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Remember, earlier in the course

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we measured the average speed
of a bullet

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which we fired from a rifle.

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That was because we had
the ability of very fast timing.

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In the old days,
fast timing was not possible

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and people measured the speed of
bullets in a very delicate way

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and all the tools that we have
learned we can apply now

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to this device which we call
the ballistic pendulum.

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We have a pendulum
with a very heavy object

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hanging here at the end--
I call it the block.

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You see it here.

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And this pendulum
has length L.

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Ours is about one meter.

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I will give you
the exact numbers later.

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And we have a bullet
of mass little m

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and the bullet comes in
with velocity v.

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It gets completely absorbed,
sticks in there.

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It's a completely
inelastic collision,

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and the pendulum will then
pick up the velocity v prime

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with the bullet inside.

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The bullet is somewhere here.

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Momentum is conserved,
so we clearly have that,

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m v equals (m plus M)
times v prime.

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So if you could measure v prime,

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then you could measure
the speed of the bullet,

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which is v.

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How do we measure v prime?

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Well, we wait for the pendulum
to come to a halt,

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let's say here,
when the speed is zero.

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When it was here,
it had a speed v prime.

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And we know that there
was kinetic energy here--

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no gravitational
potential energy.

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I can call this level U = 0,

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but right here, if this
difference in height is h,

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then all the kinetic energy
has been converted

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to gravitational
potential energy.

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So we apply the theorem,
the work-energy theorem,

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or you could say... and it's
equally valid, you could say

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we applied the conservation
of mechanical energy.

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And so this kinetic energy,

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which is one-half (m plus M)
times v prime squared

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is now converted exclusively to
gravitational potential energy,

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which equals (m plus M)
times g times that h.

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And we lose our (m plus M),

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so v prime would be
the square root of 2gh.

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And so all you would
have to measure h,

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and then you know v prime,

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and if you know v prime, you
know the speed of the bullet.

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But life is not that simple.

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It's very difficult
to measure h,

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and I can make you see that.

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Suppose this angle-- angle
theta-- when it comes to a halt

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is only two degrees.

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Then h, which is L times
(1 minus cosine theta),

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is only 0.6 millimeters

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for the dimensions
that I have chosen here

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for a length of one meter.

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And you can't even see it--
it's invisible--

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let alone that you can measure
it to any degree of accuracy.

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So what are we going to do now?

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Well, we are going
to not measure h,

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but we are going to measure x.

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I call this x = 0.

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And here, when the pendulum
comes to a halt, I call that x.

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For a two-degree angle,

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x is approximately
3½ centimeters.

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It can easily be checked by you,
of course.

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So you get a huge displacement
in this direction compared to h.

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If you use small angle
approximation--

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and you better believe
that two degrees is very small--

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then you can prove, which is
purely geometrical mathematics--

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and I leave you
with that proof--

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that this is approximately
x squared divided by 2L.

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I want you to prove that.

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You take the expansion
of the cosine,

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the theories
of the Taylor series,

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and you cut it off somewhere,

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and this is not
so difficult to prove.

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In other words, v prime squared,

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which is 2gh,
can now be replaced

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by approximately 2g times
x squared divided by 2L

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which is g times x squared
divided by L.

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And so the velocity
of the bullet, v,

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which is (m plus M)
divided by m--

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I bring the m down there--
times v prime,

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but v prime is now
the square root of this,

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so I get an x here times
the square root of g over L.

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And what you see now,
but in a very clever way,

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by getting x in here, we can now
do a quite accurate measurement

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of the speed of this bullet,

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because we can measure x
with a fair accuracy--

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maybe an uncertainty of only
one or two millimeters in x

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out of the 3½ or four
or five centimeters.

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We're going to measure
the speed of such a bullet--

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we did that before--

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to get a number which
is not too different.

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I think we got something like
200 to 250 meters per second.

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I will give you the input
for this experiment.

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The bullet mass is 2
plus or minus 0.2 grams.

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I apologize that I don't give
you all the numbers

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in mks units.

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You will have to convert them,
of course, to mks units.

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The length of the pendulum

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is 1.13 meters
plus or minus two centimeters--

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we're not certain--

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with an accuracy
of about two centimeters.

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And the mass of that block,
which is huge,

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I believe is 3,200 grams,
3.2 kilograms,

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with an uncertainty
of about two grams.

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So we have a ten percent
uncertainty in the mass,

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we have a two percent
uncertainty in the length,

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and we have
a negligible uncertainty

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in the mass of the block--
that's negligibly small.

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So if I want to know now
what the velocity is

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of the speed of the bullet,

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I can calculate what
(m plus M) divided by m is

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and I can calculate
the square root of g over L,

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and with those numbers

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I find 4.7 times
ten to the third times x.

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We have to measure x.

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Now if you look
at the uncertainties

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in the whole thing,
we're going to measure x,

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which may not be 3½ centimeters,
it may be four or five--

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but the uncertainty
in our measurement

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will probably be one
or two millimeters.

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Let's say it's 0.2 centimeters.

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So this is not
our real number yet.

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So that would be an uncertainty
of about two out of 50,

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is four out of a hundred,
is four percent.

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This is four percent,
roughly four percent.

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So I would say we get
the final speed of the bullet

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to an accuracy of about 15%

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if we combine
all the uncertainties.

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So let's give it a shot,
no pun implied.

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And we'll... we can make you see

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the pendulum right there
very shortly.

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And it is very cleverly
designed.

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When the pendulum
starts to swing,

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it's going to move
a small object.

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There you have it.

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I think I can turn this
on again.

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I think that's fine.

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There is a very small wiper,
which is this black wiper,

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and as the pendulum swings--

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this is five centimeters,
this is ten centimeters--

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the wiper will stay
at the largest extension

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when the pendulum swings back.

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Okay.

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When we fire bullets,
it's always a little risky.

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I have here the... the bolt.

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Put the bolt in.

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The bullet's in my pocket--
there's one.

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That's right.

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I take the bullet.

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And I cock the gun.

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Everything done right, yes?

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So you can look there
and then see the swing

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of this very massive block,

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and the bullet will get absorbed
in that block.

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You ready for this?

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Three, two, one (fires ), zero.

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I would say 5.2 centimeters
seems to be about right,

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so x observed
is about 5.2 centimeters,

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and we know the uncertainty
of the 15% already,

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so I have to calculate now
the 4.7 times ten to the third.

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4.7 exponent third,
multiplied by...

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I go mks, of course,
so that this .052,

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and that is 244 meters
per second.

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I remember last time we had
something very similar.

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So the speed of the bullet
is about 244 meters per second.

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It's a little under
the speed of sound,

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which is 340 meters per second,

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and we came to the same
conclusion last time.

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All right.

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We have kinetic energy
in the bullet

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before the bullet hit the block,

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and you can calculate
how much that is,

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because you know the speed now
and you know the mass--

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one-half mv squared.

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You can also calculate
how much kinetic energy there is

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when this bullet
is absorbed in here.

