1
00:00:01,000 --> 00:00:07,000
We're going to talk
about a whole new concept,

2
00:00:07,035 --> 00:00:09,000
which is the concept
of momentum.

3
00:00:11,000 --> 00:00:13,500
We've all heard the expression,

4
00:00:13,535 --> 00:00:15,965
"We have a lot
of momentum going."

5
00:00:16,000 --> 00:00:18,965
Well, in physics,
momentum is a vector,

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and it is a product
between the mass of a particle

7
00:00:22,035 --> 00:00:24,965
and its velocity.

8
00:00:25,000 --> 00:00:26,965
And so the units would be

9
00:00:27,000 --> 00:00:30,000
kilograms times meters
per second.

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00:00:30,035 --> 00:00:35,000
F = ma.

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00:00:36,000 --> 00:00:40,500
That equals m dv/dt.

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00:00:40,535 --> 00:00:44,965
That equals d(mv)/dt

13
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and that equals,
therefore, dp/dt.

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00:00:51,535 --> 00:00:57,965
So what you see--
the force is dp/dt,

15
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and what that means is if a
particle changes its momentum,

16
00:01:00,035 --> 00:01:04,965
a force must have acted upon it.

17
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It also means if a force
acts on a particle,

18
00:01:08,035 --> 00:01:11,965
it will change its momentum.

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Let us now envision that we have
a whole set of particles

20
00:01:14,035 --> 00:01:17,965
which are interacting
with each other,

21
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and the interaction could be
gravitational interaction,

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could be electrical interaction,

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but they're interacting
with each other.

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Zillions of them--
a whole star cluster.

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00:01:28,035 --> 00:01:32,965
I pick one here,
which I call mi,

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and I pick another here,
which I call mj.

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And there is an external force
on them,

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because they happen
to be exposed to forces

29
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from the outside world,

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and so on this one
is a force Fi external,

31
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and on this one is Fj external.

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But they're interacting
with each other,

33
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either attracting or repelling,

34
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and so in addition
to these external force,

35
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there is a force that j feels
due to the presence of i.

36
00:02:10,035 --> 00:02:13,965
And let's suppose
they are attracting each other.

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This would be F that j
experience in the presence of i.

38
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Actions equals minus reactions,
according to Newton's Third Law,

39
00:02:23,035 --> 00:02:28,000
so this force, Fij, will be
exactly the same as Fji,

40
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except in
the opposite direction.

41
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We call these internal forces.

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That's the interaction
between the particles.

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If these were
the only two forces,

44
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then the net force on this
object, on this particle i,

45
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would be... would be a force
in this direction.

46
00:02:51,035 --> 00:02:54,965
This would be F net.

47
00:02:55,000 --> 00:02:56,965
Now, you can do the same here.

48
00:02:57,000 --> 00:03:01,000
This would be F net
on particle j,

49
00:03:01,035 --> 00:03:04,965
and this would be F net
on particle i.

50
00:03:05,000 --> 00:03:09,500
But since there are zillions
of objects here,

51
00:03:09,535 --> 00:03:13,965
there are many of these
interacting forces,

52
00:03:14,000 --> 00:03:15,965
and so I can't tell you
what the net force will be.

53
00:03:16,000 --> 00:03:19,000
The net force is ultimately
the sum of the external force

54
00:03:19,035 --> 00:03:22,965
and all the internal forces.

55
00:03:23,000 --> 00:03:25,000
What is the total momentum
of these zillions of particles?

56
00:03:25,035 --> 00:03:30,965
Well, that's the sum
of the individual momenta.

57
00:03:31,000 --> 00:03:36,000
So that is p1 plus p2... pi...
and you have to add them all up.

58
00:03:36,035 --> 00:03:43,000
I take the derivative of this--
dp total/dt.

59
00:03:43,035 --> 00:03:50,965
That is p1/dt.

60
00:03:51,000 --> 00:03:53,500
Well, that's the force
on number one.

61
00:03:53,535 --> 00:03:55,965
It's the total force
on number one.

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00:03:56,000 --> 00:03:58,965
So it is F1, but it is
the net force on F1.

63
00:03:59,000 --> 00:04:06,000
And F... and dp/dt for this
particle equals F2 net force,

64
00:04:06,035 --> 00:04:13,000
and the force on particle number
i equals Fi net, and so on.

65
00:04:13,035 --> 00:04:17,965
And so when we add up
all these forces,

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00:04:18,000 --> 00:04:22,000
obviously that is the total
force on the entire system.

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00:04:22,035 --> 00:04:26,965
Now comes the miracle.

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The miracle is that
all these internal forces

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eat each other up.

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00:04:33,000 --> 00:04:35,000
This ji cancels this one if you
look at the system as a whole--

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00:04:35,035 --> 00:04:38,965
not if you look
at the individual particles,

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00:04:39,000 --> 00:04:42,000
but on the system as a whole,
they all cancel each other out.

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And so the total force
on the system

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is simply
the total force external.

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And all the internal forces--
you can forget about it.

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00:04:54,535 --> 00:04:59,000
And this means, then, that
we come to a key conclusion--

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00:04:59,035 --> 00:05:03,965
that dp total/dt...

78
00:05:04,000 --> 00:05:05,965
in fact, I have it
written down there.

79
00:05:06,000 --> 00:05:09,000
It's so important
that I want you to look at it

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00:05:09,035 --> 00:05:12,000
this whole lecture.

81
00:05:12,035 --> 00:05:13,965
Look at this.

82
00:05:14,000 --> 00:05:15,000
You see that dp/dt
is the total force externally--

83
00:05:15,035 --> 00:05:18,965
forget all the internal forces--

84
00:05:19,000 --> 00:05:20,965
and what it means...
that if the external force,

85
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the total external force
on the whole system is zero,

86
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it means that the momentum
cannot change.

87
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Momentum is conserved,

88
00:05:31,000 --> 00:05:32,965
and that's called
the conservation of momentum.

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It only holds if the sum
of all external forces are zero.

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We could have hundreds of stars,
like a globular cluster,

91
00:05:41,035 --> 00:05:44,965
and they could collide
with each other,

92
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they could explode,
they could break apart--

93
00:05:47,535 --> 00:05:49,965
all those forces are internal,
they don't count.

94
00:05:50,000 --> 00:05:53,000
The angular mo... the momentum
of the cluster as a whole--

95
00:05:53,035 --> 00:05:56,965
not angular momentum,
I misspoke.

96
00:05:57,000 --> 00:05:59,000
The momentum of the cluster
as a whole would not change

97
00:05:59,035 --> 00:06:03,017
if there are no external forces
on the system,

98
00:06:03,052 --> 00:06:07,000
if the net sum, the total
external force, is zero.

99
00:06:11,000 --> 00:06:13,965
So the individual stars

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will change their momentum
all the time,

101
00:06:15,000 --> 00:06:17,000
because the individual particles
in the individual stars

102
00:06:17,035 --> 00:06:20,017
will experience, of course,
the internal forces.

103
00:06:20,052 --> 00:06:23,026
I'm not saying that
the individual particles

104
00:06:23,061 --> 00:06:26,000
do not continuously experience
momentum changes.

105
00:06:26,035 --> 00:06:28,965
It's just the system as a whole

106
00:06:29,000 --> 00:06:32,000
for which the momentum
is then conserved.

107
00:06:32,035 --> 00:06:34,965
We can do a very simple example

108
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so that you get
a feeling... numerical idea.

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I would have a...
an object here as mass, m1,

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00:06:43,035 --> 00:06:46,965
and here I have one, m2,

111
00:06:47,000 --> 00:06:50,000
and let this one have
a velocity, v1,

112
00:06:50,035 --> 00:06:53,965
and this one has a velocity, v2.

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00:06:54,000 --> 00:06:56,965
One-dimensional problem,
just as a warm-up.

114
00:06:57,000 --> 00:07:01,000
They plow into each other
and they stick together.

115
00:07:01,035 --> 00:07:05,000
I put some glue on one,
and so they stick together.

116
00:07:05,035 --> 00:07:08,965
That's a given.

117
00:07:09,000 --> 00:07:09,965
You have to accept that.

118
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Let this be the direction
of increasing x,

119
00:07:11,035 --> 00:07:13,965
and so before the collision

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00:07:14,000 --> 00:07:15,965
I have a certain amount
of momentum.

121
00:07:16,000 --> 00:07:19,000
I can do this on a horizontal
table, which is frictionless.

122
00:07:19,035 --> 00:07:22,000
I can do it on an ice surface,
which is nearly frictionless

123
00:07:22,035 --> 00:07:25,965
and I ignore air drag.

124
00:07:26,000 --> 00:07:26,965
So this is the situation before,

125
00:07:27,000 --> 00:07:31,000
and so the momentum that
I have before equals m1 v1--

126
00:07:31,035 --> 00:07:35,965
this is my plus direction--

127
00:07:36,000 --> 00:07:39,000
I could put vectors over there
but that's really not necessary,

128
00:07:39,035 --> 00:07:43,000
because it's
a one-dimensional problem--

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00:07:43,035 --> 00:07:45,965
plus m2v2.

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00:07:46,000 --> 00:07:47,500
If you like that, be my guest.

131
00:07:47,535 --> 00:07:49,767
That's the momentum before.

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00:07:49,802 --> 00:07:51,965
Now they stick together

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and so when they stick together,
their total mass is m1 + m2.

134
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And then their velocity
happens to be v prime.

135
00:07:58,035 --> 00:08:02,000
We often give a prime
to the situation

136
00:08:02,035 --> 00:08:03,965
after the collision occurs.

137
00:08:04,000 --> 00:08:06,000
And so now I can apply,
for the first time,

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00:08:06,035 --> 00:08:08,965
the conservation of momentum.

139
00:08:09,000 --> 00:08:12,000
The momentum before must be the
same as the momentum afterwards

140
00:08:12,035 --> 00:08:15,000
because there are no
external forces on the system.

