1
00:00:00,415 --> 00:00:02,238
- [Instructor] So if you
powered through the last video,

2
00:00:02,238 --> 00:00:04,839
you saw that these
elastic collision problems

3
00:00:04,839 --> 00:00:08,391
can get pretty nasty, the
algebra gets pretty ugly.

4
00:00:08,391 --> 00:00:09,862
What we did, if you missed it,

5
00:00:09,862 --> 00:00:11,314
maybe you skipped right to this easy one,

6
00:00:11,314 --> 00:00:13,195
and that's cool with
me, but what you missed,

7
00:00:13,195 --> 00:00:14,867
or what you saw if you did watch it,

8
00:00:14,867 --> 00:00:16,608
is that we used conservation of momentum,

9
00:00:16,608 --> 00:00:18,884
but we had two unknown final velocities.

10
00:00:18,884 --> 00:00:20,568
We didn't know the
velocity of either object

11
00:00:20,568 --> 00:00:22,634
after the collision, so we
had to solve this expression

12
00:00:22,634 --> 00:00:24,945
for one of the velocities,
and then plug that

13
00:00:24,945 --> 00:00:27,801
into conservation of kinetic
energy, which we can do,

14
00:00:27,801 --> 00:00:30,552
because kinetic energy's conserved
for an elastic collision.

15
00:00:30,552 --> 00:00:32,259
But since we square this expression,

16
00:00:32,259 --> 00:00:35,835
it gets big and ugly, and it
gets multiplied by other stuff.

17
00:00:35,835 --> 00:00:37,691
And now you have to combine terms,

18
00:00:37,691 --> 00:00:39,445
and you end up just having
to pray that you didn't

19
00:00:39,445 --> 00:00:42,114
just make an algebra error,
or lose a sign in here.

20
00:00:42,114 --> 00:00:44,774
And even if you do get lucky
and the whole thing works out,

21
00:00:44,774 --> 00:00:47,407
and you get an answer,
if you rounded at all

22
00:00:47,407 --> 00:00:49,488
during this process, your answer's going

23
00:00:49,488 --> 00:00:51,007
to be off by a little bit.

24
00:00:51,007 --> 00:00:53,273
So the obvious question
is, is there a simpler way

25
00:00:53,273 --> 00:00:55,688
of solving these elastic
collision problems,

26
00:00:55,688 --> 00:00:58,561
where you've got two
unknown final velocities?

27
00:00:58,561 --> 00:01:00,523
And there is a simpler way,
and that's what I'm going

28
00:01:00,523 --> 00:01:01,858
to show you in this video.

29
00:01:01,858 --> 00:01:04,400
So to derive this simpler
way of solving the problem,

30
00:01:04,400 --> 00:01:05,991
we're going to do what
we always do in physics

31
00:01:05,991 --> 00:01:08,731
to derive a nice result, we're going to,

32
00:01:08,731 --> 00:01:11,691
instead of solving this problem
numerically, with numbers,

33
00:01:11,691 --> 00:01:14,629
we're going to solve it
symbolically, with symbols.

34
00:01:14,629 --> 00:01:16,080
And what I mean by that
is, instead of calling

35
00:01:16,080 --> 00:01:19,737
the mass of the golf ball .045 kilograms,

36
00:01:19,737 --> 00:01:21,734
let's just say this is
any particular mass,

37
00:01:21,734 --> 00:01:24,912
and we'll just call it mg,
for mass of the golf ball.

38
00:01:24,912 --> 00:01:26,235
And you might be like, "Well,
how's that going to help?

39
00:01:26,235 --> 00:01:27,768
"We're going to have a
bunch of variables in here,

40
00:01:27,768 --> 00:01:29,786
"instead of numbers, it's
still going to be a mess."

41
00:01:29,786 --> 00:01:30,833
Still going to get a little messy,

42
00:01:30,833 --> 00:01:33,500
but what this does, when you
solve problems symbolically,

43
00:01:33,500 --> 00:01:37,569
it often times allows you
to see patterns, symmetries,

44
00:01:37,569 --> 00:01:40,750
cancellations that are happening
here in your calculation

45
00:01:40,750 --> 00:01:42,375
that aren't so obvious when there's

46
00:01:42,375 --> 00:01:43,839
a bunch of numbers around.