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That's very easy--

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that's one-half times
the total mass, (m plus M),

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times v prime squared,
which you also know now.

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And so you will see then,
perhaps to your surprise,

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that if you compare the two,

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that 99.94% of all
available kinetic energy

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before the collision
was destroyed,

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and therefore
was converted to heat.

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That happened, of course... the
heat was produced in that block.

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Now I'm going to change
to the concept of impulse.

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It's not completely unrelated
to what we just did.

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An impulse is giving someone a
kick, that's what an impulse is.

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Our bullet gave an impulse to
this block, it gave it a kick.

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Impulse.

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Impulse is a vector,

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and it is defined
as the integral of F dt

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during a certain
amount of time--

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let's say from zero to delta t.

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Now, F equals ma,
which is also dp/dt--

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we have seen this now
several times--

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the rate of change of momentum--

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and so I can substitute
that in here,

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and so I find then the integral

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from zero to delta t
of dp/dt dt,

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and that makes me move
to the domain of momenta,

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so I have now simply
the integral over dp

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from some initial
momentum, pi,

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to some final momentum, pf.

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And so that is simply

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the final momentum
minus the initial momentum.

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So what an impulse does,
it changes the momentum.

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There is a force
that acts on something

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for a short amount of time--

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could be a little longer,
as you will see with rockets--

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and that gives it
a change in momentum.

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If we have an object
that we drop on the floor,

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so we have an object, mass m,
and we drop it on the floor

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and we let it fall
over a distance h,

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then it's going to hit the floor
with a certain speed--

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we know it's down, the velocity,

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and that equals
the square root of 2gh.

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If this were a completely
elastic collision,

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which depends, of course,
on the quality of the object,

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and it depends on the quality
of the floor--

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maybe a super ball on marble

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would be almost
completely elastic,

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then the ball would bounce back
with that same velocity.

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And if that were indeed a
completely elastic collision,

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then you can see

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that the impulse
that the ball is given to--

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that is given to the ball
as the ball hits the floor--

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the floor is giving
an impulse to the ball,

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and that impulse equals 2mv.

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The ball changes its momentum.

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It was first mv
in this direction,

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and now it's mv
in this direction,

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so the change is 2mv.

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So an impulse is given
to the ball.

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Now, if the collision
were completely inelastic,

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then the ball would just...
say, like a tomato,

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I throw a tomato on the floor,
it goes (splat ).

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No speed anymore
when it hits the ground,

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then, of course,
the impulse would only be mv,

262
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because then
it doesn't come back up,

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so there is no... the change
in momentum is then smaller.

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

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We have here two balls
that look alike.

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They have a mass
of 0.1 kilogram,

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so m equals 0.1 kilogram,

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and I will drop them from
a height of about 1½ meters,

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and that gives them a speed
when they hit the floor

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of about 5½ meters per second.

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And so the momentum change
is 2mv,

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so the impulse equals 2mv,
is about 1.1,

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and that would be
kilograms-meters per second.

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And that means that
if the collision time

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is delta t seconds,

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that the average force
acting upon this ball

277
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during the collision
with the floor

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equals the impulse
divided by delta t,

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because remember, that was
our definition of impulse.

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So if we know the impulse,

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we get a feeling
for the average force.

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And for the ball that
I will drop on the floor,

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we have done fast photography.

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I will show you some results
of the fast photography

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with a different ball,
but nevertheless, we did it

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with the ball that I will drop
on the floor very shortly--

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which is this one--

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that impact time
is only two milliseconds.

289
00:16:48 --> 00:16:54
It's hard to believe
that in two milliseconds

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the entire collision occurs.

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And so if you substitute
in here now two milliseconds,

292
00:16:55 --> 00:17:01
then you get for the average
force 550 newtons.

293
00:17:01 --> 00:17:07
Just imagine, this ball
has a mass of 0.1 kilogram,

294
00:17:06 --> 00:17:12
the weight is one newton,

295
00:17:07 --> 00:17:13
and during the impact,
it weighs 550 times more.

296
00:17:12 --> 00:17:18
What an incredible
weight increase!

297
00:17:14 --> 00:17:20
And the average acceleration

298
00:17:16 --> 00:17:22
that it experiences during
the impact is 550 times g.

299
00:17:22 --> 00:17:28
People play tennis,

300
00:17:24 --> 00:17:30
and they have speeds
of hundreds of miles per hour--

301
00:17:27 --> 00:17:33
speeds are way higher than we
have here, ten times higher--

302
00:17:30 --> 00:17:36
and so the weight increase
is even more.

303
00:17:36 --> 00:17:42
Now, if the collision
were completely inelastic,

304
00:17:40 --> 00:17:46
so that if it were a tomato
or an egg,

305
00:17:44 --> 00:17:50
and this one wouldn't come up,

306
00:17:45 --> 00:17:51
the average force would still
be approximately the same,

307
00:17:49 --> 00:17:55
the reason being that
the impulse will be half.

308
00:17:53 --> 00:17:59
But if the impact time
were also half

309
00:17:56 --> 00:18:02
and the impulse were half,
then, of course, the force...

310
00:17:59 --> 00:18:05
the average force
will be the same--

311
00:18:01 --> 00:18:07
very high, but for
a shorter amount of time.

312
00:18:04 --> 00:18:10

313
00:18:06 --> 00:18:12
So I want to show...

314
00:18:07 --> 00:18:13
oh, I first want to show you now
the... these two balls.

315
00:18:09 --> 00:18:15
One is almost complete elastic
collision with the floor.

316
00:18:14 --> 00:18:20
Whether this is a complete
elastic collision

317
00:18:16 --> 00:18:22
depends not only on this ball--
whether it is a super ball--

318
00:18:19 --> 00:18:25
it also depends on the condition
of the floor.

319
00:18:21 --> 00:18:27
This is not a very good floor,
this is not marble.

320
00:18:23 --> 00:18:29
So when I drop this one,

321
00:18:25 --> 00:18:31
it doesn't come up
to this point here.

322
00:18:27 --> 00:18:33
So it's not a completely
elastic collision,

323
00:18:30 --> 00:18:36
but it bounces pretty much.

324
00:18:31 --> 00:18:37
So it is somewhere
in completely inelastic

325
00:18:34 --> 00:18:40
and completely elastic.

326
00:18:36 --> 00:18:42
It's not bad, right?

327
00:18:38 --> 00:18:44
It's not bad.

328
00:18:39 --> 00:18:45
Now this one.

329
00:18:41 --> 00:18:47
Watch it.

330
00:18:42 --> 00:18:48
(splat )

331
00:18:43 --> 00:18:49
Looks alike, but it ain't.

332
00:18:46 --> 00:18:52
This one is completely
inelastic.

333
00:18:47 --> 00:18:53
It goes to the floor
and it goes clunk.

334
00:18:51 --> 00:18:57
You see a small bounce,
but that's it.

