 00:30:28
x = -1.

560
00:30:26 --> 00:30:32
So now I'm ready to plot x
as a function of t.

561
00:30:32 --> 00:30:38
It's, of course, a parabola

562
00:30:34 --> 00:30:40
and I use this information
that we have just derived.

563
00:30:38 --> 00:30:44
So here comes my plot.

564
00:30:40 --> 00:30:46

565
00:30:44 --> 00:30:50
Let this be
increasing value of x

566
00:30:48 --> 00:30:54
let this be eight
and let this be minus one.

567
00:30:53 --> 00:30:59

568
00:30:56 --> 00:31:02
This is the time axis.

569
00:30:58 --> 00:31:04
I have a zero here

570
00:31:00 --> 00:31:06
and so I want to cover,
let's say, about six seconds

571
00:31:05 --> 00:31:11
so I have 1, 2, 3, 4, 5, 6.

572
00:31:16 --> 00:31:22
Now I am going to use
this information

573
00:31:19 --> 00:31:25
in order to give you a curve
which is similar to that one

574
00:31:23 --> 00:31:29
except this is a simple one--
this is just a parabola.

575
00:31:27 --> 00:31:33
So I know
that at time t equals zero

576
00:31:31 --> 00:31:37
the object is at position eight.

577
00:31:35 --> 00:31:41
I know that x is zero...
that x is zero

578
00:31:39 --> 00:31:45
at the time 2 and at the time 4

579
00:31:42 --> 00:31:48
so the object is here
at this time and at this time.

580
00:31:46 --> 00:31:52
And I know that
at time t equals three

581
00:31:51 --> 00:31:57
it is at position minus one,
so the object is here.

582
00:31:57 --> 00:32:03
And I also know
that the velocity is zero

583
00:32:00 --> 00:32:06
so we can check that.

584
00:32:02 --> 00:32:08
And so if I make this plot now

585
00:32:04 --> 00:32:10
then we would get a curve
that's sort of like this

586
00:32:09 --> 00:32:15
and yes, indeed, notice,
the velocity here is zero.

587
00:32:14 --> 00:32:20
The angle alpha equals zero.

588
00:32:16 --> 00:32:22

589
00:32:20 --> 00:32:26
The object starts out
at t equals zero

590
00:32:23 --> 00:32:29
with a negative velocity.

591
00:32:25 --> 00:32:31
You can see that-- the object
at t equals zero is here.

592
00:32:30 --> 00:32:36
This is where the object is,
I hope you realize that.

593
00:32:33 --> 00:32:39
The object is never here.

594
00:32:34 --> 00:32:40
This is the road, this is
the one-dimensional track

595
00:32:37 --> 00:32:43
on which the object is sitting.

596
00:32:38 --> 00:32:44
The object is here and it starts
going in this direction.

597
00:32:42 --> 00:32:48
If it starts going
in this direction

598
00:32:44 --> 00:32:50
the velocity must be
less than zero

599
00:32:46 --> 00:32:52
and indeed it is,
it's minus six.

600
00:32:48 --> 00:32:54
But there is the acceleration

601
00:32:50 --> 00:32:56
which is plus two
inthis direction.

602
00:32:53 --> 00:32:59
The acceleration says,
"I don't want you to go down.

603
00:32:57 --> 00:33:03
I want you to go up!"

604
00:32:59 --> 00:33:05
Well, the velocity says

605
00:33:01 --> 00:33:07
"Sorry, all I can do
is slowly, slowly change"

606
00:33:04 --> 00:33:10
and that's what it's doing.

607
00:33:06 --> 00:33:12
It's slowly changing
the velocity

608
00:33:09 --> 00:33:15
and there comes a time
that the velocity is zero

609
00:33:13 --> 00:33:19
so the object goes down,
the velocity changes

610
00:33:16 --> 00:33:22
and when it is
at position minus one

611
00:33:18 --> 00:33:24
it has come to a grinding halt
and now it is returning.