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00:08:15,035 --> 00:08:18,000
When they collide,
you better believe it

142
00:08:18,035 --> 00:08:19,965
that there are internal forces.

143
00:08:20,000 --> 00:08:22,000
You better believe that there
is glue and it goes "plunk"

144
00:08:22,035 --> 00:08:24,965
and they feel those forces,
both of them.

145
00:08:25,000 --> 00:08:27,500
The momentum of each one
individually is changing,

146
00:08:27,535 --> 00:08:30,000
but not the momentum
of the system as a whole,

147
00:08:30,035 --> 00:08:35,017
and so this equals
(m1 + m2) v prime.

148
00:08:35,052 --> 00:08:40,000
And so, if you put in
some numbers--

149
00:08:40,035 --> 00:08:43,517
suppose m1 equals one kilogram,

150
00:08:43,552 --> 00:08:46,965
and v1 is five meters
per second,

151
00:08:47,000 --> 00:08:51,000
let m2 be two kilograms and let
v2 be three meters per second.

152
00:08:51,035 --> 00:08:58,000
And you see, since they are both
in the same direction

153
00:08:58,035 --> 00:09:02,965
momentum here is 5 + 6 is 11,
so this is 11.

154
00:09:03,000 --> 00:09:07,500
And this is 1 + 2 is 3,
so that is 3 times v1 prime,

155
00:09:07,535 --> 00:09:12,000
and so v1 prime is eleven-thirds
meters per second.

156
00:09:12,035 --> 00:09:16,965
So the conservation of momentum

157
00:09:17,000 --> 00:09:18,000
tells you what the speed is
after the collision.

158
00:09:23,000 --> 00:09:23,965
When we deal with collisions,

159
00:09:24,000 --> 00:09:25,000
the velocities of the objects
before the collision

160
00:09:25,035 --> 00:09:27,965
are unprimed.

161
00:09:28,000 --> 00:09:30,500
You see here v1,
and you see here v2.

162
00:09:30,535 --> 00:09:33,000
Now, the velocities
after the collisions,

163
00:09:33,035 --> 00:09:37,000
by convention, are primed.

164
00:09:37,035 --> 00:09:38,965
You see here v prime.

165
00:09:39,000 --> 00:09:40,965
Now if after the collision

166
00:09:41,000 --> 00:09:43,000
object one and object two
have a different velocity,

167
00:09:43,035 --> 00:09:47,965
we will call it v1 prime
and v2 prime.

168
00:09:48,000 --> 00:09:51,000
Now in this case, we're dealing
with an inelastic collision,

169
00:09:51,035 --> 00:09:55,000
so m1 and m2 are stuck together.

170
00:09:55,035 --> 00:09:56,965
They are merged,

171
00:09:57,000 --> 00:09:58,000
so it is sufficient
to just call it v prime.

172
00:09:58,035 --> 00:10:00,965
You could have called it
v1 prime,

173
00:10:01,000 --> 00:10:04,000
because v1 prime is the same as
v2 prime is the same as v prime.

174
00:10:04,035 --> 00:10:07,965
And during
the next four minutes,

175
00:10:08,000 --> 00:10:11,000
you will see that a few times
do I call it v1 prime.

176
00:10:11,035 --> 00:10:13,965
I wish I really hadn't.

177
00:10:14,000 --> 00:10:16,000
It would be better
if I had just used v prime.

178
00:10:16,035 --> 00:10:19,000
That is unique and sufficient.

179
00:10:21,000 --> 00:10:23,000
Now, there was a certain amount
of kinetic energy

180
00:10:23,035 --> 00:10:26,965
before the collision,

181
00:10:27,000 --> 00:10:28,000
and of course, we can
all calculate that--

182
00:10:28,035 --> 00:10:30,965
kinetic energy
before the collision.

183
00:10:31,000 --> 00:10:35,000
That is one-half m1 v1 squared
plus one-half m2 v2 squared--

184
00:10:35,035 --> 00:10:41,000
one-half m1 v1 squared,
one-half m2 v2 squared.

185
00:10:41,035 --> 00:10:47,965
How much is it?

186
00:10:48,000 --> 00:10:48,965
You can do that
as well as I can.

187
00:10:49,000 --> 00:10:52,000
If you add that up,
you will find 21.5 joules.

188
00:10:52,035 --> 00:10:55,517
Trivial to stick in the numbers.

189
00:10:55,552 --> 00:10:59,000
Now I want to hear from you.

190
00:11:02,000 --> 00:11:04,000
What do you think the situation
is with the kinetic energy

191
00:11:04,035 --> 00:11:07,965
after the collision?

192
00:11:08,000 --> 00:11:09,500
Don't look at the numbers.

193
00:11:09,535 --> 00:11:11,000
Just use your intuition.

194
00:11:11,035 --> 00:11:11,965
It may be wrong.

195
00:11:12,000 --> 00:11:12,965
That's okay, my intuition
is often wrong.

196
00:11:13,000 --> 00:11:15,500
So there was a certain amount
of kinetic energy.

197
00:11:15,535 --> 00:11:18,000
We had this collision,
and I want you to tell me

198
00:11:18,035 --> 00:11:21,000
whether you think
that the kinetic energy

199
00:11:21,035 --> 00:11:22,965
has perhaps increased
or decreased

200
00:11:23,000 --> 00:11:25,500
or maybe the kinetic energy
hasn't changed.

201
00:11:25,535 --> 00:11:28,000
Who thinks the kinetic energy
has not changed?

202
00:11:28,035 --> 00:11:31,965
Good for you.

203
00:11:32,000 --> 00:11:32,965
I see some parents even.

204
00:11:33,000 --> 00:11:36,000
Who thinks the kinetic energy
has decreased?

205
00:11:36,035 --> 00:11:39,965
Even better for you.

206
00:11:40,000 --> 00:11:41,000
Hey, there's a professor
of physics there.

207
00:11:41,035 --> 00:11:42,965
Can't go wrong there.

208
00:11:43,000 --> 00:11:44,000
And who thinks the kinetic
energy has increased?

209
00:11:44,035 --> 00:11:46,517
No one thinks that.

210
00:11:46,552 --> 00:11:48,776
Hey, it's amazing.

211
00:11:48,811 --> 00:11:50,965
Let's take a look.

212
00:11:51,000 --> 00:11:52,965
Kinetic energy
after the collision.

213
00:11:53,000 --> 00:11:59,000
That would be one-half m1
plus m2 times v prime squared.

214
00:11:59,035 --> 00:12:04,965
Would we agree?

215
00:12:05,000 --> 00:12:07,000
Let me write this
at a different location.

216
00:12:07,035 --> 00:12:09,965
So this was the 21½ joules.

217
00:12:10,000 --> 00:12:15,965
Well, you can do this
as well as I can.

218
00:12:16,000 --> 00:12:17,965
We know that v1 prime is--
that is eleven-thirds.

219
00:12:18,000 --> 00:12:22,000
And you will find that this
number equals 20.2 joules.

220
00:12:22,035 --> 00:12:25,017
So the kinetic energy went down.

221
00:12:25,052 --> 00:12:27,965
Now, you may say,
"Well, big deal.

222
00:12:28,000 --> 00:12:31,000
"Little bit of kinetic energy--
1.3 joules.

223
00:12:31,035 --> 00:12:33,965
Who cares about 1.3 joules?"

224
00:12:34,000 --> 00:12:37,000
Well, it is a real big deal
in physics, let me tell you.

225
00:12:37,035 --> 00:12:39,965
And I can appre...
I can make you appreciate

226
00:12:40,000 --> 00:12:43,000
that it is a real big deal
by doing the following.

227
00:12:43,035 --> 00:12:45,965
I don't change the masses,

228
00:12:46,000 --> 00:12:48,000
but I just am going to change
the direction of the impact.

229
00:12:48,035 --> 00:12:51,965
Here is m1 and here is m2.

230
00:12:52,000 --> 00:12:56,000
The speeds remain the same--
v1 and this is v2.

231
00:12:56,035 --> 00:13:03,965
No change in the numbers

232
00:13:04,000 --> 00:13:06,500
except that they go now
(whooshes ) head-on.

233
00:13:06,535 --> 00:13:09,000
What is the momentum
of particle number one?

234
00:13:09,035 --> 00:13:11,517
Well, that is mv.

235
00:13:11,552 --> 00:13:13,965
1 x 5 is plus 5.

236
00:13:14,000 --> 00:13:15,000
Remember, this is the increasing
direction of X.

237
00:13:15,035 --> 00:13:18,517
So the momentum is plus 5.

238
00:13:18,552 --> 00:13:21,965
What is the other one?

239
00:13:22,000 --> 00:13:23,000
That is 2 x 3,
that is 6, backwards.

240
00:13:23,035 --> 00:13:26,965
This is minus 6.

241
00:13:27,000 --> 00:13:29,000
So what is the total...
the total momentum...

242
00:13:29,035 --> 00:13:31,000
let's give it a magnitude
equals minus 1?

243
00:13:31,035 --> 00:13:33,965
The magnitude is one, of course

244
00:13:34,000 --> 00:13:36,000
and the momentum,
if I leave this off

245
00:13:36,035 --> 00:13:37,965
we would call that minus one

246
00:13:38,000 --> 00:13:39,500
because it's
a one-dimensional problem.

247
00:13:39,535 --> 00:13:41,000
The momentum before
is in this direction.

248
00:13:41,035 --> 00:13:43,965
It's minus one.

249
00:13:44,000 --> 00:13:45,500
So what now is v prime?

250
00:13:45,535 --> 00:13:47,000
They stick together.

251
00:13:47,035 --> 00:13:48,965
Here they are.

252
00:13:49,000 --> 00:13:50,500
And afterwards,
I'm not even sure

253
00:13:50,535 --> 00:13:51,965
whether they go this way
or this way.

254
00:13:52,000 --> 00:13:55,000
Yes, I am sure, because the
momentum is in this direction,

255
00:13:55,035 --> 00:13:58,517
so I predict that v1 prime
will be in this direction.

