1
00:00:03,800 --> 00:00:06,170
Now when we did
these calculation,

2
00:00:06,170 --> 00:00:11,990
we saw that the kinetic
energy in the ground frame

3
00:00:11,990 --> 00:00:16,190
was equal to the kinetic energy
in the center of mass frame

4
00:00:16,190 --> 00:00:22,460
plus a constant term, m1
plus m2 times Vcm squared.

5
00:00:22,460 --> 00:00:25,340
And this is a constant
whether you're

6
00:00:25,340 --> 00:00:27,140
in the initial or
the final picture.

7
00:00:27,140 --> 00:00:30,560
So if we look at the
change in kinetic energy

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00:00:30,560 --> 00:00:33,430
in the ground frame,
that's equal to the change

9
00:00:33,430 --> 00:00:35,910
in kinetic energy in the
center of mass frame.

10
00:00:35,910 --> 00:00:38,180
But this term is a constant.

11
00:00:38,180 --> 00:00:41,090
So there's no
change in that term.

12
00:00:41,090 --> 00:00:44,450
And so we have a crucial
result that the change

13
00:00:44,450 --> 00:00:46,130
in kinetic energy
in the ground frame

14
00:00:46,130 --> 00:00:48,680
is equal to the change
in kinetic energy

15
00:00:48,680 --> 00:00:50,330
in the center of mass frame.

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00:00:50,330 --> 00:00:53,150
Now let's look at an example
of a collision that's

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00:00:53,150 --> 00:00:56,160
completely inelastic.

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00:00:56,160 --> 00:00:58,250
So we have object one.

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00:00:58,250 --> 00:01:02,050
And it's coming in
with V1 initial.

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00:01:02,050 --> 00:01:05,750
And object 2, let's
make it at rest.

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00:01:05,750 --> 00:01:10,640
And afterwards object
1 and 2 stick together.

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00:01:10,640 --> 00:01:13,880
And they go off with V final.

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00:01:13,880 --> 00:01:18,830
So here's a collision
in an inelastic frame.

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00:01:18,830 --> 00:01:23,630
Now in the center of mass frame,
the two objects-- so here's

25
00:01:23,630 --> 00:01:27,450
our lab or ground frame.

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00:01:27,450 --> 00:01:30,080
I can call it the lab frame
because that one is at rest.

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00:01:30,080 --> 00:01:31,480
And this is the cm frame.

28
00:01:34,780 --> 00:01:38,604
And there we have V1
prime and V2 prime.

29
00:01:41,670 --> 00:01:42,930
They're both moving.

30
00:01:42,930 --> 00:01:47,789
And now in the
center of mass frame,

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00:01:47,789 --> 00:01:51,840
they're at rest at the
end of the collision.

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00:01:51,840 --> 00:01:57,240
So in this collision,
we can analyze it.

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00:01:57,240 --> 00:02:00,740
And now what I want to
show is that the change

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00:02:00,740 --> 00:02:05,850
in kinetic energy, recall
from our previous calculation,

35
00:02:05,850 --> 00:02:09,478
that the kinetic energy in
the center of mass frame

36
00:02:09,478 --> 00:02:19,840
was equal to 1/2 mu times the
relative velocity squared.

37
00:02:19,840 --> 00:02:25,710
So remember V1, 2 was
equal to V1 minus V2.

38
00:02:25,710 --> 00:02:28,800
And so the change
in kinetic energy

39
00:02:28,800 --> 00:02:32,760
in the center of mass frame
is the final kinetic energy

40
00:02:32,760 --> 00:02:40,020
is 0 minus 1/2 the kinetic
energy in the initial state.

41
00:02:40,020 --> 00:02:42,450
And what is this
quantity initial?

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00:02:42,450 --> 00:02:45,990
Well, this was a reference
frame independent concept.

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00:02:45,990 --> 00:02:48,490
So we can just look
at the lab frame.

44
00:02:48,490 --> 00:02:52,470
And right this as
V1 initial i hat

45
00:02:52,470 --> 00:02:55,960
where that's i hat because
this object is at rest.

46
00:02:55,960 --> 00:02:59,310
And so right away we have that
the change in kinetic energy

47
00:02:59,310 --> 00:03:05,430
is minus 1/2-- now remember
mu was m1 m2 over m1 plus m2

48
00:03:05,430 --> 00:03:08,490
times V1 initial mean squared.

49
00:03:08,490 --> 00:03:12,390
And that's the same change in
kinetic energy as the ground

50
00:03:12,390 --> 00:03:13,620
frame.

51
00:03:13,620 --> 00:03:18,480
So by using our result
here and our result there,

52
00:03:18,480 --> 00:03:21,120
we were able to
very quickly assess

53
00:03:21,120 --> 00:03:25,177
how much kinetic energy is lost
in this inelastic collision.

54
00:03:25,177 --> 00:03:26,760
And where does that
kinetic energy go?

55
00:03:26,760 --> 00:03:29,940
It can go to a variety of
places, deformation of objects.

56
00:03:29,940 --> 00:03:31,380
It can go into sound.

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00:03:31,380 --> 00:03:40,660
It can go into
thermal energy that

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00:03:40,660 --> 00:03:42,520
diffuses into the environment.

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00:03:42,520 --> 00:03:48,240
However, that's how
much is changed.