612
00:33:22 --> 00:33:28
This positive value of a
is now increasing the velocity

613
00:33:26 --> 00:33:32
and that's what you see.

614
00:33:28 --> 00:33:34
I therefore bet you anickel

615
00:33:29 --> 00:33:35
that if you substitute,
in that equation, t equals four

616
00:33:33 --> 00:33:39
that the velocitybetter
be positive.

617
00:33:36 --> 00:33:42
It has changed
from a minus sign to a plus sign

618
00:33:39 --> 00:33:45
because of
this positive acceleration.

619
00:33:42 --> 00:33:48
I bet you a nickel
t equals four.

620
00:33:45 --> 00:33:51
What is x... uh, what is v?

621
00:33:48 --> 00:33:54
We want to know v.

622
00:33:50 --> 00:33:56
8 - 6 + 2 meters per second.

623
00:33:54 --> 00:34:00
You see?

624
00:33:55 --> 00:34:01
Physics works-- v is now
plus two meters per second.

625
00:33:59 --> 00:34:05
So all that information
is in there

626
00:34:01 --> 00:34:07
but I want you to be able
to also digest it.

627
00:34:04 --> 00:34:10
Don't look at that curve

628
00:34:06 --> 00:34:12
as just some dumb parabola,
some dumb curve.

629
00:34:09 --> 00:34:15
Try to imagine
what is happening

630
00:34:11 --> 00:34:17
and only then
do you get some insight.

631
00:34:13 --> 00:34:19
Then you really begin
to get it in your brains.

632
00:34:18 --> 00:34:24
I now would like to write down,
in most general form

633
00:34:23 --> 00:34:29
the equation for the position
and the velocity

634
00:34:28 --> 00:34:34
as a function of time
for a one-dimensional motion

635
00:34:33 --> 00:34:39
whereby the acceleration
is constant.

636
00:34:37 --> 00:34:43
So it's going to be
one-dimensional again

637
00:34:40 --> 00:34:46
and we have a is going
to be a constant.

638
00:34:42 --> 00:34:48
And so the equation
that I write down

639
00:34:45 --> 00:34:51
is the most general way
that I can write it down.

640
00:34:48 --> 00:34:54
So we're going to get
x equals some number C1

641
00:34:54 --> 00:35:00
plus some C2 times t,
plus some C3 times t squared.

642
00:35:01 --> 00:35:07
And notice... oh,
I already erased my example.

643
00:35:03 --> 00:35:09
My example is gone

644
00:35:05 --> 00:35:11
but you would have seen
this was an eight before

645
00:35:07 --> 00:35:13
and here we had...
uh, what did we have?

646
00:35:10 --> 00:35:16
Minus... we had minus 6t
and we had plus 1t squared.

647
00:35:15 --> 00:35:21
So you recognize these three...
I can now take the derivative

648
00:35:22 --> 00:35:28
and so I get C2 plus 2C3
times t

649
00:35:28 --> 00:35:34
and then I get
the acceleration equals 2C3.

650
00:35:36 --> 00:35:42
And now we get some insight
into these quantities.

651
00:35:42 --> 00:35:48
Clearly, x1... C1
is the position of x

652
00:35:47 --> 00:35:53
at time t equals zero

653
00:35:50 --> 00:35:56
for which we often write
an x zero.

654
00:35:54 --> 00:36:00
Because when t is zero,
that is where x is.

655
00:35:58 --> 00:36:04
C2 is really the velocity
at time t equals zero

656
00:36:04 --> 00:36:10
because when t is zero,
that's when C2 is v.

657
00:36:09 --> 00:36:15
And the acceleration is
now changing with time.

658
00:36:16 --> 00:36:22
It's 2C3, therefore C3
is half the acceleration.

659
00:36:21 --> 00:36:27
So this gives you some insight

660
00:36:23 --> 00:36:29
in the meaning
of these quantities

661
00:36:24 --> 00:36:30
and you can see... you can read
now, some physics in there.