256
00:13:58,552 --> 00:14:02,000
And so this now equals m1
plus m2 times the new v1 prime.

257
00:14:02,035 --> 00:14:09,000
Well, m1 plus m2 is 1 + 2,
is 3 kilograms

258
00:14:09,035 --> 00:14:12,965
and so you see that v1 prime

259
00:14:13,000 --> 00:14:16,000
equals minus one-third meters
per second.

260
00:14:16,035 --> 00:14:18,965
So the whole system now goes off

261
00:14:19,000 --> 00:14:21,000
with one-third meters per second
in this direction.

262
00:14:21,035 --> 00:14:24,000
The kinetic energy before
has not changed--

263
00:14:24,035 --> 00:14:25,965
those 21½ joules, right?

264
00:14:26,000 --> 00:14:28,000
That is independent of how
I collide them with each other.

265
00:14:28,035 --> 00:14:32,000
What now is the kinetic energy
afterwards?

266
00:14:32,035 --> 00:14:35,965
This is a great tragedy,

267
00:14:36,000 --> 00:14:39,000
because now you get one-half
the sum of the masses,

268
00:14:39,035 --> 00:14:42,000
which is three times
this small number squared.

269
00:14:42,035 --> 00:14:45,965
Goes with V squared, right?

270
00:14:46,000 --> 00:14:47,965
And now there
is only 0.17 joules left.

271
00:14:48,000 --> 00:14:55,000
Almost all kinet... kinetic
energy has been destroyed.

272
00:14:55,035 --> 00:15:00,000
So what you see here
in front of your eyes--

273
00:15:00,035 --> 00:15:02,000
that kinetic energy
can be destroyed,

274
00:15:02,035 --> 00:15:04,517
but momentum cannot be destroyed

275
00:15:04,552 --> 00:15:06,965
in the absence
of external forces.

276
00:15:07,000 --> 00:15:09,500
Kinetic energy and momentum
is a totally different thing.

277
00:15:09,535 --> 00:15:12,000
The momentum of the individual
particles gets changed,

278
00:15:12,035 --> 00:15:14,517
but the net momentum
did not change

279
00:15:14,552 --> 00:15:16,965
but the kinetic energy
was destroyed.

280
00:15:17,000 --> 00:15:20,000
I can destroy the kinetic energy
completely if I want that.

281
00:15:20,035 --> 00:15:23,965
I can arrange a collision

282
00:15:24,000 --> 00:15:25,965
so that all kinetic energy
has been removed.

283
00:15:26,000 --> 00:15:30,000
Suppose this particle has a mass
five and the velocity is one.

284
00:15:30,035 --> 00:15:34,965
This is my shorthand notation.

285
00:15:35,000 --> 00:15:37,000
And this particle has a mass one
but it has a velocity five.

286
00:15:37,035 --> 00:15:41,965
The momentum of this system
is zero--

287
00:15:42,000 --> 00:15:44,000
plus five in this direction,
minus five in this direction.

288
00:15:44,035 --> 00:15:47,017
But you bet your life
there is kinetic energy.

289
00:15:47,052 --> 00:15:49,965
After they collision...
they hit each other,

290
00:15:50,000 --> 00:15:53,000
we agreed that they would stick,
there was glue on it.

291
00:15:53,035 --> 00:15:56,017
Momentum afterwards must
therefore still be zero.

292
00:15:56,052 --> 00:15:59,000
Internal forces don't matter
no matter what happens.

293
00:15:59,035 --> 00:16:02,965
Kinetic energy is zero.

294
00:16:03,000 --> 00:16:04,965
So the whole system collides--
(bing )--

295
00:16:05,000 --> 00:16:08,000
And afterwards you have here
the sum of the total masses

296
00:16:08,035 --> 00:16:11,965
and it just stands still,

297
00:16:12,000 --> 00:16:14,000
because I told you that they are
going to stick together.

298
00:16:14,035 --> 00:16:18,000
I used glue.

299
00:16:19,000 --> 00:16:23,000
If you have a car collision,
and two cars hit each other,

300
00:16:23,035 --> 00:16:27,000
and I compare the situation
just before the collision

301
00:16:27,035 --> 00:16:31,965
with just after the collision--

302
00:16:32,000 --> 00:16:34,000
here are the two cars and one
has a speed in this direction,

303
00:16:34,035 --> 00:16:37,000
the other has a speed
in this direction,

304
00:16:37,035 --> 00:16:39,965
and they hit each other.

305
00:16:40,000 --> 00:16:41,500
And they stick together,
it's a given.

306
00:16:41,535 --> 00:16:43,000
You go (whooshes )--
one big clump.

307
00:16:43,035 --> 00:16:46,000
So this is before.

308
00:16:47,000 --> 00:16:50,000
And I can give them
some velocities.

309
00:16:50,035 --> 00:16:52,965
This is the speed of one, v1,

310
00:16:53,000 --> 00:16:54,965
and this is the speed
of the other, v2.

311
00:16:55,000 --> 00:16:59,000
Afterwards you see something
like this: v prime.

312
00:16:59,035 --> 00:17:04,000
Just a wreck!

313
00:17:05,000 --> 00:17:07,965
The impact time is so short

314
00:17:08,000 --> 00:17:11,000
that the change in momentum
due to friction with the road--

315
00:17:11,035 --> 00:17:14,000
that would be an external force,
friction with the road.

316
00:17:14,035 --> 00:17:18,000
But that can be ignored,
is negligibly small.

317
00:17:18,035 --> 00:17:20,965
The cars hit.

318
00:17:21,000 --> 00:17:22,965
There is a huge internal force
going on.

319
00:17:23,000 --> 00:17:27,000
One slams on the other
and the other slams on one.

320
00:17:27,035 --> 00:17:31,000
There is even fireworks--
metal scrapes over metal.

321
00:17:31,035 --> 00:17:34,000
Friction, but that's
internal friction,

322
00:17:34,035 --> 00:17:36,965
not friction with the road.

323
00:17:37,000 --> 00:17:38,965
So momentum is approximately
conserved

324
00:17:39,000 --> 00:17:43,000
if we can ignore the friction
from the road during the impact

325
00:17:43,035 --> 00:17:49,000
because the impact time
is so short.

326
00:17:49,035 --> 00:17:51,965
So let here be car with mass m1

327
00:17:52,000 --> 00:17:53,965
and here is the other car
with mass m2.

328
00:17:54,000 --> 00:17:56,965
And we'll make it
a two-dimensional problem,

329
00:17:57,000 --> 00:18:00,000
because we have seen only
one-dimensional problems now.

330
00:18:00,035 --> 00:18:03,517
Let's make it
a two-dimensional problem.

331
00:18:03,552 --> 00:18:06,776
So this is the direction
for this car,

332
00:18:06,811 --> 00:18:09,965
and let's say
this is the direction

333
00:18:10,000 --> 00:18:13,000
in which the other car is going,
velocity v2.

334
00:18:13,035 --> 00:18:16,000
And tragedy has it
that this is the place

335
00:18:16,035 --> 00:18:19,000
where they're going to collide.

336
00:18:21,000 --> 00:18:24,000
In what direction
and what will be the speed

337
00:18:24,035 --> 00:18:26,965
after the collision?

338
00:18:27,000 --> 00:18:28,000
I only compare just the moment
before they hit

339
00:18:28,035 --> 00:18:30,517
with just the moment
after they hit.

340
00:18:30,552 --> 00:18:33,000
What comes later
is a different story.

341
00:18:33,035 --> 00:18:34,965
When you have formed the wreck,

342
00:18:35,000 --> 00:18:36,965
clearly it's going to slide
on the road,

343
00:18:37,000 --> 00:18:40,000
and then there is an external
force which is friction,

344
00:18:40,035 --> 00:18:42,965
which will slow it down.

345
00:18:43,000 --> 00:18:44,000
It's just during the impact
I claim that

346
00:18:44,035 --> 00:18:46,965
to a reasonable approximation,

347
00:18:47,000 --> 00:18:48,965
momentum will have
to be conserved.

348
00:18:49,000 --> 00:18:52,000
And so, what is the momentum
of this one?

349
00:18:52,035 --> 00:18:55,017
Well, this may be
the momentum of this one.

350
00:18:55,052 --> 00:18:58,000
It may have a very small mass,
small speed.

351
00:18:58,035 --> 00:18:59,965
And what is the momentum
of this one?

352
00:19:00,000 --> 00:19:03,000
Well, this may be the momentum
of number one

353
00:19:03,035 --> 00:19:05,965
and this could be the momentum
of number two.

354
00:19:06,000 --> 00:19:09,000
The net momentum is
the vectorial sum of these two,

355
00:19:09,035 --> 00:19:14,965
which is this.

356
00:19:15,000 --> 00:19:18,000
That is p total of the system.

357
00:19:18,035 --> 00:19:20,965
That is never going to change.

358
00:19:21,000 --> 00:19:23,000
That's before and after
the collision exactly the same.

359
00:19:23,035 --> 00:19:26,965
And therefore if you knew
this angle theta

360
00:19:27,000 --> 00:19:30,000
and you know p1 and you know p2,
then you can calculate

361
00:19:30,035 --> 00:19:34,000
in what direction the objects
are going to slide,

362
00:19:34,035 --> 00:19:37,965
but of course you can
also calculate then

363
00:19:38,000 --> 00:19:41,000
the velocity after the impact,
because this total momentum

364
00:19:41,035 --> 00:19:46,000
must be the sum of the total
of the two cars--

365
00:19:46,035 --> 00:19:49,000
the mass of the two cars--
times v prime.

366
00:19:49,035 --> 00:19:51,517
And so you can calculate
everything.

367
00:19:51,552 --> 00:19:54,276
And that's what
the police is doing

368
00:19:54,311 --> 00:19:56,655
when they find wrecks
on the road.