47
00:01:43,839 --> 00:01:45,975
When it's a bunch of numbers,
it just looks like a big mess.

48
00:01:45,975 --> 00:01:47,777
When you've got symbolic
expressions in here,

49
00:01:47,777 --> 00:01:50,110
sometimes something magical happens,

50
00:01:50,110 --> 00:01:51,747
and that's what's going to happen here,

51
00:01:51,747 --> 00:01:54,348
and it's going to give us a
result that's way simpler.

52
00:01:54,348 --> 00:01:56,646
So let's do this, let's solve
this problem symbolically.

53
00:01:56,646 --> 00:01:57,992
We're going to get rid
of all these numbers,

54
00:01:57,992 --> 00:01:59,954
and we're going to turn these variables,

55
00:01:59,954 --> 00:02:02,091
instead of calling it 40 meters per second

56
00:02:02,091 --> 00:02:04,424
for the initial velocity
of the tennis ball,

57
00:02:04,424 --> 00:02:05,492
we're going to leave it symbolic.

58
00:02:05,492 --> 00:02:07,907
So instead of giving a number,
we're going to call this,

59
00:02:07,907 --> 00:02:10,031
I'll call it vt, for the tennis ball,

60
00:02:10,031 --> 00:02:11,785
then I'll write i, for initial.

61
00:02:11,785 --> 00:02:14,211
So this is the initial
velocity of the tennis ball,

62
00:02:14,211 --> 00:02:16,568
and we'll do the same
thing for the golf ball.

63
00:02:16,568 --> 00:02:19,088
Instead of calling it 50
meters per second to the left,

64
00:02:19,088 --> 00:02:23,639
we'll call it vgi, the initial
velocity of the golf ball.

65
00:02:23,639 --> 00:02:26,715
So when I wrote 50 before, I
meant the size of the velocity.

66
00:02:26,715 --> 00:02:29,722
But this time, when I write
vg, I mean the velocity.

67
00:02:29,722 --> 00:02:31,870
In other words, this
vgi might be negative.

68
00:02:31,870 --> 00:02:33,727
In fact, if the golf
ball's going leftward,

69
00:02:33,727 --> 00:02:35,440
it's going to be a negative number.

70
00:02:35,440 --> 00:02:36,863
But that's okay, it's symbolic.

71
00:02:36,863 --> 00:02:39,417
We're going to treat
this vgi as the velocity,

72
00:02:39,417 --> 00:02:40,972
so it could be a negative number,

73
00:02:40,972 --> 00:02:42,133
or it could be a positive number

74
00:02:42,133 --> 00:02:43,759
if the golf ball were going to the right.

75
00:02:43,759 --> 00:02:45,280
We're going to do the
same thing for the masses.

76
00:02:45,280 --> 00:02:47,047
We already wrote the mass
of the golf ball's mg,

77
00:02:47,047 --> 00:02:51,402
now we're going to write the
mass of the tennis ball as mt.

78
00:02:51,402 --> 00:02:53,560
And now we're gong to solve,
same way we did before,

79
00:02:53,560 --> 00:02:55,373
we're going to use
conservation of momentum.

80
00:02:55,373 --> 00:02:58,542
So our initial momentum would
be, mass of the tennis ball

81
00:02:58,542 --> 00:03:01,862
times the initial velocity
of the tennis ball, vti.

82
00:03:01,862 --> 00:03:04,855
And then plus mass of the golf ball,

83
00:03:04,855 --> 00:03:08,373
times the initial velocity
of the golf ball, vgi.

84
00:03:08,373 --> 00:03:09,337
And you might be like, "Wait, this should

85
00:03:09,337 --> 00:03:10,916
"be a negative sign, right?"

86
00:03:10,916 --> 00:03:12,773
No, I know this golf ball's going left,

87
00:03:12,773 --> 00:03:15,606
but I'm letting this vgi be the velocity.