662
00:36:27 --> 00:36:33
C1, C2, and C3
can independently be

663
00:36:30 --> 00:36:36
either zero, or larger
than zero, or negative.

664
00:36:33 --> 00:36:39
It makes no difference--
each one of these combinations

665
00:36:37 --> 00:36:43
is a valid possibility
in physics.

666
00:36:40 --> 00:36:46

667
00:36:42 --> 00:36:48
When we have gravity

668
00:36:45 --> 00:36:51
an object is influenced by
the gravitational acceleration

669
00:36:51 --> 00:36:57
and the gravitational
acceleration is a constant.

670
00:36:56 --> 00:37:02
And we write, often

671
00:36:57 --> 00:37:03
for that gravitational
acceleration, the letter "g".

672
00:37:01 --> 00:37:07
Whether I drop an object
or throw it vertically up

673
00:37:05 --> 00:37:11
or I throw it vertically down,
it's all one-dimensional.

674
00:37:09 --> 00:37:15
It becomes two-dimensional
when I throw it at an angle.

675
00:37:13 --> 00:37:19
I keep it one-dimensional

676
00:37:15 --> 00:37:21
the acceleration is
always the same

677
00:37:18 --> 00:37:24
and that g--
gravitational acceleration--

678
00:37:21 --> 00:37:27
in Boston is 9.80 meters
per second squared

679
00:37:25 --> 00:37:31
and it varies a little bit
for different places on Earth.

680
00:37:29 --> 00:37:35

681
00:37:31 --> 00:37:37
This gravitational acceleration
is independent

682
00:37:34 --> 00:37:40
of the mass
of the object that I drop

683
00:37:37 --> 00:37:43
of the speed of the object

684
00:37:39 --> 00:37:45
of the chemical
composition of the object

685
00:37:42 --> 00:37:48
of the size of the object
and of the shape of the object

686
00:37:47 --> 00:37:53
assuming that we have
no air drag

687
00:37:49 --> 00:37:55
assuming that these experiments
are done in... in vacuum.

688
00:37:53 --> 00:37:59
Is it obvious that
the gravitational acceleration

689
00:37:57 --> 00:38:03
is independent
of all these quantities?

690
00:38:00 --> 00:38:06
By no means.

691
00:38:02 --> 00:38:08
Is it true?

692
00:38:03 --> 00:38:09
We think so, but I want you
to appreciate

693
00:38:07 --> 00:38:13
that it is not obvious

694
00:38:09 --> 00:38:15
and it can not be proven
from first principles.

695
00:38:12 --> 00:38:18

696
00:38:15 --> 00:38:21
Remember, last time we dropped
an apple from three meters

697
00:38:19 --> 00:38:25
and we dropped another one
from one and a half meters.

698
00:38:23 --> 00:38:29
And in your second assignment,
which you haven't seen yet

699
00:38:27 --> 00:38:33
I'm asking you to calculate

700
00:38:29 --> 00:38:35
the gravitational
acceleration for me

701
00:38:31 --> 00:38:37
using these both experiments.

702
00:38:33 --> 00:38:39
And, of course,
I want you to also tell me

703
00:38:36 --> 00:38:42
what the uncertainty is
in your final answer.

704
00:38:39 --> 00:38:45
And I'd like to help you
a little bit to set it up

705
00:38:45 --> 00:38:51
and also to get these equations
in terms of gravity.

706
00:38:51 --> 00:38:57
Whenever we deal with gravity,
we get the g in there.

707
00:38:55 --> 00:39:01
So suppose here is the object
at time t equals zero.

708
00:39:00 --> 00:39:06
It was the apple, and I call
that position x zero.

709
00:39:04 --> 00:39:10
I call that zero, I'm free
to choose my zero position

710
00:39:07 --> 00:39:13
and I drop it zero speed.

711
00:39:09 --> 00:39:15
I just let it go, because that's
the way we did it in class.

712
00:39:12 --> 00:39:18
The object goes down
and it hits the floor.