335
00:18:53 --> 00:18:59
And so the impact times
are very short--

336
00:18:59 --> 00:19:05
two milliseconds in the case
of the one that bounces back,

337
00:19:03 --> 00:19:09
one millisecond in the case
of the one that went clunk,

338
00:19:06 --> 00:19:12
and their average weight

339
00:19:07 --> 00:19:13
is about 550 times
their normal weight.

340
00:19:11 --> 00:19:17

341
00:19:13 --> 00:19:19
I'd like to show you
fast photography

342
00:19:16 --> 00:19:22
on not this very same ball,

343
00:19:18 --> 00:19:24
but on another one that we did.

344
00:19:20 --> 00:19:26
Let me take this... this out.

345
00:19:23 --> 00:19:29
And that is a ball
that comes down

346
00:19:27 --> 00:19:33
with a speed of four meters
per second.

347
00:19:31 --> 00:19:37
And each frame
is one millisecond,

348
00:19:34 --> 00:19:40
so you will see a ruler,

349
00:19:36 --> 00:19:42
and the ruler indicates...
has marks in centimeters,

350
00:19:39 --> 00:19:45
and so you will see it go...

351
00:19:41 --> 00:19:47
in four milliseconds
it will go one centimeter,

352
00:19:45 --> 00:19:51
so it has a speed
of 2½ meters per second.

353
00:19:47 --> 00:19:53
It will hit the floor,

354
00:19:48 --> 00:19:54
and then we can count
the number of milliseconds

355
00:19:50 --> 00:19:56
that it takes contact
and going back up again.

356
00:19:54 --> 00:20:00
It's not going to be
two milliseconds,

357
00:19:56 --> 00:20:02
it's a little longer,
but, again, impressively short.

358
00:20:00 --> 00:20:06
All right, we're going
to make it rather dark

359
00:20:03 --> 00:20:09
in order to get decent quality.

360
00:20:06 --> 00:20:12
I'm going to turn
five of these off

361
00:20:10 --> 00:20:16
and I'm going to set
the TV at two,

362
00:20:14 --> 00:20:20
and now I will start this.

363
00:20:17 --> 00:20:23
There we go.

364
00:20:20 --> 00:20:26
Let's hope that that will go.

365
00:20:22 --> 00:20:28
Okay, there comes the ball down.

366
00:20:25 --> 00:20:31
These marks are in centimeters.

367
00:20:30 --> 00:20:36
So there's a ruler
in centimeters.

368
00:20:34 --> 00:20:40
This is one centimeter.

369
00:20:35 --> 00:20:41
I'll rewind a little, because
we were a little too late.

370
00:20:41 --> 00:20:47
Okay, let's start again.

371
00:20:43 --> 00:20:49
So watch when it passes
this mark-- one, two, three--

372
00:20:49 --> 00:20:55
you see four milliseconds
for about one centimeter.

373
00:20:52 --> 00:20:58
So that's 2½ meters per second.

374
00:20:55 --> 00:21:01
And now we'll count the number
of milliseconds at impact.

375
00:20:57 --> 00:21:03

376
00:21:00 --> 00:21:06
One, two, three, four,
five, six, and it's off.

377
00:21:06 --> 00:21:12
About six, maybe
seven milliseconds.

378
00:21:08 --> 00:21:14

379
00:21:10 --> 00:21:16
And this is no special ball.

380
00:21:12 --> 00:21:18
These impact times
are amazingly short.

381
00:21:15 --> 00:21:21

382
00:21:32 --> 00:21:38
Now I have something
very special for you--

383
00:21:35 --> 00:21:41
something really special,

384
00:21:38 --> 00:21:44
something that
has kept me awake--

385
00:21:40 --> 00:21:46
a lot of things keep me awake
in physics,

386
00:21:42 --> 00:21:48
and not only in physics...

387
00:21:44 --> 00:21:50
(scattered laughter )

388
00:21:45 --> 00:21:51
but this... but this
is very special.

389
00:21:48 --> 00:21:54
This is very special.

390
00:21:49 --> 00:21:55
I have here a basketball.

391
00:21:52 --> 00:21:58
(ball bouncing )

392
00:21:56 --> 00:22:02
Not completely elastic,
but not bad.

393
00:21:58 --> 00:22:04
A tennis ball-- not completely
elastic, but not bad.

394
00:22:03 --> 00:22:09
Now I'm going to drop them
together vertically down,

395
00:22:07 --> 00:22:13
and then this ball
will bounce up somehow.

396
00:22:12 --> 00:22:18
And so the question
that I have for you is,

397
00:22:14 --> 00:22:20
do you think if I drop it
from this height

398
00:22:17 --> 00:22:23
that this tennis ball
will sort of come up

399
00:22:19 --> 00:22:25
at most to this height,

400
00:22:21 --> 00:22:27
or do you think
it will be lower,

401
00:22:23 --> 00:22:29
or do you think
it will be higher?

402
00:22:25 --> 00:22:31
So use your intuition.

403
00:22:26 --> 00:22:32
(chuckling ):
In the worst case it can be...
it can be wrong--

404
00:22:30 --> 00:22:36
he is already
pushing his finger up--

405
00:22:32 --> 00:22:38
so what do you think?

406
00:22:33 --> 00:22:39
Will the tennis ball reach
about the same height?

407
00:22:36 --> 00:22:42
Who is in favor of that?

408
00:22:38 --> 00:22:44
Who is in favor of higher?

409
00:22:40 --> 00:22:46

410
00:22:41 --> 00:22:47
Wow.

411
00:22:42 --> 00:22:48
Who is in favor of alot higher?

412
00:22:45 --> 00:22:51
Okay.

413
00:22:46 --> 00:22:52
Okay, I'll try it.

414
00:22:47 --> 00:22:53
Now, I cannot guarantee you that
this ball will go straight up

415
00:22:51 --> 00:22:57
after the impact, because
clearly that's impossible.

416
00:22:54 --> 00:23:00
That has zero chance,

417
00:22:55 --> 00:23:01
so it probably go up
in some direction.

418
00:22:58 --> 00:23:04
But you will see the effect
that I had in mind.

419
00:23:01 --> 00:23:07

420
00:23:03 --> 00:23:09
So there we go.

421
00:23:06 --> 00:23:12
(balls bounce )

422
00:23:07 --> 00:23:13
And you see that, indeed,
that tennis ball...

423
00:23:13 --> 00:23:19
goesway higher.

424
00:23:14 --> 00:23:20
I'll try it once more to see

425
00:23:15 --> 00:23:21
whether I can get it to go up
a little bit more vertically,

426
00:23:18 --> 00:23:24
but that is very difficult.

427
00:23:20 --> 00:23:26
It goes way higher,

428
00:23:21 --> 00:23:27
and this is something that you
should be able to calculate,

429
00:23:24 --> 00:23:30
and you can, and you will.

430
00:23:26 --> 00:23:32
Believe me, it's part
of assignment number six.

431
00:23:28 --> 00:23:34
You haven't seen it yet.

432
00:23:30 --> 00:23:36

433
00:23:31 --> 00:23:37
There we go.

434
00:23:33 --> 00:23:39
(ball bounces )

435
00:23:34 --> 00:23:40
Oh, boy, that was better.