369
00:19:56,690 --> 00:19:58,965
They actually use
the... the track,

370
00:19:59,000 --> 00:20:01,000
the... the skidding tracks of
the wreck to calculate v prime,

371
00:20:01,035 --> 00:20:05,000
and then they can try
to reconstruct the situation

372
00:20:05,035 --> 00:20:07,965
as it was before the collision.

373
00:20:08,000 --> 00:20:11,500
Now, in all these cases,
the objects stuck together.

374
00:20:11,535 --> 00:20:15,000
We've only considered cases
where they stick together,

375
00:20:15,035 --> 00:20:18,017
in which case we call those
in physics

376
00:20:18,052 --> 00:20:20,965
"completely inelastic
collisions."

377
00:20:21,000 --> 00:20:23,000
Now, next lecture I will
also deal with situations

378
00:20:23,035 --> 00:20:25,965
whereby during the collision,

379
00:20:26,000 --> 00:20:28,500
the particles bounce off
each other.

380
00:20:28,535 --> 00:20:30,767
In completely
inelastic collisions,

381
00:20:30,802 --> 00:20:32,965
you always lose kinetic energy--

382
00:20:33,000 --> 00:20:35,500
sometimes a little,
as you see there;

383
00:20:35,535 --> 00:20:37,965
sometimes a lot,
as you see there;

384
00:20:38,000 --> 00:20:40,965
sometimes everything,
as you see there.

385
00:20:41,000 --> 00:20:45,500
Always in inelastic collisions
do you lose kinetic energy.

386
00:20:45,535 --> 00:20:50,000
Can we have, in a collision,
an increase of kinetic energy?

387
00:20:50,035 --> 00:20:56,000
Well, depends on how you define
the word "collision."

388
00:20:56,035 --> 00:20:58,965
The answer is yes.

389
00:20:59,000 --> 00:21:00,000
And in fact,
I will show you an example

390
00:21:00,035 --> 00:21:03,000
and even do a demonstration.

391
00:21:05,000 --> 00:21:07,500
In the most simple case
that I can think of,

392
00:21:07,535 --> 00:21:10,000
I have here a block
which has a certain mass, m.

393
00:21:10,035 --> 00:21:12,965
But there is an explosive inside

394
00:21:13,000 --> 00:21:16,000
and all of a sudden,
it goes "Boom!"

395
00:21:16,035 --> 00:21:17,965
It explodes.

396
00:21:18,000 --> 00:21:22,000
Well, before, the speed is zero
and so the momentum is zero.

397
00:21:22,035 --> 00:21:25,517
And there comes the bang--
(whooshes )

398
00:21:25,552 --> 00:21:29,776
and one piece flies
in this direction

399
00:21:29,811 --> 00:21:34,000
with a certain velocity,
v2 prime.

400
00:21:34,035 --> 00:21:38,965
This is m2.

401
00:21:39,000 --> 00:21:40,000
And another piece flies off
in this direction

402
00:21:40,035 --> 00:21:44,517
with mass m1,
with velocity v1 prime.

403
00:21:44,552 --> 00:21:48,276
Clearly, momentum
must be conserved.

404
00:21:48,311 --> 00:21:52,000
This explosion
is only internal forces,

405
00:21:52,035 --> 00:21:54,965
and so you can write down...

406
00:21:55,000 --> 00:21:56,965
if you call this
the increasing direction of x,

407
00:21:57,000 --> 00:22:00,000
so this momentum is positive
and this momentum is negative,

408
00:22:00,035 --> 00:22:04,017
the total momentum
will never change.

409
00:22:04,052 --> 00:22:07,965
It's the same before
as it is afterwards,

410
00:22:08,000 --> 00:22:12,000
so this must be m2 times v2
prime minus m1 times v1 prime.

411
00:22:12,035 --> 00:22:17,965
What happened
with the kinetic energy?

412
00:22:18,000 --> 00:22:21,000
Well, kinetic energy
has clearly increased.

413
00:22:21,035 --> 00:22:23,965
There was zero kinetic energy
to start with.

414
00:22:24,000 --> 00:22:27,000
This one now has kinetic energy
and this one has kinetic energy.

415
00:22:27,035 --> 00:22:29,965
Where did that come from?

416
00:22:30,000 --> 00:22:31,000
Well, it was the chemical
reaction of the explosion.

417
00:22:31,035 --> 00:22:33,965
Momentum, however,
was conserved.

418
00:22:34,000 --> 00:22:36,000
See, momentum really doesn't
care about these explosions.

419
00:22:36,035 --> 00:22:40,000
That's all internal forces.

420
00:22:40,035 --> 00:22:41,965
So be very careful.

421
00:22:42,000 --> 00:22:44,000
Never confuse momentum
with energy.

422
00:22:44,035 --> 00:22:45,965
Energy can change
or cannot change,

423
00:22:46,000 --> 00:22:48,000
can increase or decrease or
remain the same kinetic energy,

424
00:22:48,035 --> 00:22:52,000
but if there are no external
forces on the system,

425
00:22:52,035 --> 00:22:55,000
momentum is always conserved.

426
00:22:57,000 --> 00:22:57,965
And this is what I can show you.

427
00:22:58,000 --> 00:23:00,965
I have a demonstration set up
on this air track,

428
00:23:01,000 --> 00:23:04,000
and we do it on an air track so
there is a minimum of friction.

429
00:23:04,035 --> 00:23:08,017
And we're going to push
two cars together

430
00:23:08,052 --> 00:23:12,000
and we hold them together
with a spring.

431
00:23:12,035 --> 00:23:17,965
The spring is like
the explosion,

432
00:23:18,000 --> 00:23:21,000
like the dynamite
that can push these apart.

433
00:23:21,035 --> 00:23:24,965
So here is one car
on the air track,

434
00:23:25,000 --> 00:23:28,000
then there is a spring
and there is another car.

435
00:23:28,035 --> 00:23:31,517
Let this mass be m1
and this mass be m2,

436
00:23:31,552 --> 00:23:34,776
and this surface
is nearly frictionless.

437
00:23:34,811 --> 00:23:38,000
And I hold them together
with a string

438
00:23:38,035 --> 00:23:43,000
so that I can pull
the spring in.

439
00:23:43,035 --> 00:23:46,965
So this spring is compressed,

440
00:23:47,000 --> 00:23:50,000
and the whole system is at rest,
v equals zero.

441
00:23:50,035 --> 00:23:53,965
There is potential energy
in that spring.

442
00:23:54,000 --> 00:23:56,000
I take a burner (whooshes ),
and I burn away this thread.

443
00:23:56,035 --> 00:24:01,000
This is the situation before.

444
00:24:01,035 --> 00:24:04,965
Momentum before is zero.

445
00:24:05,000 --> 00:24:09,000
Total momentum afterwards
would also be zero.

446
00:24:09,035 --> 00:24:11,965
It's identical
to what I did there.

447
00:24:12,000 --> 00:24:15,000
And so this object will fly
in this direction,

448
00:24:15,035 --> 00:24:18,000
and this object will fly
in that direction.

449
00:24:18,035 --> 00:24:21,965
And you already can tell

450
00:24:22,000 --> 00:24:22,965
what the ratio
of the velocities will be,

451
00:24:23,000 --> 00:24:27,000
because you will see that
v2 prime divided by v1 prime--

452
00:24:27,035 --> 00:24:33,000
you see it right there--
equals m1 divided by m2.

453
00:24:33,035 --> 00:24:38,517
And so I will make
some predictions

454
00:24:38,552 --> 00:24:43,965
about the speed
of these two objects.

455
00:24:44,000 --> 00:24:47,000
The largest mass, by the way,
would get the smallest velocity,

456
00:24:47,035 --> 00:24:53,000
and the smallest mass would get
the largest velocity,

457
00:24:53,035 --> 00:24:56,965
the largest speed.

458
00:24:57,000 --> 00:24:58,965
What you see in here
are really speeds.

459
00:24:59,000 --> 00:25:02,000
We don't have the information
on the direction anymore.

460
00:25:02,035 --> 00:25:04,965
How do we measure the speed?

461
00:25:05,000 --> 00:25:07,500
Well, we actually measure
the time

462
00:25:07,535 --> 00:25:10,267
for the cars
to move ten centimeters.

463
00:25:10,302 --> 00:25:13,151
Each car has
a little metal plate

464
00:25:13,186 --> 00:25:15,965
which is ten centimeters long,

465
00:25:16,000 --> 00:25:18,965
and it blanks out
a light-emitting diode.

466
00:25:19,000 --> 00:25:22,000
And the moment that the light-
emitting diode is occluded,

467
00:25:22,035 --> 00:25:25,965
the timer starts,

468
00:25:26,000 --> 00:25:27,000
and the moment that the light-
emitting diode emerges again,

469
00:25:27,035 --> 00:25:29,965
the timer stops.

470
00:25:30,000 --> 00:25:32,500
So that's the way
we will measure the time

471
00:25:32,535 --> 00:25:34,767
for each car to move
by ten centimeters.

472
00:25:34,802 --> 00:25:37,000
The first experiment
that I will be doing,

473
00:25:37,035 --> 00:25:40,965
m1 is 244 plus or minus one gram

474
00:25:41,000 --> 00:25:46,500
and m2 is also 244
plus or minus one gram.

475
00:25:46,535 --> 00:25:52,000
In other words,
the uncertainty in the masses

476
00:25:52,035 --> 00:25:55,017
is something like 0.4%.

477
00:25:55,052 --> 00:25:57,526
That's just part of life.

478
00:25:57,561 --> 00:26:00,000
I can't do much better.

479
00:26:02,000 --> 00:26:03,965
The time that it takes
for object number one

480
00:26:04,000 --> 00:26:07,500
to go ten centimeters
is obviously ten divided by v1,

481
00:26:07,535 --> 00:26:11,000
which is the velocity that
we want to compare them with.

482
00:26:11,035 --> 00:26:15,000
Now, the ten centimeters
is really only known

483
00:26:15,035 --> 00:26:18,965
to an accuracy
of about one millimeters.