713
00:39:19 --> 00:39:25
Well, the general equations,
now, which deal in gravity...

714
00:39:24 --> 00:39:30
If I call this
the increasing value of x...

715
00:39:29 --> 00:39:35
You can choose it differently.

716
00:39:32 --> 00:39:38
This is my choice today...
is the following.

717
00:39:38 --> 00:39:44
x equals x zero plus v zero t
plus one-half g t squared

718
00:39:44 --> 00:39:50
and g now is 9.80 meters
per second squared.

719
00:39:53 --> 00:39:59
The velocity,
at any moment in time

720
00:39:57 --> 00:40:03
equals v zero plus gt

721
00:40:00 --> 00:40:06
and the acceleration
is constant-- it's simply g.

722
00:40:05 --> 00:40:11
Now, in my case, I have chosen
t equals zero, x zero, zero

723
00:40:09 --> 00:40:15
and I have chosen this zero,
so these go.

724
00:40:12 --> 00:40:18
And so you see
that when the object is here--

725
00:40:16 --> 00:40:22
which is three meters
below this point--

726
00:40:19 --> 00:40:25
and you know the time,
how long it took to get there

727
00:40:23 --> 00:40:29
that you can now calculate "g"

728
00:40:25 --> 00:40:31
because x would be then
three meters.

729
00:40:27 --> 00:40:33
That's when it's here.

730
00:40:28 --> 00:40:34
We made a measurement in class

731
00:40:30 --> 00:40:36
how long it took,
so you know the time

732
00:40:32 --> 00:40:38
and so you can come up
with a value for g.

733
00:40:35 --> 00:40:41
And you can do that
for both measurements

734
00:40:38 --> 00:40:44
and, of course, I want you
to tell me, also

735
00:40:41 --> 00:40:47
what the uncertainty is
in those measurements.

736
00:40:44 --> 00:40:50
Remember that we derived,
last time, that C...

737
00:40:48 --> 00:40:54
that the time that it takes
for the apple to fall

738
00:40:52 --> 00:40:58
was C times the square root
of h over g

739
00:40:55 --> 00:41:01
and we never knew
what that C was.

740
00:40:58 --> 00:41:04
I did a demonstration
to show you

741
00:41:00 --> 00:41:06
that the time is proportional
to the square root of h.

742
00:41:03 --> 00:41:09
We never knew what that C was.

743
00:41:05 --> 00:41:11
Now you know, because now you
have the equations here

744
00:41:08 --> 00:41:14
and you see that that C simply
was the square root of two.

745
00:41:11 --> 00:41:17
But I could not derive that
from my dimensional analysis.

746
00:41:15 --> 00:41:21

747
00:41:18 --> 00:41:24
Now I want you to relax
and, at the same time

748
00:41:24 --> 00:41:30
get a little bit alert
for a change.

749
00:41:28 --> 00:41:34
Look at this situation,
v equals gt.

750
00:41:31 --> 00:41:37
That means
when I drop an apple--

751
00:41:33 --> 00:41:39
and I'm going to drop
another one today--

752
00:41:36 --> 00:41:42
that the velocity
increases with time.

753
00:41:40 --> 00:41:46
So if I strobe this apple
while it was falling

754
00:41:44 --> 00:41:50
I would see the separation,
when it strobes

755
00:41:47 --> 00:41:53
to increase with time, because
the velocity goes up with time.

756
00:41:53 --> 00:41:59

757
00:41:56 --> 00:42:02
I have here an apple, or
I am going to put an apple up

758
00:42:01 --> 00:42:07
about three meters from
the floor-- three meters.

759
00:42:05 --> 00:42:11
So the height is three meters,
approximately.

760
00:42:09 --> 00:42:15
We know from last time,
remember, we did it

761
00:42:12 --> 00:42:18
it was about 780 milliseconds
to hit the floor.

762
00:42:16 --> 00:42:22
I will just round it off
and I think about it...

763
00:42:19 --> 00:42:25
about eight-tenths of a second,
just to get an idea.