88
00:03:15,606 --> 00:03:17,708
So there's a hidden negative sign in here.

89
00:03:17,708 --> 00:03:20,784
If it's going leftward,
this vgi would equal

90
00:03:20,784 --> 00:03:22,677
some negative number, in other words,

91
00:03:22,677 --> 00:03:24,277
so I wouldn't want to
put another negative,

92
00:03:24,277 --> 00:03:26,856
or I'd be canceling off the
negative that would be in here,

93
00:03:26,856 --> 00:03:28,226
that's why I get to write a plus here.

94
00:03:28,226 --> 00:03:30,606
And this is going to
equal the final momentum

95
00:03:30,606 --> 00:03:35,274
of the tennis ball, mt times
vt final, plus the final

96
00:03:35,274 --> 00:03:39,742
momentum of the golf
ball, mg, times vg final.

97
00:03:39,742 --> 00:03:41,543
And it's good to keep
track of your unknowns.

98
00:03:41,543 --> 00:03:43,668
Right now, I don't know
the final velocities.

99
00:03:43,668 --> 00:03:45,595
I'm given all these initial values

100
00:03:45,595 --> 00:03:47,592
of the velocity, and the masses.

101
00:03:47,592 --> 00:03:49,972
So those we can consider "given" up here.

102
00:03:49,972 --> 00:03:52,375
The things I don't know
are the final velocities.

103
00:03:52,375 --> 00:03:55,080
So again, just like before,
when we did this numerically,

104
00:03:55,080 --> 00:03:57,147
I can't solve for either one of these,

105
00:03:57,147 --> 00:03:58,192
because there's two of them.

106
00:03:58,192 --> 00:03:59,422
So often times, what you do

107
00:03:59,422 --> 00:04:01,301
when you're solving problems symbolically,

108
00:04:01,301 --> 00:04:03,161
since you're going to want to
clean them up at some point,

109
00:04:03,161 --> 00:04:05,378
you try to look any ways
that you can simplify.

110
00:04:05,378 --> 00:04:06,969
There aren't many ways we can simplify,

111
00:04:06,969 --> 00:04:09,766
but I could bring this
mt term over to the left,

112
00:04:09,766 --> 00:04:12,320
and bring this mg term over to the right,

113
00:04:12,320 --> 00:04:15,228
so I could write this as mt times vti,

114
00:04:15,228 --> 00:04:17,228
minus mt times vt final.

115
00:04:18,507 --> 00:04:20,283
And similarly, on the right hand side,

116
00:04:20,283 --> 00:04:23,270
I'd get mg, vg final, and then

117
00:04:23,270 --> 00:04:26,044
I'm subtracting this term from both sides.

118
00:04:26,044 --> 00:04:29,284
So, I'd get minus mg vg initial,

119
00:04:29,284 --> 00:04:31,537
and you notice, we can
pull out a common factor.

120
00:04:31,537 --> 00:04:33,615
You might be like, "Why
are we doing this?"

121
00:04:33,615 --> 00:04:35,134
Well, if I was doing
this for the first time,

122
00:04:35,134 --> 00:04:37,249
I wouldn't necessarily know, either.

123
00:04:37,249 --> 00:04:38,828
But it's often good practice to try

124
00:04:38,828 --> 00:04:40,488
to simplify as much as possible.

125
00:04:40,488 --> 00:04:42,671
And in this case, this
is going to be crucial.

126
00:04:42,671 --> 00:04:44,343
This is going to be an important step

127
00:04:44,343 --> 00:04:47,105
to trying to simplify this entire process.

128
00:04:47,105 --> 00:04:49,321
Right now, I can see how
that wouldn't be obvious,

129
00:04:49,321 --> 00:04:51,169
but you got to just
trust in me for a minute.

130
00:04:51,169 --> 00:04:52,307
We're going to want to do this,

131
00:04:52,307 --> 00:04:55,024
because it'll make our lives
much better here in a second.

132
00:04:55,024 --> 00:04:56,241
And on the right hand
side, I'll write it as

133
00:04:56,241 --> 00:04:59,241
mg times vg final, minus vg initial.