484
00:26:19,000 --> 00:26:23,000
So that is an uncertainty of 1%
in my timing in my velocity,

485
00:26:23,035 --> 00:26:27,017
so this is a 1% uncertainty.

486
00:26:27,052 --> 00:26:30,965
So if we measure the times

487
00:26:31,000 --> 00:26:34,000
that object number one
and object number two

488
00:26:34,035 --> 00:26:37,017
will go through
this timing device--

489
00:26:37,052 --> 00:26:39,965
a measurement
of ten centimeters--

490
00:26:40,000 --> 00:26:42,500
you would expect the times
to be the same

491
00:26:42,535 --> 00:26:45,267
to within roughly 1 + .5...
say 1½%.

492
00:26:45,302 --> 00:26:48,151
There is, however,
an additional problem

493
00:26:48,186 --> 00:26:51,000
that is very difficult
for me to evaluate

494
00:26:51,035 --> 00:26:53,965
and that's the following.

495
00:26:54,000 --> 00:26:55,965
If here is
the light-emitting diode,

496
00:26:56,000 --> 00:26:58,000
and here is that metal plate
that I mentioned to you,

497
00:26:58,035 --> 00:27:00,965
which comes over this diode,

498
00:27:01,000 --> 00:27:03,000
occludes the diode,
and says "start the timer,"

499
00:27:03,035 --> 00:27:05,517
and then the other end,
of course,

500
00:27:05,552 --> 00:27:07,965
moves off
and then the timer stops--

501
00:27:08,000 --> 00:27:10,500
that criterion for on and off
for one diode

502
00:27:10,535 --> 00:27:13,000
may not be exactly the same
as for the other.

503
00:27:13,035 --> 00:27:15,965
Here is one system.

504
00:27:16,000 --> 00:27:17,000
One car will fly
in this direction.

505
00:27:17,035 --> 00:27:18,965
And here is another system,

506
00:27:19,000 --> 00:27:21,000
and the car will fly
in that direction.

507
00:27:21,035 --> 00:27:23,517
And that's not so easy
to evaluate

508
00:27:23,552 --> 00:27:26,000
unless you do
a lot of experiments.

509
00:27:26,035 --> 00:27:27,965
So I would roughly guess--

510
00:27:28,000 --> 00:27:29,965
I'm always on
the conservative side--

511
00:27:30,000 --> 00:27:32,000
that this may add an uncertainty
in the criterion of the diode

512
00:27:32,035 --> 00:27:34,017
of another millimeter.

513
00:27:34,052 --> 00:27:35,965
And so effectively,

514
00:27:36,000 --> 00:27:38,000
I really don't know any better
than two millimeters

515
00:27:38,035 --> 00:27:40,965
what the ten centimeters is.

516
00:27:41,000 --> 00:27:43,000
So I would say we really
have to allow here

517
00:27:43,035 --> 00:27:45,965
for two percent plus .4%,

518
00:27:46,000 --> 00:27:48,000
so I would predict that
the times of these two cars

519
00:27:48,035 --> 00:27:52,000
will be the same
to within roughly 5%.

520
00:27:52,035 --> 00:27:54,965
That would be my prediction.

521
00:27:55,000 --> 00:27:57,000
So we're going to write down
here the times,

522
00:27:57,035 --> 00:28:00,000
and then we're going
to measure them.

523
00:28:00,035 --> 00:28:02,965
So time one and time two,

524
00:28:03,000 --> 00:28:05,000
and we will see whether
these two numbers are the same

525
00:28:05,035 --> 00:28:09,000
within the uncertainty
of our measurements.

526
00:28:09,035 --> 00:28:11,965
Here is the air track.

527
00:28:12,000 --> 00:28:13,965
Have to turn on the airflow.

528
00:28:14,000 --> 00:28:17,000
Oh, let me tell,
for those of you...

529
00:28:20,000 --> 00:28:22,000
For the parents in my audience
who haven't seen this one,

530
00:28:22,035 --> 00:28:25,965
this is a marvelous device.

531
00:28:26,000 --> 00:28:27,000
It is a track
which is well-constructed,

532
00:28:27,035 --> 00:28:29,965
so that these cars
fit beautifully

533
00:28:30,000 --> 00:28:32,000
on this V-shaped track, and then
we blow air out of this track,

534
00:28:32,035 --> 00:28:35,517
which lifts these cars up
so that they float,

535
00:28:35,552 --> 00:28:39,000
and so we can move them
in horizontal direction

536
00:28:39,035 --> 00:28:41,017
with extremely little friction.

537
00:28:41,052 --> 00:28:42,965
There is no metal
touching metal.

538
00:28:43,000 --> 00:28:45,000
The only friction you have
is the air drag.

539
00:28:45,035 --> 00:28:46,965
That you can never avoid.

540
00:28:47,000 --> 00:28:49,000
When you go through the air
with a car, there is wind--

541
00:28:49,035 --> 00:28:51,000
the same wind that you feel
when you drive your bicycle.

542
00:28:51,035 --> 00:28:53,965
But there's very little,

543
00:28:54,000 --> 00:28:55,000
and that's why these experiments
are done on these air tracks.

544
00:28:59,000 --> 00:29:02,000
These are not cheap, by the way,
these air tracks--

545
00:29:02,035 --> 00:29:04,965
they're very expensive.

546
00:29:05,000 --> 00:29:05,965
Well, you... you pay
$25,000 tuition,

547
00:29:06,000 --> 00:29:09,000
so you might as well get
something for it, right?

548
00:29:12,000 --> 00:29:14,000
So I hope that these timers
are on.

549
00:29:14,035 --> 00:29:16,000
They are.

550
00:29:17,000 --> 00:29:18,965
And now we have here one car,

551
00:29:19,000 --> 00:29:23,000
and notice how frictionless
it goes when I touch it.

552
00:29:23,035 --> 00:29:27,517
Very nice, almost frictionless.

553
00:29:27,552 --> 00:29:31,965
And here we have another one,

554
00:29:32,000 --> 00:29:34,000
and they have equal mass
to within one gram.

555
00:29:34,035 --> 00:29:37,017
And there is a spring
between them.

556
00:29:37,052 --> 00:29:39,965
Some of you may be able
to see it.

557
00:29:40,000 --> 00:29:42,000
And I'm going to attach now that
string to hold them together.

558
00:29:42,035 --> 00:29:46,965
And so I put potential energy
in the spring

559
00:29:47,000 --> 00:29:51,000
by pushing them together,
and they are lined up now here.

560
00:29:51,035 --> 00:29:54,017
They have no momentum.

561
00:29:54,052 --> 00:29:56,526
The timer's not on?

562
00:29:56,561 --> 00:29:59,000
Has to be reset.

563
00:29:59,035 --> 00:30:01,000
Thank you.

564
00:30:02,000 --> 00:30:04,000
They're now both zero.

565
00:30:04,035 --> 00:30:06,000
Yeah, you're sure?

566
00:30:06,035 --> 00:30:06,965
Okay.

567
00:30:07,000 --> 00:30:07,965
So now I'm going
to burn off that wire.

568
00:30:08,000 --> 00:30:09,965
The momentum of each one
of those cars will change.

569
00:30:10,000 --> 00:30:13,000
One will go in this direction,
the other one in this direction.

570
00:30:13,035 --> 00:30:16,965
Kinetic energy will increase.

571
00:30:17,000 --> 00:30:19,500
That's the potential energy
from the spring.

572
00:30:19,535 --> 00:30:22,267
But the momentum
of the system as a whole

573
00:30:22,302 --> 00:30:25,000
after I burn the wire
will remain zero.

574
00:30:25,035 --> 00:30:28,000
Ready for this?

575
00:30:29,000 --> 00:30:32,500
Make me happy.

576
00:30:32,535 --> 00:30:36,000
What do you see?

577
00:30:37,000 --> 00:30:40,000
You're going to make me unhappy?

578
00:30:40,035 --> 00:30:42,000
I better walk around.

579
00:30:42,035 --> 00:30:42,965
I can't wait.

580
00:30:43,000 --> 00:30:45,000
220... oh, fantastic,
absolutely fantastic!

581
00:30:45,035 --> 00:30:49,965
223, 219 milliseconds.

582
00:30:50,000 --> 00:30:52,000
That is well
within the uncertainties.

583
00:30:55,000 --> 00:30:56,000
What was the number, 219?

584
00:30:59,000 --> 00:31:04,000
219 milliseconds
and 223 milliseconds.

585
00:31:08,000 --> 00:31:10,000
If I repeat the experiment
and I make the string shorter,

586
00:31:10,035 --> 00:31:13,017
I will get different times.

587
00:31:13,052 --> 00:31:15,965
I can never predict the times,

588
00:31:16,000 --> 00:31:17,965
because if I make the spring
more squeezed,

589
00:31:18,000 --> 00:31:21,000
then obviously I will have
more potential energy.

590
00:31:21,035 --> 00:31:24,000
And so the speeds of the cars
will be higher,

591
00:31:24,035 --> 00:31:25,965
but the times will be the same

592
00:31:26,000 --> 00:31:28,000
within the uncertainty
of the measurements.

593
00:31:28,035 --> 00:31:31,517
So I can never predict the time.

594
00:31:31,552 --> 00:31:34,965
Now we're going
to do an experiment

595
00:31:35,000 --> 00:31:39,000
whereby I'm going to make m2
twice the mass.

596
00:31:39,035 --> 00:31:45,000
m2 is 488 plus or minus
one gram.

597
00:31:48,000 --> 00:31:50,000
Now, that's going
to be interesting,

598
00:31:50,035 --> 00:31:52,000
because now you really
begin to test

599
00:31:52,035 --> 00:31:54,017
the conservation of momentum.

600
00:31:54,052 --> 00:31:55,965
The momentum before is zero.

601
00:31:56,000 --> 00:31:58,000
I burn the wire, momentum
afterwards is also zero.

602
00:31:58,035 --> 00:32:02,965
But the velocities...
oh, I erased this--

603
00:32:03,000 --> 00:32:07,000
v2 prime divided by v1 prime
equals m1 divided by m2.

