1
00:00:03,450 --> 00:00:06,150
Let's examine the
velocities of objects

2
00:00:06,150 --> 00:00:08,260
in the center of
mass reference frame.

3
00:00:08,260 --> 00:00:11,240
So we have object
1 and object 2.

4
00:00:11,240 --> 00:00:12,790
And we have CM.

5
00:00:12,790 --> 00:00:15,070
And we have r1 prime.

6
00:00:15,070 --> 00:00:16,920
And we have r2 prime.

7
00:00:16,920 --> 00:00:21,060
And recall that we found that
the position vector, r1 prime,

8
00:00:21,060 --> 00:00:23,880
was equal to the
reduced mass over mu

9
00:00:23,880 --> 00:00:28,050
m1 times the relative
position to r1, 2.

10
00:00:28,050 --> 00:00:32,640
I'll draw that up here,
r1, 2 from 2 to 1.

11
00:00:32,640 --> 00:00:36,930
And just to remind
you, r1 minus r2.

12
00:00:36,930 --> 00:00:40,440
And likewise the
position vector r2 prime

13
00:00:40,440 --> 00:00:45,210
is minus mu m2 over r1, 2.

14
00:00:45,210 --> 00:00:47,940
Well, now we can
differentiate the positions

15
00:00:47,940 --> 00:00:49,650
in the center of mass frame.

16
00:00:49,650 --> 00:00:52,780
And we find that
the V1, which is,

17
00:00:52,780 --> 00:00:55,350
of course, the derivative
of the position,

18
00:00:55,350 --> 00:01:00,420
is just mu over m1 V1, 2.

19
00:01:00,420 --> 00:01:09,450
And V2 prime is equal to mu
over m2 with a minus sign V1, 2.

20
00:01:09,450 --> 00:01:11,730
Now the significance
of this result

21
00:01:11,730 --> 00:01:14,850
is that the velocity
of the object

22
00:01:14,850 --> 00:01:20,070
in the center of mass frame is
equal to some constant times

23
00:01:20,070 --> 00:01:22,440
the relative velocity.

24
00:01:22,440 --> 00:01:26,430
Likewise, velocity V2 prime
in the center of mass frame

25
00:01:26,430 --> 00:01:29,730
is also proportional
to a constant.

26
00:01:29,730 --> 00:01:32,770
And it's in minus the
relative velocities.

27
00:01:32,770 --> 00:01:35,039
So the significance
of this result

28
00:01:35,039 --> 00:01:40,020
is that when we looked at
an elastic one-dimensional

29
00:01:40,020 --> 00:01:43,380
collision, we already
have the result

30
00:01:43,380 --> 00:01:48,060
that the relative
velocities, V1, 2 initial,

31
00:01:48,060 --> 00:01:52,800
is minus the final
relative velocity.

32
00:01:52,800 --> 00:01:55,770
And we also know
from what we saw

33
00:01:55,770 --> 00:02:01,390
that the relative velocity is
independent of the reference

34
00:02:01,390 --> 00:02:04,110
frame.

35
00:02:04,110 --> 00:02:09,389
So this condition
about relative velocity

36
00:02:09,389 --> 00:02:11,340
is true in any reference frame.

37
00:02:11,340 --> 00:02:14,760
And in particular, it's true
in the center of mass frame.

38
00:02:14,760 --> 00:02:18,920
So what that shows us is
that these conditions,

39
00:02:18,920 --> 00:02:23,070
this statement, is
that the magnitude

40
00:02:23,070 --> 00:02:27,770
of the relative
velocity is constant.

41
00:02:32,370 --> 00:02:37,710
And the conclusion is because
in the center of mass frame,

42
00:02:37,710 --> 00:02:43,560
V1 prime is proportional
to the relative velocity.

43
00:02:43,560 --> 00:02:47,910
That tells us that in a
center of mass reference frame

44
00:02:47,910 --> 00:02:52,079
that the initial
velocity of object

45
00:02:52,079 --> 00:02:59,160
1 in the center of mass frame
is equal to the final velocity

46
00:02:59,160 --> 00:03:00,960
of object 1.

47
00:03:00,960 --> 00:03:06,930
So the quantity, the speed
of objects 1 and objects 2

48
00:03:06,930 --> 00:03:10,680
in an elastic collision in the
center of mass reference frame

49
00:03:10,680 --> 00:03:12,870
do not change magnitude.

50
00:03:12,870 --> 00:03:15,690
We can call this
quantity V2 prime

51
00:03:15,690 --> 00:03:19,350
and likewise V
prime 2 initial is

52
00:03:19,350 --> 00:03:24,360
equal to V2 final
prime V2 prime.

53
00:03:24,360 --> 00:03:27,660
And this will make our
analysis of collisions

54
00:03:27,660 --> 00:03:31,320
in the center of mass
reference frame very easy.

55
00:03:31,320 --> 00:03:33,300
Two objects, when
they're colliding

56
00:03:33,300 --> 00:03:35,090
in the center of
mass reference frame,

57
00:03:35,090 --> 00:03:37,870
their speeds do not change.

58
00:03:37,870 --> 00:03:41,090
Only the direction of
their velocities change.