436
00:23:35 --> 00:23:41

437
00:23:37 --> 00:23:43
Well... oh!

438
00:23:40 --> 00:23:46
Do any one of you know

439
00:23:41 --> 00:23:47
approximately how much
higher it goes,

440
00:23:43 --> 00:23:49
if this ball has way higher mass
than this one?

441
00:23:46 --> 00:23:52
Of course, the mass ratio
comes into it.

442
00:23:48 --> 00:23:54
Any idea?

443
00:23:49 --> 00:23:55
Twice as high?

444
00:23:50 --> 00:23:56
You'll be surprised
when you do assignment six.

445
00:23:54 --> 00:24:00
Much higher.

446
00:23:56 --> 00:24:02
Okay, in fact, you could
even see it here all right

447
00:23:59 --> 00:24:05
that it was quite a bit higher
than twice.

448
00:24:01 --> 00:24:07
Great... a great experiment,

449
00:24:03 --> 00:24:09
and it's something you can do
yourself in your dormitory.

450
00:24:09 --> 00:24:15
Now I want to discuss in the
remaining time about rockets.

451
00:24:13 --> 00:24:19
A rocket experience an impulse
from the engine,

452
00:24:17 --> 00:24:23
and that changes
the momentum of the rocket.

453
00:24:21 --> 00:24:27
But before we go into
the details of the rocket,

454
00:24:26 --> 00:24:32
think about, again, this idea of
throwing objects on the floor.

455
00:24:31 --> 00:24:37
And let's turn to tomatoes.

456
00:24:33 --> 00:24:39
I want tomatoes, because I want
a complete inelastic collision,

457
00:24:37 --> 00:24:43
so tomatoes hit the floor,
and it's not that I...

458
00:24:40 --> 00:24:46
I'm not only going to throw
one tomato on the floor,

459
00:24:44 --> 00:24:50
but I'm very angry today,

460
00:24:46 --> 00:24:52
I'm going to throw a lot
of these tomatoes on the floor--

461
00:24:50 --> 00:24:56
n, as in Nancy, tomatoes
on the floor.

462
00:24:52 --> 00:24:58
If one tomato hits the floor,

463
00:24:55 --> 00:25:01
the change of momentum is mv,
if m is the mass of the tomato.

464
00:25:00 --> 00:25:06
But I'm going to throw n
on the floor,

465
00:25:03 --> 00:25:09
so the change of momentum
is n, as in Nancy,

466
00:25:06 --> 00:25:12
times the mass of the tomato
times v.

467
00:25:09 --> 00:25:15
And this is the number
of kilograms per second

468
00:25:13 --> 00:25:19
of tomatoes that I throw
on the floor.

469
00:25:17 --> 00:25:23
So this is a change of momentum.

470
00:25:20 --> 00:25:26
And so this equals delta p
divided by delta t,

471
00:25:26 --> 00:25:32
and that's an average force.

472
00:25:28 --> 00:25:34

473
00:25:31 --> 00:25:37
So the floor will experience
a force in down direction.

474
00:25:36 --> 00:25:42
Of course that was
also the case here

475
00:25:40 --> 00:25:46
when the ball experienced
a force up,

476
00:25:43 --> 00:25:49
which we calculated here.

477
00:25:45 --> 00:25:51
The floor experienced,
of course,

478
00:25:47 --> 00:25:53
the same force
in down direction--

479
00:25:50 --> 00:25:56
action equals minus reaction--
Newton's third law.

480
00:25:53 --> 00:25:59
So the force experiences...
the floor experiences a force

481
00:25:57 --> 00:26:03
in down direction.

482
00:25:58 --> 00:26:04
And I can write it down in a
somewhat more civilized form:

483
00:26:04 --> 00:26:10
F equals dm/dt
times the velocity,

484
00:26:07 --> 00:26:13
if the velocity of the tomatoes
that hit the floor is constant.

485
00:26:14 --> 00:26:20
And we're going
to apply this to rockets,

486
00:26:16 --> 00:26:22
whereby the exhaust
out of rockets

487
00:26:18 --> 00:26:24
has, relative to the rocket,
a constant speed.

488
00:26:20 --> 00:26:26
And this is then the number
of kilograms per second

489
00:26:25 --> 00:26:31
that I throw on the floor.

490
00:26:29 --> 00:26:35
This is very real.

491
00:26:30 --> 00:26:36
I can throw these tomatoes
on a bathroom scale,

492
00:26:33 --> 00:26:39
and if I threw four kilograms
per second on a bathroom scale,

493
00:26:37 --> 00:26:43
and they hit the bathroom scale
with five meters per second,

494
00:26:41 --> 00:26:47
you better believe it that you
will see that the bathroom scale

495
00:26:45 --> 00:26:51
will reach an average force

496
00:26:46 --> 00:26:52
of about 20 newtons--
four times five.

497
00:26:49 --> 00:26:55
If it weren't tomatoes,

498
00:26:51 --> 00:26:57
but if they were super balls
which would bounce up,

499
00:26:53 --> 00:26:59
then the momentum change
would be double,

500
00:26:56 --> 00:27:02
and so the bathroom scale
would indicate 40 newtons.

501
00:26:59 --> 00:27:05
So this is a real thing, it's a
real force that you can record.

502
00:27:02 --> 00:27:08

503
00:27:04 --> 00:27:10
Now I'm going to be
a little bit unpleasant to you.

504
00:27:08 --> 00:27:14
I'm going to throw
rotten tomatoes at you.

505
00:27:11 --> 00:27:17

506
00:27:14 --> 00:27:20
Here you are,
and I have here a tomato,

507
00:27:18 --> 00:27:24
and this tomato has zero speed
to start with.

508
00:27:23 --> 00:27:29

509
00:27:26 --> 00:27:32
Let me make you a little bigger,

510
00:27:28 --> 00:27:34
otherwise I won't even...
I won't even hit you.

511
00:27:30 --> 00:27:36

512
00:27:33 --> 00:27:39
So this is you.

513
00:27:34 --> 00:27:40

514
00:27:36 --> 00:27:42
And so I give this tomato
a certain velocity, v of x.

515
00:27:42 --> 00:27:48
The tomato hits you, (splat ).

516
00:27:46 --> 00:27:52
Maybe it stays there,
it's possible.

517
00:27:48 --> 00:27:54
Maybe (squish ),
it will drip down.

518
00:27:51 --> 00:27:57
But in any case, the velocity
in the x direction is gone.

519
00:27:58 --> 00:28:04
So it hits you with a velocity
vx and then v of x equals zero.

520
00:28:05 --> 00:28:11
And it may make a mess.

521
00:28:07 --> 00:28:13

522
00:28:09 --> 00:28:15
You will experience a force.

523
00:28:10 --> 00:28:16
If I keep throwing these
tomatoes at you all the time...

524
00:28:14 --> 00:28:20
and this is the force
that you will experience--

525
00:28:17 --> 00:28:23
and that force is, of course,
in this direction.