604
00:32:07,035 --> 00:32:13,000
And so twice the mass
will get half the speed.

605
00:32:13,035 --> 00:32:17,965
And so now you're going to see

606
00:32:18,000 --> 00:32:19,965
that one car will go twice
as slow as the other.

607
00:32:20,000 --> 00:32:23,000
That's the only way that nature
can conserve momentum.

608
00:32:23,035 --> 00:32:25,965
Nature has no choice.

609
00:32:26,000 --> 00:32:28,000
Nature can deal with kinetic
energy one way or another,

610
00:32:28,035 --> 00:32:31,965
but it cannot finagle momentum.

611
00:32:32,000 --> 00:32:33,965
So it must give
the more massive car

612
00:32:34,000 --> 00:32:37,000
half the speed
than it gives the other car.

613
00:32:37,035 --> 00:32:40,517
And so we're going
to redo this experiment

614
00:32:40,552 --> 00:32:44,000
now with a car here
which is twice the mass.

615
00:32:50,000 --> 00:32:52,000
And this one equals...

616
00:32:56,000 --> 00:32:59,000
Where is my other car?

617
00:32:59,035 --> 00:33:02,000
Oh, my car is here.

618
00:33:03,000 --> 00:33:06,500
And this one is one mass,
so this is the 244,

619
00:33:06,535 --> 00:33:09,965
and the one on the...
on your right is the 488.

620
00:33:10,000 --> 00:33:14,000
So I have to bring them together
again with a string.

621
00:33:18,000 --> 00:33:19,965
So I... I do the work now.

622
00:33:20,000 --> 00:33:22,000
I always do the work here
in 26 100.

623
00:33:22,035 --> 00:33:24,965
I do the work.

624
00:33:25,000 --> 00:33:25,965
I squeeze these springs.

625
00:33:26,000 --> 00:33:27,965
I get paid for that, by the way.

626
00:33:28,000 --> 00:33:30,000
And that's potential energy
that goes into the spring.

627
00:33:30,035 --> 00:33:33,965
There it is.

628
00:33:34,000 --> 00:33:34,965
Momentum is zero.

629
00:33:35,000 --> 00:33:37,000
Kinetic energy is zero.

630
00:33:37,035 --> 00:33:39,000
I have to zero the timers.

631
00:33:42,000 --> 00:33:44,000
Oh, and I would like
to make a prediction.

632
00:33:44,035 --> 00:33:47,965
I would like to see...

633
00:33:48,000 --> 00:33:51,000
we'll measure T1
and then we'll measure T2,

634
00:33:51,035 --> 00:33:54,000
and then we get a number
and we get a number,

635
00:33:54,035 --> 00:33:57,017
and if we multiply this number
by 2.00,

636
00:33:57,052 --> 00:33:59,965
then I would like
to get this number

637
00:34:00,000 --> 00:34:02,965
within the uncertainties
of the measurements.

638
00:34:03,000 --> 00:34:06,000
There is always an uncertainty
in any measurement that you do.

639
00:34:06,035 --> 00:34:11,965
All right?

640
00:34:12,000 --> 00:34:13,000
Make sure that...
are the timers zero?

641
00:34:15,000 --> 00:34:18,000
Okay, there we go.

642
00:34:19,000 --> 00:34:25,000
What do you see?

643
00:34:25,035 --> 00:34:31,000
Ah! 406, 193-- whoo!

644
00:34:31,035 --> 00:34:33,517
On the button!

645
00:34:33,552 --> 00:34:35,965
406 and 193.

646
00:34:36,000 --> 00:34:41,000
0,406-- is that what you see?

647
00:34:41,035 --> 00:34:46,965
Yeah?

648
00:34:47,000 --> 00:34:47,965
And 0,193.

649
00:34:48,000 --> 00:34:50,000
Now, I don't even need
my calculator

650
00:34:50,035 --> 00:34:51,517
to multiply this by two.

651
00:34:51,552 --> 00:34:53,000
I can even do that by heart.

652
00:34:53,035 --> 00:34:57,017
9-3-- ,396, ,406.

653
00:35:05,298 --> 00:35:09,211
Close enough.

654
00:35:09,246 --> 00:35:10,211
The two are
in excellent agreement

655
00:35:10,246 --> 00:35:13,746
within the 2½% uncertainty
in each measurement.

656
00:35:13,781 --> 00:35:17,246
So, you see, that's exactly
how you can demonstrate

657
00:35:17,281 --> 00:35:20,763
the conservation of momentum.

658
00:35:20,798 --> 00:35:24,211
They picked up kinetic energy,

659
00:35:24,246 --> 00:35:28,246
but after the wire was burned,
the momentum remained zero.

660
00:35:36,246 --> 00:35:44,211
Now I change the topic.

661
00:35:44,246 --> 00:35:46,211
It... it appears
that I change the topic,

662
00:35:46,246 --> 00:35:49,246
but you will see in a few weeks
that it's really related--

663
00:35:49,281 --> 00:35:53,211
not even a few weeks--

664
00:35:53,246 --> 00:35:54,246
you will see
that it's really related

665
00:35:54,281 --> 00:35:56,763
to the conservation of momentum.

666
00:35:56,798 --> 00:35:59,211
I'm going to explain to you now

667
00:35:59,246 --> 00:36:03,746
what we physicists mean
by center of mass of a system.

668
00:36:03,781 --> 00:36:08,246
The center of mass of a system
is defined as follows:

669
00:36:08,281 --> 00:36:14,211
I have here some kind
of an object--

670
00:36:14,246 --> 00:36:18,246
not just a point mass,
but it has a finite size.

671
00:36:18,281 --> 00:36:23,211
It could be a hammer.

672
00:36:23,246 --> 00:36:26,246
It could be something
like this-- a squash racquet.

673
00:36:26,281 --> 00:36:29,246
And let the center of mass
be, for instance, here.

674
00:36:29,281 --> 00:36:32,763
This is the center of mass.

675
00:36:32,798 --> 00:36:36,211
I pick any origin I want to.

676
00:36:36,246 --> 00:36:38,246
You're totally free
to choose this origin--

677
00:36:38,281 --> 00:36:42,211
doesn't matter
where you take it.

678
00:36:42,246 --> 00:36:46,246
And this, now, is the position
vector of the center of mass.

679
00:36:46,281 --> 00:36:50,246
And now I carve out a zillion
little mass elements, m of i,

680
00:36:50,281 --> 00:36:55,211
covering the entire object,

681
00:36:55,246 --> 00:36:59,246
and then this is
the position vector, r of i.

682
00:36:59,281 --> 00:37:03,246
And the center of mass
is defined as follows:

683
00:37:03,281 --> 00:37:09,211
the total mass of this object

684
00:37:09,246 --> 00:37:12,246
times the position vector
of the center of mass

685
00:37:12,281 --> 00:37:16,211
is the sum, summed over i--

686
00:37:16,246 --> 00:37:19,211
over all these
little elements, i--

687
00:37:19,246 --> 00:37:22,246
times the position vectors of
the individual little particles

688
00:37:22,281 --> 00:37:27,246
that make up this object.

689
00:37:29,246 --> 00:37:32,211
v center of mass-- I find that

690
00:37:32,246 --> 00:37:35,246
by taking the derivative
of this equation--

691
00:37:35,281 --> 00:37:40,211
equals one over the mass total.

692
00:37:40,246 --> 00:37:44,246
I bring this on this side,
I take the derivative here,

693
00:37:44,281 --> 00:37:48,246
and so here I get the sum
of m of i times the dri/dt.

694
00:37:48,281 --> 00:37:53,246
And so this is the sum
of i... of mi vi,

695
00:37:53,281 --> 00:37:57,211
simply taking
the first derivative,

696
00:37:57,246 --> 00:38:01,246
which changes positions
into velocities.

697
00:38:05,246 --> 00:38:08,746
Now, this is also 1 over m total
times the total momentum,

698
00:38:08,781 --> 00:38:12,246
because I've all these momenta
of these individual particles.

699
00:38:12,281 --> 00:38:19,246
They have a...
together, a total momentum.

700
00:38:19,281 --> 00:38:23,211
And so now you come

701
00:38:23,246 --> 00:38:25,746
to another very important
statement in physics,

702
00:38:25,781 --> 00:38:28,211
and that is that the total
momentum of an object--

703
00:38:28,246 --> 00:38:32,246
which could be a hammer, which
could be a squash racquet--

704
00:38:32,281 --> 00:38:37,211
is the total mass of that object

705
00:38:37,246 --> 00:38:40,246
times the velocity
of the center of mass.

706
00:38:40,281 --> 00:38:44,263
And if I take
the derivative of this,

707
00:38:44,298 --> 00:38:48,211
then dp/dt of
the total momentum--

708
00:38:48,246 --> 00:38:53,246
of which we learned that
the total momentum, dp/dt,

709
00:38:53,281 --> 00:38:57,211
is the total external force,

710
00:38:57,246 --> 00:38:59,211
we already learned
that earlier--

711
00:38:59,246 --> 00:39:02,211
that now is the total mass
of the system

712
00:39:02,246 --> 00:39:05,246
times the acceleration
of the center of mass,

713
00:39:05,281 --> 00:39:08,246
because I take the derivative
of this equation.

714
00:39:08,281 --> 00:39:11,211
That gives me dP/dt here,

715
00:39:11,246 --> 00:39:13,246
and the velocity changes
to acceleration.

716
00:39:13,281 --> 00:39:16,211
And look at this!

717
00:39:16,246 --> 00:39:18,246
This is really an amazing
statement.

718
00:39:18,281 --> 00:39:22,211
This says F = ma.

719
00:39:22,246 --> 00:39:24,246
But if I have here
this squash racquet

720
00:39:24,281 --> 00:39:26,211
and here is the center of mass,

721
00:39:26,246 --> 00:39:29,246
then if I throw this object up
in 26 100, as I will do later,

722
00:39:29,281 --> 00:39:33,211
then the center of mass behaves

723
00:39:33,246 --> 00:39:36,246
as if all the mass
of the entire squash racquet

724
00:39:36,281 --> 00:39:39,211
was right at that
center of mass.