764
00:42:22 --> 00:42:28
If I flash it, if I strobe it
twice per second--

765
00:42:29 --> 00:42:35
we call that two hertz--

766
00:42:31 --> 00:42:37
so my strobe is
two times per second.

767
00:42:36 --> 00:42:42

768
00:42:38 --> 00:42:44
Then I should hit that ball,
when it's falling

769
00:42:41 --> 00:42:47
twice with my strobe light.

770
00:42:43 --> 00:42:49
I don't know
where it is, though

771
00:42:45 --> 00:42:51
because when we strobe it
and when I let the apple go

772
00:42:48 --> 00:42:54
the two are not synchronized,
so maybe the first time

773
00:42:51 --> 00:42:57
that the light blinks,
it may be here

774
00:42:53 --> 00:42:59
and the second time,
it may be here.

775
00:42:55 --> 00:43:01
But it's also possible
that the first time it's here

776
00:42:58 --> 00:43:04
and the second time, it's there.

777
00:43:00 --> 00:43:06
And so the first thing
I want to do

778
00:43:03 --> 00:43:09
is to test your alertness.

779
00:43:05 --> 00:43:11
We will blink.

780
00:43:06 --> 00:43:12
You will tell me
where you see them.

781
00:43:09 --> 00:43:15
But we will take a picture.

782
00:43:11 --> 00:43:17
We will take a picture
which will show us

783
00:43:14 --> 00:43:20
exactly where
those two balls were.

784
00:43:16 --> 00:43:22
So that's
the first alertness test.

785
00:43:19 --> 00:43:25
So get ready for this,
and then we will do a second one

786
00:43:23 --> 00:43:29
which is even more intriguing.

787
00:43:25 --> 00:43:31
So now I have
to first lower this velvet

788
00:43:32 --> 00:43:38
so that we get
a nice dark background.

789
00:43:38 --> 00:43:44

790
00:43:45 --> 00:43:51
There we go.

791
00:43:46 --> 00:43:52
(whooshes )

792
00:43:47 --> 00:43:53

793
00:43:53 --> 00:43:59
Wow, with my fingerprints on it,
it's not so black any more.

794
00:43:59 --> 00:44:05
There it is...
that's the background.

795
00:44:02 --> 00:44:08

796
00:44:10 --> 00:44:16
Oh, what am I doing?

797
00:44:11 --> 00:44:17
I need the ladder again--
I have to bring the apple up!

798
00:44:14 --> 00:44:20

799
00:44:16 --> 00:44:22
Friday's always
a bad day for me.

800
00:44:19 --> 00:44:25
Okay... so now I am going
to bring the apple up.

801
00:44:23 --> 00:44:29
There's some metal here,
there are electromagnets

802
00:44:27 --> 00:44:33
and so I throw a switch here

803
00:44:30 --> 00:44:36
so that the electromagnet
is activated.

804
00:44:33 --> 00:44:39
Very similar
to what we did last time.

805
00:44:36 --> 00:44:42
We have to put the apple up
and the apple is hanging there.

806
00:44:40 --> 00:44:46

807
00:44:41 --> 00:44:47
There we go.

808
00:44:42 --> 00:44:48

809
00:44:50 --> 00:44:56
So now I have to start
the, uh...

810
00:44:53 --> 00:44:59

811
00:44:58 --> 00:45:04
The strobe.

812
00:45:01 --> 00:45:07

813
00:45:05 --> 00:45:11
That's about two hertz, that's
about two flashes per second

814
00:45:08 --> 00:45:14
and I'm going to make it
pitch black.

815
00:45:10 --> 00:45:16

816
00:45:13 --> 00:45:19
Pitch black.

817
00:45:15 --> 00:45:21

818
00:45:21 --> 00:45:27
All the lights go off.

819
00:45:24 --> 00:45:30
I will count down 3, 2, 1, 0

820
00:45:27 --> 00:45:33
and Bob, there,
who is behind the camera

821
00:45:33 --> 00:45:39
will open the shutter
when I say "one."