134
00:05:00,666 --> 00:05:02,593
So just so you're keeping track,
the variables I don't know

135
00:05:02,593 --> 00:05:06,076
are this vt final, and this vg final.

136
00:05:06,076 --> 00:05:07,882
So I'm still stuck on
this right hand side.

137
00:05:07,882 --> 00:05:10,344
It looks a little nicer, because
I've got terms grouped up,

138
00:05:10,344 --> 00:05:12,596
but I'm still, if this
collision's elastic,

139
00:05:12,596 --> 00:05:15,926
going to have to use
conservation of kinetic energy.

140
00:05:15,926 --> 00:05:17,087
So, we'll do this over here.

141
00:05:17,087 --> 00:05:19,225
So if I take the total
initial kinetic energy,

142
00:05:19,225 --> 00:05:22,023
and I set that equal to the
total final kinetic energy,

143
00:05:22,023 --> 00:05:24,763
I'll have 1/2 mass of the tennis ball,

144
00:05:24,763 --> 00:05:28,582
vti, the initial velocity
of the tennis ball, squared,

145
00:05:28,582 --> 00:05:30,557
so it's really just the initial
speed of the tennis ball,

146
00:05:30,557 --> 00:05:34,051
squared, plus the initial
kinetic energy of the golf ball,

147
00:05:34,051 --> 00:05:38,033
which would be 1/2 mg, vgi squared.

148
00:05:38,033 --> 00:05:40,646
And this is going to have to
equal the final kinetic energy

149
00:05:40,646 --> 00:05:42,037
of all the objects, so I'll have

150
00:05:42,037 --> 00:05:44,291
a 1/2 mass of the tennis ball,

151
00:05:44,291 --> 00:05:46,740
final velocity of the
tennis ball, squared,

152
00:05:46,740 --> 00:05:49,971
plus 1/2 mass of the golf ball,

153
00:05:49,971 --> 00:05:52,344
final velocity of the golf ball squared.

154
00:05:52,344 --> 00:05:54,074
And again, we're doing
this to clean this up.

155
00:05:54,074 --> 00:05:55,722
We want to get a nice
expression at the end,

156
00:05:55,722 --> 00:05:56,918
so I'm going to cancel some terms.

157
00:05:56,918 --> 00:05:58,416
Look at, 1/2s in everything.

158
00:05:58,416 --> 00:06:00,830
So I can cancel 1/2 from every term here

159
00:06:00,830 --> 00:06:03,442
by just dividing both sides by 1/2,

160
00:06:03,442 --> 00:06:05,845
or I can imagine multiplying
both sides by two.

161
00:06:05,845 --> 00:06:07,598
And that would get rid of all these 1/2s.

162
00:06:07,598 --> 00:06:09,363
And then I'm going to the
same trick I played over here.

163
00:06:09,363 --> 00:06:11,499
I'm going to get all my
mt terms on one side,

164
00:06:11,499 --> 00:06:14,004
and all my mg terms on another side.

165
00:06:14,004 --> 00:06:16,294
Again, it might not be obvious
why we're going to do this,

166
00:06:16,294 --> 00:06:18,603
but I'm telling you, something
magical's about to happen,

167
00:06:18,603 --> 00:06:20,091
so you got to take my word for it.

168
00:06:20,091 --> 00:06:22,758
If I write mt times vti squared,

169
00:06:23,666 --> 00:06:25,582
and then I'm going to subtract this term

170
00:06:25,582 --> 00:06:28,125
from both sides to get the mts together,

171
00:06:28,125 --> 00:06:31,898
so I'll have a minus mt vt final squared.

172
00:06:31,898 --> 00:06:33,243
And then that's going to equal,

173
00:06:33,243 --> 00:06:35,613
because what I'm going to
do is subtract this term

174
00:06:35,613 --> 00:06:37,912
from both sides, so I
get the mgs together.