526
00:28:20 --> 00:28:26

527
00:28:25 --> 00:28:31
You've got all these tomatoes,
and you feel that as a force.

528
00:28:28 --> 00:28:34
But now look at the symmetry
of the problem.

529
00:28:32 --> 00:28:38
Here the velocity goes
from vx to zero.

530
00:28:35 --> 00:28:41
But I, throwing the tomatoes,

531
00:28:38 --> 00:28:44
have to increase the velocity
from zero to vx.

532
00:28:42 --> 00:28:48
So for obvious reasons,

533
00:28:44 --> 00:28:50
I must then feel a force
in this direction--

534
00:28:47 --> 00:28:53
think of it as a recoil
when you fire a bullet.

535
00:28:51 --> 00:28:57
So I experience exactly the same
force, but in this direction,

536
00:28:55 --> 00:29:01
and that now is the idea
behind a rocket.

537
00:28:58 --> 00:29:04
A rocket is spewing out
tomatoes--

538
00:29:01 --> 00:29:07
well, not quite tomatoes--

539
00:29:03 --> 00:29:09
it's spewing out hot gas
in this direction,

540
00:29:06 --> 00:29:12
and then the rocket
will experience a force

541
00:29:10 --> 00:29:16
in that direction.

542
00:29:12 --> 00:29:18
That is the basic concept
behind a rocket.

543
00:29:15 --> 00:29:21
And the higher the speed
of the gas that it throws out--

544
00:29:20 --> 00:29:26
the higher that velocity--

545
00:29:23 --> 00:29:29
the more kilograms per second
it spits out,

546
00:29:27 --> 00:29:33
the higher dm/dt, the higher
will be the force on the rocket,

547
00:29:31 --> 00:29:37
and this force on the rocket is
called the thrust of the rocket.

548
00:29:36 --> 00:29:42

549
00:29:39 --> 00:29:45
So if we have a rocket
in space--

550
00:29:43 --> 00:29:49
here's the rocket--

551
00:29:49 --> 00:29:55
and the rocket is spewing out
gas with a velocity, u,

552
00:29:56 --> 00:30:02
which is fixed relative
to the rocket--

553
00:30:00 --> 00:30:06
it's a burning
of chemical energy.

554
00:30:02 --> 00:30:08
Chemicals are burned, it comes
out with a certain speed,

555
00:30:07 --> 00:30:13
and the rocket will then
experience a force,

556
00:30:11 --> 00:30:17
which we call the thrust,

557
00:30:13 --> 00:30:19
and that is given
by this equation.

558
00:30:15 --> 00:30:21
If you know how many kilograms
per second are spewed out,

559
00:30:18 --> 00:30:24
and you know what the velocity
is, which I've called u here,

560
00:30:21 --> 00:30:27
this will tell you what
the thrust is of that rocket.

561
00:30:24 --> 00:30:30
If we take the case
of the Saturn rockets

562
00:30:29 --> 00:30:35
that were used for the landing
on the Moon...

563
00:30:34 --> 00:30:40

564
00:30:36 --> 00:30:42
Saturn.

565
00:30:37 --> 00:30:43

566
00:30:40 --> 00:30:46
For the Saturn rockets,

567
00:30:42 --> 00:30:48
the speed u was about
2½ kilometers per second.

568
00:30:48 --> 00:30:54
So the gas came out
with 2½ kilometers per second

569
00:30:51 --> 00:30:57
relative to the rocket,

570
00:30:52 --> 00:30:58
and the rate at which this gas
was coming out is phenomenal--

571
00:30:56 --> 00:31:02
15 tons of material per second.

572
00:31:01 --> 00:31:07
dm/dt is about 15,000 kilograms
per second--

573
00:31:07 --> 00:31:13
it's almost unimaginable.

574
00:31:11 --> 00:31:17
And that would give it
then a thrust

575
00:31:16 --> 00:31:22
of about 35 million newtons,

576
00:31:18 --> 00:31:24
which, of course, was higher
than the weight of the rocket,

577
00:31:24 --> 00:31:30
otherwise the rocket
would never go up.

578
00:31:28 --> 00:31:34
An incredible thrust.

579
00:31:29 --> 00:31:35
I have also a nice problem
for you in assignment six

580
00:31:32 --> 00:31:38
with the Saturn rockets.

581
00:31:34 --> 00:31:40
You'll see these numbers again.

582
00:31:36 --> 00:31:42
These are rounded off.

583
00:31:37 --> 00:31:43

584
00:31:39 --> 00:31:45
So rockets obtain an impulse
from their engines,

585
00:31:43 --> 00:31:49
a force acts upon the rocket
for a certain amount of time--

586
00:31:47 --> 00:31:53
we call that the burn time--

587
00:31:50 --> 00:31:56
but as they burn the fuel, the
mass of the rocket goes down,

588
00:31:55 --> 00:32:01
because the fuel leaves.

589
00:31:57 --> 00:32:03
And therefore the acceleration
during the burn goes up,

590
00:32:01 --> 00:32:07
because the mass goes down.

591
00:32:03 --> 00:32:09
And this makes it
somewhat complicated

592
00:32:06 --> 00:32:12
to derive the velocity change

593
00:32:09 --> 00:32:15
that you get during the burn,
during this impulse.

594
00:32:14 --> 00:32:20
It's done in your book--
Ohanian.

595
00:32:17 --> 00:32:23
I find that derivation
a little bit complicated.

596
00:32:22 --> 00:32:28
I looked in other books,

597
00:32:24 --> 00:32:30
and I found one in the book
by Tipler on physics,

598
00:32:28 --> 00:32:34
and his derivation--

599
00:32:29 --> 00:32:35
and I will go through that
in general terms...

600
00:32:33 --> 00:32:39
I put that on the Web for you.

601
00:32:34 --> 00:32:40
It should be there tonight,

602
00:32:36 --> 00:32:42
so you may want to take
very few notes.

603
00:32:38 --> 00:32:44
The entire derivation,

604
00:32:40 --> 00:32:46
of which I will just highlight
the important issues,

605
00:32:45 --> 00:32:51
is on the Web.

606
00:32:46 --> 00:32:52

607
00:32:48 --> 00:32:54
And the way that Tipler
is doing it

608
00:32:51 --> 00:32:57
is exclusively
from your frame of reference.

609
00:32:55 --> 00:33:01
You're sitting in 26.100,

610
00:32:58 --> 00:33:04
and you are seeing
this rocket go up.

611
00:33:02 --> 00:33:08

612
00:33:04 --> 00:33:10
Let's keep that equation, which
is the thrust of the rocket,

613
00:33:11 --> 00:33:17
except that in the case of
a rocket we would call this u,

614
00:33:16 --> 00:33:22
which is the speed of the
exhaust relative to the rocket,

615
00:33:20 --> 00:33:26
and this is what
we call the thrust.

616
00:33:22 --> 00:33:28
So let's now take a rocket,
which at time t

617
00:33:27 --> 00:33:33
as seen from your
frame of reference--

618
00:33:31 --> 00:33:37
where you are sitting.