822
00:45:35 --> 00:45:41
And when I say "zero",
the ball will fall.

823
00:45:38 --> 00:45:44
So you may only see the ball
in its highest position.

824
00:45:42 --> 00:45:48
That may not count there,
of course

825
00:45:44 --> 00:45:50
because it makes two flashes
in the time

826
00:45:47 --> 00:45:53
that the shutter is open
and that I drop it.

827
00:45:50 --> 00:45:56
Okay, if you're ready,
I'm ready.

828
00:45:53 --> 00:45:59
Make it as dark as we can.

829
00:45:56 --> 00:46:02
Bob, are you ready?

830
00:45:58 --> 00:46:04
Class ready?

831
00:45:59 --> 00:46:05
CLASS:
Yes.

832
00:46:00 --> 00:46:06
LEWIN:
Everyone ready?

833
00:46:02 --> 00:46:08
You don't look ready.

834
00:46:05 --> 00:46:11
LEWIN:
Okay... three, two, one.

835
00:46:09 --> 00:46:15
That was zero.

836
00:46:10 --> 00:46:16
So let's look at this again
in slow motion.

837
00:46:16 --> 00:46:22

838
00:46:21 --> 00:46:27
Where's the ball?

839
00:46:22 --> 00:46:28
Oh, boy, you try
that trick ten times.

840
00:46:26 --> 00:46:32
You'll never do that again.

841
00:46:29 --> 00:46:35
So now we are developing
the picture

842
00:46:33 --> 00:46:39
and I would like you to tell me
where you saw the balls.

843
00:46:39 --> 00:46:45
Where were they, roughly?

844
00:46:41 --> 00:46:47
Where was the first one?

845
00:46:44 --> 00:46:50
How much... how much below
the highest point?

846
00:46:48 --> 00:46:54
Only this much?

847
00:46:49 --> 00:46:55
The first one.

848
00:46:50 --> 00:46:56
And then the second one
was pretty low, then.

849
00:46:54 --> 00:47:00
(class murmurs )

850
00:46:55 --> 00:47:01
Okay, sounds interesting.

851
00:46:57 --> 00:47:03
We'll take a look.

852
00:46:59 --> 00:47:05
While the picture is developing

853
00:47:03 --> 00:47:09
I'm now going to test
your real alertness.

854
00:47:08 --> 00:47:14
I'm going to strobe it
with an unknown frequency...

855
00:47:14 --> 00:47:20
unknown to you.

856
00:47:15 --> 00:47:21
I will tell you a secret--
it's a higher frequency.

857
00:47:19 --> 00:47:25
You're going to see
more balls on the way down.

858
00:47:23 --> 00:47:29
I'm not going to ask you
where they are, exactly.

859
00:47:27 --> 00:47:33
All I want you to tell me,
afterwards, how many you saw.

860
00:47:31 --> 00:47:37
That's all.

861
00:47:32 --> 00:47:38
So count them as it falls.

862
00:47:34 --> 00:47:40
You know we have
only 0.8 seconds to count.

863
00:47:40 --> 00:47:46
Bob, how did
the picture come out?

864
00:47:42 --> 00:47:48

865
00:47:46 --> 00:47:52
Wow, you're good!

866
00:47:47 --> 00:47:53
Whoa, you're good.

867
00:47:49 --> 00:47:55
It was very high, actually...

868
00:47:52 --> 00:47:58
the first... the first flash,
very high.

869
00:47:56 --> 00:48:02

870
00:48:03 --> 00:48:09
You see, it's...
you did very well.

871
00:48:06 --> 00:48:12
We're going to start, now,
with the second part.

872
00:48:11 --> 00:48:17

873
00:48:20 --> 00:48:26
Is the audio restored?

874
00:48:22 --> 00:48:28
Should be.

875
00:48:23 --> 00:48:29
So, I activated
the magnet again.

876
00:48:26 --> 00:48:32

877
00:48:34 --> 00:48:40
There it is.