175
00:06:37,912 --> 00:06:39,349
So I got this term over here all ready,

176
00:06:39,349 --> 00:06:43,585
mg vg final squared
minus, and then this term

177
00:06:43,585 --> 00:06:45,236
that I'm subtracting from both sides,

178
00:06:45,236 --> 00:06:49,264
mg vg initial squared,
and I play the same game

179
00:06:49,264 --> 00:06:51,389
I played over here, I
pull out a common factor.

180
00:06:51,389 --> 00:06:54,036
I can pull out a common
factor of mt, I get mt,

181
00:06:54,036 --> 00:06:57,453
times vti squared minus vt final squared.

182
00:06:59,317 --> 00:07:02,999
And that's going to equal
mg, times the quantity

183
00:07:02,999 --> 00:07:07,260
vg final squared minus vg initial squared.

184
00:07:07,260 --> 00:07:08,850
And now, things are getting interesting.

185
00:07:08,850 --> 00:07:12,182
If you look at this left
equation, and this right equation,

186
00:07:12,182 --> 00:07:15,236
they're looking a lot more like
each other, which is great.

187
00:07:15,236 --> 00:07:18,475
Because what I want to do is
plug this right hand equation

188
00:07:18,475 --> 00:07:21,238
into the left hand
equation in a clever way,

189
00:07:21,238 --> 00:07:23,049
that causes things to cancel.

190
00:07:23,049 --> 00:07:24,489
That's why I'm doing it this way.

191
00:07:24,489 --> 00:07:26,289
I mean, we could have done brute force,

192
00:07:26,289 --> 00:07:27,728
just like we did numerically.

193
00:07:27,728 --> 00:07:29,261
Solve for one of these velocities,

194
00:07:29,261 --> 00:07:31,757
plus it straight into
one of these velocities,

195
00:07:31,757 --> 00:07:34,915
get a huge mess, and try
to like combine terms.

196
00:07:34,915 --> 00:07:36,287
But this way we're doing it right here's

197
00:07:36,287 --> 00:07:38,541
going to be much cleaner,
so what do we do?

198
00:07:38,541 --> 00:07:40,827
At this point, I want to
make this left hand equation

199
00:07:40,827 --> 00:07:42,674
look more like this right hand equation.

200
00:07:42,674 --> 00:07:45,864
So is there any way I can change
this difference of squares

201
00:07:45,864 --> 00:07:48,699
into a difference of just velocities?

202
00:07:48,699 --> 00:07:50,023
And there is, if you remember,

203
00:07:50,023 --> 00:07:52,798
I can write this squared
term, I can write this as

204
00:07:52,798 --> 00:07:56,215
mt times the quantity vti minus vt final,

205
00:07:58,486 --> 00:08:01,153
multiplied by vti plus vt final.

206
00:08:03,235 --> 00:08:04,721
Because when I multiply these together,

207
00:08:04,721 --> 00:08:08,506
I'm going to get vti squared,
minus vt final squared,

208
00:08:08,506 --> 00:08:10,769
and then the cross terms
are going to cancel.

209
00:08:10,769 --> 00:08:12,860
If you don't believe
me, try it on your own.

210
00:08:12,860 --> 00:08:15,263
Check to make sure this
multiplies out to get that.

211
00:08:15,263 --> 00:08:18,000
And it will, so I'm going to
replace this term with this,

212
00:08:18,000 --> 00:08:19,407
and that's going to equal, I'm going to do

213
00:08:19,407 --> 00:08:20,685
the same thing on the right hand side.

214
00:08:20,685 --> 00:08:22,703
I'm going to write this as mg times

215
00:08:22,703 --> 00:08:25,953
the quantity vg final minus vg initial,

216
00:08:26,919 --> 00:08:28,696
and then that multiplied by the quantity

217
00:08:28,696 --> 00:08:31,696
vg final plus vg initial, and again,

218
00:08:32,748 --> 00:08:35,289
this will multiply to give me that.

219
00:08:35,289 --> 00:08:37,100
If you don't believe me,
try it out on your own.

220
00:08:37,101 --> 00:08:39,260
And now, we're in
business, check this out.