725
00:39:39,246 --> 00:39:42,246
So the behavior of the center of
mass is extremely predictable,

726
00:39:42,281 --> 00:39:46,246
whereas the behavior of
the squash racquet is not.

727
00:39:46,281 --> 00:39:50,211
It may start tumbling.

728
00:39:50,246 --> 00:39:52,211
If the external force on
the squash racquet were zero,

729
00:39:52,246 --> 00:39:56,246
then it would continue to go
always with the same velocity,

730
00:39:56,281 --> 00:40:00,211
the center of mass.

731
00:40:00,246 --> 00:40:02,246
If I had here a hammer--
this is a hammer--

732
00:40:02,281 --> 00:40:06,263
and there were
no external forces--

733
00:40:06,298 --> 00:40:09,272
I would be somewhere
in outer space--

734
00:40:09,307 --> 00:40:12,211
and it would have
a certain velocity.

735
00:40:12,246 --> 00:40:15,246
Then the center of mass,
but only the center of mass,

736
00:40:15,281 --> 00:40:18,246
would have a velocity
that never changes,

737
00:40:18,281 --> 00:40:21,763
because if there's
no external force

738
00:40:21,798 --> 00:40:25,246
then there is only
a constant velocity.

739
00:40:25,281 --> 00:40:28,211
There's no acceleration.

740
00:40:28,246 --> 00:40:31,211
But the hammer itself
may be tumbling.

741
00:40:31,246 --> 00:40:33,246
A little later in time, the
hammer may have this position.

742
00:40:33,281 --> 00:40:36,211
A little earlier in time,

743
00:40:36,246 --> 00:40:38,211
the hammer may have had
this position.

744
00:40:38,246 --> 00:40:40,211
But the center of mass
is just one smooth motion.

745
00:40:40,246 --> 00:40:44,246
That is very mysterious that
there is one and only one point

746
00:40:44,281 --> 00:40:48,246
in any one of us--
in you, in me, in any object--

747
00:40:48,281 --> 00:40:52,211
that the center of mass behaves

748
00:40:52,246 --> 00:40:55,746
as if all the matter
were together in one point.

749
00:40:55,781 --> 00:40:59,246
And so this is another
quite important statement:

750
00:40:59,281 --> 00:41:03,246
For the center of mass,
the total momentum

751
00:41:03,281 --> 00:41:06,211
is the total (no audio)

752
00:41:06,246 --> 00:41:08,211
...the velocity of
the center of mass.

753
00:41:08,246 --> 00:41:11,246
And you take the derivative
of that equation

754
00:41:11,281 --> 00:41:14,211
and you get F = ma.

755
00:41:14,246 --> 00:41:15,246
And that means if
the external force is zero--

756
00:41:15,281 --> 00:41:19,246
you can go back
to the upper line again--

757
00:41:19,281 --> 00:41:21,211
if the external total force
is zero,

758
00:41:21,246 --> 00:41:24,246
then the momentum of the system
is conserved,

759
00:41:24,281 --> 00:41:27,211
and so the center of mass

760
00:41:27,246 --> 00:41:29,246
will then keep
its velocity unchanged.

761
00:41:32,246 --> 00:41:36,246
Let's do one calculation
to give you some experience

762
00:41:36,281 --> 00:41:41,246
on how you derive
the center of mass.

763
00:41:41,281 --> 00:41:43,763
I'll take a simple case.

764
00:41:43,798 --> 00:41:46,211
I won't take a hammer.

765
00:41:46,246 --> 00:41:47,211
It's a little bit complicated.

766
00:41:47,246 --> 00:41:51,246
I will take three masses
which are held together

767
00:41:51,281 --> 00:41:55,246
by very unphysical...
by massless rods, say.

768
00:41:55,281 --> 00:42:00,211
Then I have three point masses

769
00:42:00,246 --> 00:42:03,246
and that makes my life
a little easier.

770
00:42:03,281 --> 00:42:06,246
Let this be the Y axis
and this the X axis,

771
00:42:06,281 --> 00:42:11,211
and here at zero
I have a mass m.

772
00:42:11,246 --> 00:42:14,246
At a distance l,
I have here a mass 2m.

773
00:42:19,246 --> 00:42:21,746
This is also l
and this is also l,

774
00:42:21,781 --> 00:42:24,211
so this is equilateral triangle.

775
00:42:24,246 --> 00:42:26,746
These are massless rods,
and here I have a mass m.

776
00:42:26,781 --> 00:42:29,246
And I'm asking you,
where is the center of mass?

777
00:42:29,281 --> 00:42:34,211
So you have three point masses,

778
00:42:34,246 --> 00:42:37,211
and they are connected
with massless rods.

779
00:42:37,246 --> 00:42:41,246
Well, for those of you who have
a good feeling for symmetry,

780
00:42:41,281 --> 00:42:45,246
they would say certainly it has
to lie somewhere on this line.

781
00:42:45,281 --> 00:42:49,246
And it's probably slanted
in the direction of the 2m,

782
00:42:49,281 --> 00:42:52,211
so it will probably
be somewhere here.

783
00:42:52,246 --> 00:42:56,246
And so this will be the position
vector r center of mass,

784
00:42:56,281 --> 00:43:00,263
and the individual
position vectors

785
00:43:00,298 --> 00:43:04,211
from the origin here
will be this,

786
00:43:04,246 --> 00:43:07,246
and this will be a position
vector to this object,

787
00:43:07,281 --> 00:43:10,246
and the position vector
to this object is zero.

788
00:43:10,281 --> 00:43:14,211
And so now we can calculate

789
00:43:14,246 --> 00:43:16,246
the position of the center
of mass as follows.

790
00:43:16,281 --> 00:43:20,211
We know that the total mass,

791
00:43:20,246 --> 00:43:22,246
which is 4m times the position
vector, r center of mass--

792
00:43:22,281 --> 00:43:26,263
I go all the way up there
on the blackboard;

793
00:43:26,298 --> 00:43:30,272
there's my definition
for center of mass--

794
00:43:30,307 --> 00:43:34,246
equals the sum
of the individual masses

795
00:43:34,281 --> 00:43:37,211
times their position vectors.

796
00:43:37,246 --> 00:43:40,246
So it is the sum of i
m of i times r i.

797
00:43:40,281 --> 00:43:44,246
And this i goes
from one to three.

798
00:43:48,246 --> 00:43:49,211
Now, this is
a vectorial equation,

799
00:43:49,246 --> 00:43:51,246
and whenever we have a vectorial
equation, it sometimes pays off

800
00:43:51,281 --> 00:43:56,246
to split it
into two components--

801
00:43:56,281 --> 00:43:57,763
a y and to an x component.

802
00:43:57,798 --> 00:43:59,211
And so in the x direction...

803
00:43:59,246 --> 00:44:01,246
of course,
the same equation holds

804
00:44:01,281 --> 00:44:03,246
for the x component
of these vectors.

805
00:44:03,281 --> 00:44:06,211
So now I have that 4m

806
00:44:06,246 --> 00:44:08,246
times the x component
of the center of mass

807
00:44:08,281 --> 00:44:12,211
equals this mass

808
00:44:12,246 --> 00:44:13,246
times the x component
of its position vector,

809
00:44:13,281 --> 00:44:17,211
which is zero, plus this mass,

810
00:44:17,246 --> 00:44:21,246
which is 2m times
the x position, which is l--

811
00:44:21,281 --> 00:44:27,211
so plus 2m times l--
plus this mass

812
00:44:27,246 --> 00:44:31,246
times the x component of this
mass, which is one-half l.

813
00:44:31,281 --> 00:44:35,763
So that gives me
plus one-half m times l.

814
00:44:35,798 --> 00:44:40,246
My m goes and so I get
that x center of mass

815
00:44:40,281 --> 00:44:46,263
equals 2½ divided by four.

816
00:44:46,298 --> 00:44:52,211
That is five-eighths l.

817
00:44:52,246 --> 00:44:55,211
So we were not too far off
where we put it.

818
00:44:55,246 --> 00:44:59,246
Now, in the y direction,
you can do exactly the same.

819
00:44:59,281 --> 00:45:03,211
You can split it up
into the position vector

820
00:45:03,246 --> 00:45:06,211
of this object, which is
one-half l square root three.

821
00:45:06,246 --> 00:45:11,246
This one has no y component
and this one has no y component,

822
00:45:11,281 --> 00:45:16,211
so this is very easy.

823
00:45:16,246 --> 00:45:17,246
So you're going to get that 4m
times y of the center of mass

824
00:45:17,281 --> 00:45:24,211
equals this mass m

825
00:45:24,246 --> 00:45:27,746
times the y component
of that position vector,

826
00:45:27,781 --> 00:45:31,211
and that is one-half l
square root of three.

827
00:45:31,246 --> 00:45:36,246
And so you lose your m, and so
you see that y center of mass

828
00:45:36,281 --> 00:45:41,246
then becomes the square root of
three divided by eight times l.

829
00:45:41,281 --> 00:45:47,246
And I think that's about 0.22L,
very roughly.

830
00:45:47,281 --> 00:45:51,211
Yes.

831
00:45:51,246 --> 00:45:52,246
And so you see that
we didn't put it in so badly.

832
00:45:52,281 --> 00:45:56,211
It's about one-fifth
of this distance.

833
00:45:56,246 --> 00:45:58,746
It's about one-fifth higher
than this distance.

834
00:45:58,781 --> 00:46:01,211
And so you can calculate
the center of mass.

835
00:46:01,246 --> 00:46:04,246
That's really not too hard,
if you have discrete points.

836
00:46:04,281 --> 00:46:07,246
If you have a car or if you have
an object like this,

837
00:46:07,281 --> 00:46:10,211
then it is, of course,

838
00:46:10,246 --> 00:46:12,246
much harder to calculate
the center of mass.