878
00:48:35 --> 00:48:41

879
00:48:43 --> 00:48:49
Oh, goodness!

880
00:48:51 --> 00:48:57
Working?

881
00:48:53 --> 00:48:59
Okay, thank you, Bob.

882
00:48:55 --> 00:49:01
Okay, Bob,
if you're ready, I'm ready.

883
00:48:58 --> 00:49:04
We're going to make it
as dark as we can.

884
00:49:01 --> 00:49:07
So all I want you to tell me,
how many balls will you see?

885
00:49:05 --> 00:49:11
Oh, oh, oh, oh, I have
to change-- oh, my goodness!

886
00:49:08 --> 00:49:14
(class laughing )

887
00:49:12 --> 00:49:18
(class murmuring )

888
00:49:14 --> 00:49:20

889
00:49:20 --> 00:49:26
Come on, you're now at MIT!

890
00:49:21 --> 00:49:27
(class laughs )

891
00:49:23 --> 00:49:29
What do you think?

892
00:49:24 --> 00:49:30
All right, ready?

893
00:49:26 --> 00:49:32
Bob, you're okay?

894
00:49:28 --> 00:49:34
BOB:
Okay.

895
00:49:29 --> 00:49:35
LEWIN:
Three, two, one...

896
00:49:31 --> 00:49:37

897
00:49:33 --> 00:49:39
(class laughs )

898
00:49:36 --> 00:49:42
Well?

899
00:49:37 --> 00:49:43

900
00:49:41 --> 00:49:47
Who saw three?

901
00:49:44 --> 00:49:50
STUDENT:
Four.

902
00:49:45 --> 00:49:51
Four.

903
00:49:46 --> 00:49:52
(class calls out
different answers )

904
00:49:48 --> 00:49:54
LEWIN:
Four, I want to know four.

905
00:49:49 --> 00:49:55
STUDENT:
Seven.

906
00:49:50 --> 00:49:56
Five?

907
00:49:51 --> 00:49:57
Five, here's a five,
there's a five.

908
00:49:57 --> 00:50:03
Another five?

909
00:49:58 --> 00:50:04
Who saw six?

910
00:49:59 --> 00:50:05
STUDENT:
Six.

911
00:50:00 --> 00:50:06
LEWIN:
Wow... seven?

912
00:50:03 --> 00:50:09
Eight?

913
00:50:04 --> 00:50:10
Nine?

914
00:50:05 --> 00:50:11
Ten?

915
00:50:06 --> 00:50:12
Eleven?

916
00:50:09 --> 00:50:15
Who just saw a blur?

917
00:50:11 --> 00:50:17
(class laughs )

918
00:50:13 --> 00:50:19
Those are the real winners,
I think.

919
00:50:15 --> 00:50:21
Well, I'll tell you,
it was ten hertz.

920
00:50:17 --> 00:50:23
Since it was 0.8 seconds,
depending upon where you hit it

921
00:50:22 --> 00:50:28
how lucky you are,
I will show you.

922
00:50:25 --> 00:50:31
You will either see seven
or maybe eight balls

923
00:50:29 --> 00:50:35
but it was a good test.

924
00:50:31 --> 00:50:37
And for those of you
who thought that it was only...

925
00:50:37 --> 00:50:43
that only saw five, there you
see them, let's count them.

926
00:50:43 --> 00:50:49
Let's count them together.

927
00:50:45 --> 00:50:51

928
00:50:49 --> 00:50:55
One, this is one.

929
00:50:50 --> 00:50:56
Two, three, four, five, six,
seven, this is a bounce.

930
00:50:55 --> 00:51:01
So for those who saw five,
I would say

931
00:50:58 --> 00:51:04
"Take some rest this weekend,
you need it"

932
00:51:00 --> 00:51:06
and I'll need it, too.

933
00:51:01 --> 00:51:07
See you Monday.

934
00:51:02 --> 00:51:08

935
00:51:08 --> 00:51:14.000