221
00:08:39,260 --> 00:08:40,664
This is where the magic's going to happen.

222
00:08:40,664 --> 00:08:44,008
I've got mt, vti minus vt final, here.

223
00:08:44,009 --> 00:08:47,167
And I've got that exact
same expression over here.

224
00:08:47,167 --> 00:08:50,244
So what I'm going to do is
take this entire expression,

225
00:08:50,244 --> 00:08:53,959
mg times the quantity of
vg final minus vg initial,

226
00:08:53,959 --> 00:08:56,303
and I'm going to plug it in for that.

227
00:08:56,303 --> 00:08:58,603
And you might be like, "How
come, what, why can we do that?"

228
00:08:58,603 --> 00:09:01,633
And it's because this term
here, mt times this difference,

229
00:09:01,633 --> 00:09:03,711
is the exact same as this term here,

230
00:09:03,711 --> 00:09:06,324
mt times this difference,
and I know that this term

231
00:09:06,324 --> 00:09:09,133
equals that term, they're the same thing.

232
00:09:09,133 --> 00:09:11,304
I can replace this, anywhere I see this,

233
00:09:11,304 --> 00:09:13,963
I can replace it with that,
because they're equal.

234
00:09:13,963 --> 00:09:16,889
So that's allowed, I
don't affect the equality

235
00:09:16,889 --> 00:09:19,907
if I just plug this term in for that term,

236
00:09:19,907 --> 00:09:21,138
because they're equal.

237
00:09:21,138 --> 00:09:21,974
So that's what I'm going to do.

238
00:09:21,974 --> 00:09:23,959
I'm going to take this term for mg,

239
00:09:23,959 --> 00:09:25,734
I'm going to plus this all the way in

240
00:09:25,734 --> 00:09:27,791
over to this hand side,
and what am I going to get?

241
00:09:27,791 --> 00:09:31,958
I'm going to get mg, times
vg final minus vg initial.

242
00:09:33,052 --> 00:09:34,922
So that's what this whole term equals.

243
00:09:34,922 --> 00:09:36,953
But it still multiplied by that,

244
00:09:36,953 --> 00:09:38,022
so I got to bring this down,

245
00:09:38,022 --> 00:09:39,426
and still multiply by this one,

246
00:09:39,426 --> 00:09:42,259
which is vt initial plus vt final.

247
00:09:43,246 --> 00:09:45,556
And that's going to equal
this right hand side,

248
00:09:45,556 --> 00:09:47,693
just stays the same, I
didn't do anything to that.

249
00:09:47,693 --> 00:09:49,441
And now, do you see it?

250
00:09:49,441 --> 00:09:51,095
Do you see how wonderful this is?

251
00:09:51,095 --> 00:09:54,867
I can divide both sides
by mg, that cancels out,

252
00:09:54,867 --> 00:09:57,805
and that's kind of weird,
there's going to be no mass left

253
00:09:57,805 --> 00:09:59,326
in this expression that we find.

254
00:09:59,326 --> 00:10:01,606
So the relationship
that we're about to get

255
00:10:01,606 --> 00:10:04,326
doesn't depend on the masses
of the objects colliding,

256
00:10:04,326 --> 00:10:06,036
which is a little weird, and cool.

257
00:10:06,036 --> 00:10:07,673
But even better, look at this term,

258
00:10:07,673 --> 00:10:10,866
vg final minus vg initial,
that's right over here,

259
00:10:10,866 --> 00:10:12,629
vg final minus vg initial.

260
00:10:12,629 --> 00:10:16,241
So I can divide both sides by
that, and that cancels out.

261
00:10:16,241 --> 00:10:18,169
And we're going to get one
of the simplest expressions

262
00:10:18,169 --> 00:10:20,165
you could imagine, let
me make some room for it.