839
00:46:12,281 --> 00:46:14,246
I will teach you in a few weeks
a very easy way

840
00:46:14,281 --> 00:46:17,211
to determine experimentally

841
00:46:17,246 --> 00:46:19,746
where the center of mass
is located--

842
00:46:19,781 --> 00:46:22,013
experimentally,
which is different

843
00:46:22,048 --> 00:46:24,246
from calculating
it analytically,

844
00:46:24,281 --> 00:46:26,211
as we just did.

845
00:46:26,246 --> 00:46:27,246
So I mentioned to you that
the motion of the center of mass

846
00:46:27,281 --> 00:46:33,211
is very uniform in the absence
of external forces,

847
00:46:33,246 --> 00:46:37,246
and that I can demonstrate for
you again with the air track.

848
00:46:37,281 --> 00:46:41,263
We have a system here
of two cars

849
00:46:41,298 --> 00:46:44,772
which I connected by a spring.

850
00:46:44,807 --> 00:46:48,211
I will turn on the air shortly

851
00:46:48,246 --> 00:46:51,211
because the air makes
a lot of noise.

852
00:46:51,246 --> 00:46:54,246
This is... these are two cars
connected by a spring,

853
00:46:54,281 --> 00:46:59,246
and I will give these cars
a certain motion

854
00:46:59,281 --> 00:47:02,263
and they will go
in this direction.

855
00:47:02,298 --> 00:47:05,272
And they will oscillate
in a weird way

856
00:47:05,307 --> 00:47:08,246
because they are connected
with a spring

857
00:47:08,281 --> 00:47:12,211
and I keep them connected.

858
00:47:12,246 --> 00:47:14,211
And it will be nearly impossible

859
00:47:14,246 --> 00:47:16,246
for us to evaluate the motion
of these two cars individually.

860
00:47:16,281 --> 00:47:20,246
But if I give the whole system
a certain velocity

861
00:47:20,281 --> 00:47:23,211
and then they go like this,

862
00:47:23,246 --> 00:47:24,246
and they keep going like this
and making crazy things,

863
00:47:24,281 --> 00:47:27,246
momentum of the center of mass
will not change--

864
00:47:27,281 --> 00:47:31,211
only of the center of mass.

865
00:47:31,246 --> 00:47:33,211
Not of this car,
not of that car.

866
00:47:33,246 --> 00:47:35,746
That's the uniqueness
of the center of mass.

867
00:47:35,781 --> 00:47:38,246
And so the center of mass
will just laugh at us

868
00:47:38,281 --> 00:47:42,211
and ignore all these motions

869
00:47:42,246 --> 00:47:44,746
and will travel at
a constant speed very nicely.

870
00:47:44,781 --> 00:47:47,246
So if you concentrate
on that little object,

871
00:47:47,281 --> 00:47:50,211
you may be able to see that.

872
00:47:50,246 --> 00:47:52,246
You may need a lot of
imagination to see that,

873
00:47:52,281 --> 00:47:55,211
because you're going
to be distracted

874
00:47:55,246 --> 00:47:58,246
by the weird motion
of the other two objects.

875
00:47:58,281 --> 00:48:00,763
This is the center of mass.

876
00:48:00,798 --> 00:48:03,211
It's right in the middle.

877
00:48:03,246 --> 00:48:04,246
The objects have the same mass.

878
00:48:06,246 --> 00:48:08,246
There we go.

879
00:48:09,246 --> 00:48:16,246
You see how complicated
this motion is

880
00:48:16,281 --> 00:48:19,211
of the individual cars?

881
00:48:19,246 --> 00:48:20,246
Can you see that the center of
mass is moving very uniformly,

882
00:48:20,281 --> 00:48:24,211
or can you not see that?

883
00:48:24,246 --> 00:48:26,246
Oh, you think you can see that?

884
00:48:26,281 --> 00:48:28,246
You have a lot of imagination.

885
00:48:28,281 --> 00:48:31,211
But I will help you.

886
00:48:31,246 --> 00:48:32,246
I will turn on...
I will turn off all the lights

887
00:48:32,281 --> 00:48:36,263
and only turn on
ultraviolet light.

888
00:48:36,298 --> 00:48:40,211
And ultraviolet light
will interact

889
00:48:40,246 --> 00:48:43,246
with that little ball,
the center of mass,

890
00:48:43,281 --> 00:48:48,211
and when I then make it dark,

891
00:48:48,246 --> 00:48:51,246
you will only see the motion
of the center of mass.

892
00:48:51,281 --> 00:48:56,211
And then you can really see

893
00:48:56,246 --> 00:48:58,246
that the center of mass moves
in a very civilized way.

894
00:48:58,281 --> 00:49:03,246
I'll bring it out here again.

895
00:49:05,246 --> 00:49:12,211
Okay.

896
00:49:12,246 --> 00:49:13,246
So now I'm going to help you
to concentrate.

897
00:49:13,281 --> 00:49:15,763
Are the timers off?

898
00:49:15,798 --> 00:49:18,246
Yes, timers are off?

899
00:49:19,246 --> 00:49:21,211
Okay, dark.

900
00:49:21,246 --> 00:49:22,246
Let your eyes get used
to the darkness.

901
00:49:27,246 --> 00:49:31,746
Okay, I'm going to do
the same thing,

902
00:49:31,781 --> 00:49:36,211
and now look at
that center of mass.

903
00:49:36,246 --> 00:49:39,211
And we know that these cars
are doing crazy things.

904
00:49:39,246 --> 00:49:43,246
Hard to predict but the center
of mass behaves decently.

905
00:49:43,281 --> 00:49:50,211
Beautifully!

906
00:49:50,246 --> 00:49:51,211
Constant velocity.

907
00:49:51,246 --> 00:49:53,246
I can let it go backwards,
so you can enjoy this once more.

908
00:49:58,246 --> 00:50:06,211
Center of mass motion,

909
00:50:06,246 --> 00:50:09,246
in the absence of external
forces has a constant velocity.

910
00:50:09,281 --> 00:50:17,211
When I throw up a hammer,

911
00:50:17,246 --> 00:50:21,746
then the hammer will do
very weird things.

912
00:50:21,781 --> 00:50:26,211
The hammer will start
to tumble and rotate,

913
00:50:26,246 --> 00:50:29,246
but the center of mass will
behave in a civilized way.

914
00:50:29,281 --> 00:50:34,211
If I throw up a hammer,

915
00:50:34,246 --> 00:50:36,246
then the center of mass
and only the center of mass

916
00:50:36,281 --> 00:50:39,763
will just go along
a perfect parabola

917
00:50:39,798 --> 00:50:43,246
as if it were just
a tennis ball.

918
00:50:43,281 --> 00:50:46,211
Now at one point...

919
00:50:46,246 --> 00:50:47,211
I will do it
with a squash racquet--

920
00:50:47,246 --> 00:50:50,211
at one point the squash racquet
may be like this,

921
00:50:50,246 --> 00:50:53,246
and at another point, the squash
racquet may be like this,

922
00:50:53,281 --> 00:50:56,263
and a little later,
it may be like this,

923
00:50:56,298 --> 00:50:59,246
but the center of mass
of the squash racquet

924
00:50:59,281 --> 00:51:01,763
will perfectly go
along a parabola.

925
00:51:01,798 --> 00:51:04,522
And so we have here
a squash racquet

926
00:51:04,557 --> 00:51:07,246
and we have here
the center of mass.

927
00:51:07,281 --> 00:51:09,211
I have also here a regular...

928
00:51:09,246 --> 00:51:11,246
well, it's not quite
a tennis ball, but close enough.

929
00:51:11,281 --> 00:51:15,211
This one, you would expect it

930
00:51:15,246 --> 00:51:17,246
to behave perfectly
like a parabola.

931
00:51:17,281 --> 00:51:19,246
From this one,
you would not expect it

932
00:51:19,281 --> 00:51:22,211
to behave like a parabola.

933
00:51:22,246 --> 00:51:23,211
Let me throw this one up
in light,

934
00:51:23,246 --> 00:51:26,246
and you will see that it has
very strange motion.

935
00:51:29,246 --> 00:51:31,211
If I show the whole thing in UV,

936
00:51:31,246 --> 00:51:33,246
then you will see the same kind
of beautiful parabola

937
00:51:33,281 --> 00:51:36,246
as you would see
with this ball--

938
00:51:36,281 --> 00:51:37,246
something like this.

939
00:51:38,246 --> 00:51:41,246
Forget the fact
that it lights up.

940
00:51:41,281 --> 00:51:44,246
You remember it was
the last lecture

941
00:51:44,281 --> 00:51:46,211
that I wanted to remind you of.

942
00:51:46,246 --> 00:51:48,211
So now we're going
to turn off the lights

943
00:51:48,246 --> 00:51:51,246
and I want to show you the
motion of the center of mass.

944
00:51:51,281 --> 00:51:55,246
You see the center of mass here?

945
00:51:55,281 --> 00:51:58,263
Can you all see it?

946
00:51:58,298 --> 00:52:01,246
Okay, there we go.

947
00:52:01,281 --> 00:52:04,211
You ready?

948
00:52:04,246 --> 00:52:05,246
Concentrate only
on the center of mass.

949
00:52:07,246 --> 00:52:09,246
Nice parabola or not?

950
00:52:09,281 --> 00:52:11,246
I'll do it again.

951
00:52:13,246 --> 00:52:16,246
You see the center of mass?

952
00:52:16,281 --> 00:52:17,211
Can you see it?

953
00:52:17,246 --> 00:52:18,246
You can still see it, right?

954
00:52:20,246 --> 00:52:22,246
Wonderful parabola for me!

955
00:52:22,281 --> 00:52:24,211
All right.

956
00:52:24,246 --> 00:52:25,246
Enjoy the presence
of your parents.

957
00:52:25,281 --> 00:52:28,246
Have a good weekend.

958
00:52:28,281 --> 00:52:29,246
See you Monday.