619
00:33:33 --> 00:33:39
I use v-- it's your
frame of reference.

620
00:33:36 --> 00:33:42
It's going up with a velocity v.

621
00:33:38 --> 00:33:44

622
00:33:40 --> 00:33:46
The mass of the rocket is m.

623
00:33:43 --> 00:33:49
And now we're going to look
at time t plus delta t.

624
00:33:47 --> 00:33:53
The rocket has increased
its speed, v plus delta v.

625
00:33:56 --> 00:34:02

626
00:33:58 --> 00:34:04
The mass is m-- let me do that
in... not in color.

627
00:34:03 --> 00:34:09
The mass is now m minus delta m,

628
00:34:06 --> 00:34:12
and here is
a little bit of exhaust

629
00:34:09 --> 00:34:15
that was spewed out
with velocity u

630
00:34:13 --> 00:34:19
relative to the rocket--
u relative to the rocket.

631
00:34:18 --> 00:34:24
So you in 26.100
will see the velocity

632
00:34:22 --> 00:34:28
of this little piece delta m--

633
00:34:25 --> 00:34:31
you will see this
to be v minus u.

634
00:34:29 --> 00:34:35

635
00:34:31 --> 00:34:37
If the velocity of the rocket
is larger than u-- could be--

636
00:34:34 --> 00:34:40
you will see the exhaust going
up from your frame of reference.

637
00:34:38 --> 00:34:44
If the velocity of the exhaust

638
00:34:40 --> 00:34:46
is larger than the velocity
of the rocket,

639
00:34:42 --> 00:34:48
from you frame of reference,
you will see it go down.

640
00:34:46 --> 00:34:52
That's taken care of
in the signs.

641
00:34:48 --> 00:34:54

642
00:34:51 --> 00:34:57
Now I'm going to compare
the momentum here

643
00:34:56 --> 00:35:02
with the momentum there,

644
00:34:59 --> 00:35:05
and we take the situation

645
00:35:00 --> 00:35:06
that there are no external
forces acting upon it--

646
00:35:02 --> 00:35:08
somewhere in outer space,
this rocket burns.

647
00:35:05 --> 00:35:11
Momentum must be conserved.

648
00:35:09 --> 00:35:15
The momentum at time t
equals m times v.

649
00:35:15 --> 00:35:21
That is very simple.

650
00:35:17 --> 00:35:23
That is time t.

651
00:35:18 --> 00:35:24
The momentum at time t plus
delta t of the entire system

652
00:35:23 --> 00:35:29
including the exhaust--

653
00:35:25 --> 00:35:31
if I'm telling you
that momentum is conserved,

654
00:35:29 --> 00:35:35
you can't ignore the exhaust.

655
00:35:31 --> 00:35:37
It's momentum of the system
that is conserved.

656
00:35:33 --> 00:35:39
The momentum of the rocket
is going to change

657
00:35:36 --> 00:35:42
and the momentum of the exhaust
is going to change,

658
00:35:39 --> 00:35:45
but not of the system.

659
00:35:40 --> 00:35:46

660
00:35:41 --> 00:35:47
So we're going to get
mass times velocity--

661
00:35:48 --> 00:35:54
(m minus delta m) times
(v plus delta v) plus delta m,

662
00:35:57 --> 00:36:03
which has a velocity v minus u.

663
00:36:02 --> 00:36:08

664
00:36:03 --> 00:36:09
And how large is this?

665
00:36:06 --> 00:36:12
Well, there is
mv plus m delta v.

666
00:36:09 --> 00:36:15

667
00:36:14 --> 00:36:20
Here you see the m delta v,
here you see the mv,

668
00:36:18 --> 00:36:24
and then you get
minus u delta m,

669
00:36:22 --> 00:36:28
and the delta mv here
cancels minus delta mv here,

670
00:36:26 --> 00:36:32
and this term, delta m delta v,

671
00:36:28 --> 00:36:34
is the product of two incredibly
small numbers-- I ignore that.

672
00:36:33 --> 00:36:39

673
00:36:35 --> 00:36:41
So this is the momentum
at t plus delta t

674
00:36:38 --> 00:36:44
and this is the momentum
at time t,

675
00:36:41 --> 00:36:47
and so the change
in momentum, delta p--

676
00:36:44 --> 00:36:50
which must be zero
because momentum is conserved--

677
00:36:48 --> 00:36:54
is m delta v minus u delta m.

678
00:36:52 --> 00:36:58
m delta v minus u delta m.

679
00:36:57 --> 00:37:03

680
00:36:59 --> 00:37:05
I can take the derivative
of this equation,

681
00:37:04 --> 00:37:10
so I get on the left side dp/dt.

682
00:37:09 --> 00:37:15
dp/dt is going to be zero,

683
00:37:13 --> 00:37:19
so I get zero
equals m times dv/dt,

684
00:37:18 --> 00:37:24
but dv/dt is the acceleration
of the rocket minus u dm/dt.

685
00:37:27 --> 00:37:33

686
00:37:29 --> 00:37:35
And that is the thrust
on the rocket.

687
00:37:34 --> 00:37:40
So what you see here

688
00:37:35 --> 00:37:41
is something that
is very easy to digest--

689
00:37:38 --> 00:37:44
that ma, which is the... this is
the acceleration of the rocket,

690
00:37:42 --> 00:37:48
this is the mass of the rocket
at time t--

691
00:37:44 --> 00:37:50
equals the thrust of the rocket,
and that equals u dm/dt.

692
00:37:51 --> 00:37:57
And some people call this
the... the "rocket equation."

693
00:37:59 --> 00:38:05
Now, this is true if there is
no external force on the system.

694
00:38:06 --> 00:38:12
It is interesting to include
a real launch from Earth,

695
00:38:11 --> 00:38:17
and if you have a real launch
from Earth,

696
00:38:15 --> 00:38:21
then the rocket is going up
in this direction,

697
00:38:19 --> 00:38:25
but then gravity is exactly
in the opposite direction.

698
00:38:24 --> 00:38:30
In other words, only when you
launch vertically from Earth

699
00:38:28 --> 00:38:34
would you have
a thrust like this,

700
00:38:31 --> 00:38:37
and you would have mg like this.

701
00:38:34 --> 00:38:40
In that case, this equation
has to be adjusted,

702
00:38:40 --> 00:38:46
and then you get
ma equals m thrust minus mg.

703
00:38:46 --> 00:38:52
Only if you have a launch
from Earth vertically up.

704
00:38:51 --> 00:38:57
Now you have to do
a little bit of massaging,

705
00:38:55 --> 00:39:01
and I will leave you
with that massaging.

706
00:38:58 --> 00:39:04
A couple of integrals are
necessary to convert this

707
00:39:02 --> 00:39:08
into the final velocity
of the rocket after the burn

708
00:39:06 --> 00:39:12
compared to
the initial velocity.