263
00:10:20,165 --> 00:10:21,639
Let me clear this up, we're going to get

264
00:10:21,639 --> 00:10:25,135
that vt initial, the initial
velocity of the tennis ball,

265
00:10:25,135 --> 00:10:29,419
plus vt final, the final
velocity of the tennis ball,

266
00:10:29,419 --> 00:10:33,517
has to equal vg initial, I'm
going to switch the order here,

267
00:10:33,517 --> 00:10:35,804
because they're adding, and
you can switch the order

268
00:10:35,804 --> 00:10:36,884
of things you're adding, just so

269
00:10:36,884 --> 00:10:38,219
it looks like the left hand side,

270
00:10:38,219 --> 00:10:41,202
vg initial, the initial
velocity of the golf ball,

271
00:10:41,202 --> 00:10:45,116
plus vg final, the final
velocity of the golf ball.

272
00:10:45,116 --> 00:10:46,961
Look at how beautiful this is.

273
00:10:46,961 --> 00:10:49,400
It says that in an elastic collision,

274
00:10:49,400 --> 00:10:51,640
if you take the initial and final velocity

275
00:10:51,640 --> 00:10:54,218
of one of the objects, that has to equal

276
00:10:54,218 --> 00:10:57,585
the initial plus final
velocity of the other object,

277
00:10:57,585 --> 00:11:01,554
regardless of what the masses
of the objects colliding are.

278
00:11:01,554 --> 00:11:03,309
And I would have never seen this

279
00:11:03,309 --> 00:11:05,793
unless we would have
solved this symbolically

280
00:11:05,793 --> 00:11:08,998
to see that stuff cancels,
this would not be obvious.

281
00:11:08,998 --> 00:11:11,587
I could have solved a million
of these elastic problems,

282
00:11:11,587 --> 00:11:12,969
and probably never would have guessed

283
00:11:12,969 --> 00:11:14,443
that this was the case.

284
00:11:14,443 --> 00:11:16,416
And the big reason why this is useful

285
00:11:16,416 --> 00:11:18,971
is because now we can use
this simple expression,

286
00:11:18,971 --> 00:11:21,781
as opposed to using
conservation of kinetic energy.

287
00:11:21,781 --> 00:11:24,161
Conservation of kinetic
energy was the thing

288
00:11:24,161 --> 00:11:25,786
that was giving us all the problems,

289
00:11:25,786 --> 00:11:28,004
because it had the square of the speeds.

290
00:11:28,004 --> 00:11:30,639
And when we plugged an
expression in and squared it,

291
00:11:30,639 --> 00:11:32,845
we got this nasty algebraic expression

292
00:11:32,845 --> 00:11:34,076
that we had to deal with.

293
00:11:34,076 --> 00:11:36,878
But now, with this simple
expression between velocities,

294
00:11:36,878 --> 00:11:39,794
we can simply solve for one
of these unknown velocities

295
00:11:39,794 --> 00:11:41,639
in this equation, and plug it

296
00:11:41,639 --> 00:11:44,100
into the conservation
of momentum equation.

297
00:11:44,100 --> 00:11:46,074
There will be no squaring
of an expression.

298
00:11:46,074 --> 00:11:46,989
You're still going to have to plug

299
00:11:46,989 --> 00:11:48,466
one equation into the other,

300
00:11:48,466 --> 00:11:50,567
but the process will be much cleaner,

301
00:11:50,567 --> 00:11:53,945
much simpler, and much less
prone to algebra errors.

302
00:11:53,945 --> 00:11:55,977
And in the next video,
I'll show you an example

303
00:11:55,977 --> 00:11:59,100
of how to use this process to quickly find

304
00:11:59,100 --> 00:12:03,477
the final velocity of either
object in an elastic collision.

305
00:12:03,477 --> 00:12:06,043
So recapping, we used
a symbolic expression

306
00:12:06,043 --> 00:12:08,328
for conservation of momentum, plugged that

307
00:12:08,328 --> 00:12:10,731
into the conservation of energy formula,

308
00:12:10,731 --> 00:12:13,320
and ended up with a
beautiful, simple result

309
00:12:13,320 --> 00:12:15,015
that we're going to be
able to use to solve

310
00:12:15,015 --> 00:12:17,964
elastic collision problems
in a way that avoids

311
00:12:17,964 --> 00:00:00,000
having to use conservation
of energy every single time.