709
00:39:10 --> 00:39:16
And that part
I will leave you with,

710
00:39:12 --> 00:39:18
but you will see that
worked out in all detail

711
00:39:14 --> 00:39:20
on the notes that I left
on the Web.

712
00:39:16 --> 00:39:22
And then you come up
with a very famous equation

713
00:39:20 --> 00:39:26
that the final velocity
of the rocket

714
00:39:22 --> 00:39:28
minus the initial velocity
of the rocket

715
00:39:25 --> 00:39:31
equals minus u
times the logarithm

716
00:39:27 --> 00:39:33
of the final mass of the rocket

717
00:39:30 --> 00:39:36
divided by the initial mass
of the rocket.

718
00:39:33 --> 00:39:39
This is if there were
no gravity at all.

719
00:39:36 --> 00:39:42
Just in case, and only in case
of a vertical launch from Earth,

720
00:39:42 --> 00:39:48
there is also here
a term, minus gt--

721
00:39:45 --> 00:39:51
only if you have
a vertical launch from Earth.

722
00:39:50 --> 00:39:56

723
00:39:52 --> 00:39:58
When I watched my lecture,
I noticed that in my enthusiasm

724
00:39:55 --> 00:40:01
I put brackets
around the minus gt.

725
00:39:57 --> 00:40:03
This is quite misleading,

726
00:39:59 --> 00:40:05
because it may give you
the wrong impression

727
00:40:01 --> 00:40:07
that there's a product
at stake here, which is not so.

728
00:40:06 --> 00:40:12
So the brackets really should
not be around the minus gt.

729
00:40:10 --> 00:40:16
It is this term,

730
00:40:12 --> 00:40:18
minus u logarithm of mf
divided by m of i minus gt.

731
00:40:16 --> 00:40:22

732
00:40:17 --> 00:40:23
Let's now look at this equation
in a little bit more detail,

733
00:40:21 --> 00:40:27
so that we get a little bit
of feeling for it.

734
00:40:23 --> 00:40:29
Suppose we had
a vertical launch from Earth,

735
00:40:26 --> 00:40:32
but we had no rocket.

736
00:40:27 --> 00:40:33
It's possible.

737
00:40:30 --> 00:40:36
So this term does not exist.

738
00:40:32 --> 00:40:38
What do you see?

739
00:40:33 --> 00:40:39
That the velocity
equals v initial--

740
00:40:36 --> 00:40:42
we have called that before
in 801 v zero--

741
00:40:40 --> 00:40:46
that's the initial speed--
minus gt.

742
00:40:44 --> 00:40:50
Ha!

743
00:40:45 --> 00:40:51
We had that result
during our first lecture--

744
00:40:47 --> 00:40:53
completely consistent
with this equation.

745
00:40:49 --> 00:40:55
If you have no rocket, you get
that v equals v zero minus gt,

746
00:40:53 --> 00:40:59
if you throw the object
vertically up

747
00:40:55 --> 00:41:01
and we had a vertical launch.

748
00:40:56 --> 00:41:02
So that looks good.

749
00:40:59 --> 00:41:05
We launch from the Earth,

750
00:41:02 --> 00:41:08
and we now have an initial speed
which is zero.

751
00:41:07 --> 00:41:13
The rocket is standing there,
and we fire the rocket.

752
00:41:11 --> 00:41:17
t, by the way, is the burn time
of the rocket.

753
00:41:14 --> 00:41:20
Let me write that down.

754
00:41:17 --> 00:41:23
t is the burn time.

755
00:41:19 --> 00:41:25

756
00:41:21 --> 00:41:27
So initial speed is zero.

757
00:41:25 --> 00:41:31
So now, this final velocity...
if we want this final velocity

758
00:41:30 --> 00:41:36
to be anything physically
meaningful-- a positive number--

759
00:41:34 --> 00:41:40
this thing has
to come out positive.

760
00:41:36 --> 00:41:42
And you will say,
"But how can that be?"

761
00:41:39 --> 00:41:45
Because we have
a minus sign here

762
00:41:41 --> 00:41:47
and we have a minus sign there.

763
00:41:43 --> 00:41:49
How can that ever
become positive?

764
00:41:45 --> 00:41:51
Well, don't forget
that the final mass

765
00:41:48 --> 00:41:54
is smaller always
than the initial mass,

766
00:41:51 --> 00:41:57
because you burn fuel,

767
00:41:53 --> 00:41:59
and so this logarithm
is always going to be negative,

768
00:41:57 --> 00:42:03
and so this term is always
going to be positive,

769
00:42:00 --> 00:42:06
if you burn any fuel.

770
00:42:01 --> 00:42:07
Now, of course for this velocity

771
00:42:03 --> 00:42:09
to be a real value--
physically meaningful--

772
00:42:06 --> 00:42:12
this term has to kill this one,
has to be larger than this one,

773
00:42:11 --> 00:42:17
otherwise the rocket
won't even go up.

774
00:42:13 --> 00:42:19
You could be spewing out fuel
all the time,

775
00:42:15 --> 00:42:21
and the rocket would just be
sitting there,

776
00:42:18 --> 00:42:24
because the thrust up
effectively

777
00:42:20 --> 00:42:26
is not enough to lift it off.

778
00:42:21 --> 00:42:27
So the first term has to win
from the second term

779
00:42:26 --> 00:42:32
in case of a vertical launch.

780
00:42:28 --> 00:42:34

781
00:42:30 --> 00:42:36
Let's take a example
with some numbers--

782
00:42:33 --> 00:42:39
it always gives
a little bit of insight.

783
00:42:36 --> 00:42:42
We have a burn time
of about 100 seconds,

784
00:42:40 --> 00:42:46
and the initial speed is zero,

785
00:42:43 --> 00:42:49
and let us take a case whereby u
is 1,000 meters per second.

786
00:42:49 --> 00:42:55
That is less
than the Saturn rocket,

787
00:42:51 --> 00:42:57
but it's still a sizable...
it's a kilometer per second.

788
00:42:54 --> 00:43:00
So the exhaust is coming out
relative to the rocket

789
00:42:57 --> 00:43:03
with one kilometer per second.

790
00:42:59 --> 00:43:05
And the final mass divided by
the initial mass of the rocket

791
00:43:04 --> 00:43:10
is 0.1, so 90% was burned away,
the fuel that is gone.

792
00:43:09 --> 00:43:15
You have only ten percent left
when the burn is over.

793
00:43:13 --> 00:43:19

794
00:43:15 --> 00:43:21
So now we can calculate--

795
00:43:17 --> 00:43:23
if this was a launch from Earth,
a vertical launch up--

796
00:43:20 --> 00:43:26
we can calculate
how large this term is,

797
00:43:23 --> 00:43:29
take the logarithm
of this value,

798
00:43:25 --> 00:43:31
multiply this by minus u,

799
00:43:28 --> 00:43:34
and then we find
that the final velocity...

800
00:43:32 --> 00:43:38
The first term is 2,300,

801
00:43:35 --> 00:43:41
and the second term,
100 seconds,

802
00:43:38 --> 00:43:44
and this is ten,
is minus 1000.

803
00:43:41 --> 00:43:47
So you pay a price
for the gravitational field.

804
00:43:46 --> 00:43:52
And so you have about
1.3 kilometers per second

805
00:43:50 --> 00:43:56
is the final speed
of that rocket.

806
00:43:53 --> 00:43:59
These are meters per second,

807
00:43:54 --> 00:44:00
and that is in
kilometers per second.

808
00:43:58 --> 00:44:04
Now, if there were no gravity,

809
00:44:00 --> 00:44:06
then of course the gain
in speed-- this difference--

810
00:44:04 --> 00:44:10
would be 2,300 meters
per second.

811
00:44:06 --> 00:44:12
Also, if there
is no vertical launch

812
00:44:10 --> 00:44:16
but if, for instance, the rocket
were in orbit around the Earth--

813
00:44:16 --> 00:44:22
here we have the rocket

814
00:44:18 --> 00:44:24
and going in orbit
around the Earth--

815
00:44:21 --> 00:44:27
then, of course,
even though there is gravity,

816
00:44:26 --> 00:44:32
gravity is now
not doing any work.

817
00:44:28 --> 00:44:34
And therefore if you fire this
rocket for 100 seconds,

818
00:44:33 --> 00:44:39
the change in velocity,
the tangential change,

819
00:44:36 --> 00:44:42
will also be 2,300 meters
per second.

820
00:44:39 --> 00:44:45
You cannot say
now there is gravity

821
00:44:41 --> 00:44:47
and therefore we have to take
the gt term into account.

822
00:44:45 --> 00:44:51
It will be the same
for this equation.

823
00:44:47 --> 00:44:53
If you had an object
going in this direction,

824
00:44:50 --> 00:44:56
this minus gt term
wouldn't be there either.

825
00:44:53 --> 00:44:59
It's only, of course, when you
deal with a vertical motion.

826
00:44:57 --> 00:45:03

827
00:44:58 --> 00:45:04
It is very important to realize,
and very non-intuitive,

828
00:45:03 --> 00:45:09
that when you burn
a certain amount of fuel

829
00:45:06 --> 00:45:12
for a given amount of time

830
00:45:08 --> 00:45:14
you obtain a fixed change
in the velocity.

831
00:45:11 --> 00:45:17
This is fixed
for a given amount of fuel.

832
00:45:15 --> 00:45:21
The change in kinetic energy
is not fixed,

833
00:45:17 --> 00:45:23
and I'll give you some numbers

834
00:45:19 --> 00:45:25
so that you can immediately
check that for yourself.

835
00:45:23 --> 00:45:29
And it is very non-intuitive.

836
00:45:25 --> 00:45:31
It is, in fact, a mistake that
is made by many physicists

837
00:45:28 --> 00:45:34
who think that if you burn
the same amount of fuel

838
00:45:31 --> 00:45:37
of the same rocket
for the same amount of time

839
00:45:34 --> 00:45:40
that the increase
in kinetic energy is a given.

840
00:45:37 --> 00:45:43
That is not true.

841
00:45:38 --> 00:45:44
It's the change in velocity
that is a given.

842
00:45:42 --> 00:45:48
But suppose that v final
minus v initial

843
00:45:44 --> 00:45:50
is 100 meters per second.

844
00:45:46 --> 00:45:52
That's just a given.

845
00:45:48 --> 00:45:54
I have a rocket, I have
a certain amount of fuel,

846
00:45:50 --> 00:45:56
I burn it, and that is
my change in velocity.

847
00:45:54 --> 00:46:00
I start off with v initial zero,

848
00:45:57 --> 00:46:03
so my kinetic energy increase
is one-half m times 100 squared,

849
00:46:04 --> 00:46:10
which is ten to the fourth.

850
00:46:07 --> 00:46:13
That's what I got out
of this burn.

851
00:46:10 --> 00:46:16
Now I use the very same rocket,

852
00:46:12 --> 00:46:18
the same amount of fuel,
the same burn time,

853
00:46:15 --> 00:46:21
so I get again that
the final velocity

854
00:46:17 --> 00:46:23
minus the initial velocity
is still 100.

855
00:46:20 --> 00:46:26
That's exactly the same.

856
00:46:22 --> 00:46:28
But before the burn, this rocket
had an initial velocity

857
00:46:27 --> 00:46:33
of 1,000 meters per second.

858
00:46:29 --> 00:46:35
What is now the gain
of kinetic energy?

859
00:46:33 --> 00:46:39
The final velocity is now 1,100.

860
00:46:35 --> 00:46:41
That's non-negotiable, because
the rocket changes the momentum

861
00:46:39 --> 00:46:45
changes the velocity
by a fixed amount.

862
00:46:42 --> 00:46:48
So now the gain
of kinetic energy,

863
00:46:45 --> 00:46:51
the increase of kinetic energy

864
00:46:48 --> 00:46:54
equals one-half m times 1,100
squared minus 1,000 squared.

865
00:46:54 --> 00:47:00
This is the new velocity
and this was the old one.

866
00:47:00 --> 00:47:06
And this number is
one-half m times 200,000.

867
00:47:05 --> 00:47:11
This could be in joules,
and this is also in joules.

868
00:47:10 --> 00:47:16
This number is 20 times higher.

869
00:47:12 --> 00:47:18
So you see that the change in
velocity was exactly the same.

870
00:47:17 --> 00:47:23
The rocket burned
the same amount of time,

871
00:47:20 --> 00:47:26
the same amount of fuel,

872
00:47:21 --> 00:47:27
but the kinetic energy increase
is way more

873
00:47:25 --> 00:47:31
if the rocket has a higher speed
to start with.

874
00:47:28 --> 00:47:34

875
00:47:30 --> 00:47:36
We here in 26.100
made our own rocket.

876
00:47:34 --> 00:47:40
It's a very down-to-earth model,
no pun implied.

877
00:47:40 --> 00:47:46
But it is quite powerful, and
I would like to show it to you,

878
00:47:46 --> 00:47:52
because we are
quite proud of it.

879
00:47:49 --> 00:47:55
As a rocket, we use a fire
extinguisher-- carbon dioxide--

880
00:47:54 --> 00:48:00
and this fantastic invention
I have here.

881
00:47:58 --> 00:48:04

882
00:48:05 --> 00:48:11
And this powerful rocket

883
00:48:07 --> 00:48:13
is enough to reach
the escape velocity of...

884
00:48:11 --> 00:48:17
(loud whooshing )

885
00:48:12 --> 00:48:18
...of 26.100.

886
00:48:14 --> 00:48:20
(students laughing )

887
00:48:17 --> 00:48:23
I almost reached the escape
velocity, but I crashed.

888
00:48:21 --> 00:48:27
See you next Friday.

889
00:48:24 --> 00:48:30
(laughter and
scattered applause )

890
00:48:27 --> 00:48:33

891
00:48:29 --> 00:48:35
(more whooshing )

892
00:48:31 --> 00:48:37

893
00:48:38 --> 00:48:44.000