1 00:00:00,322 --> 00:00:02,072 - [Narrator] Alright, here's pretty much the fastest way 2 00:00:02,072 --> 00:00:04,610 you can solve one of these elastic collision problems, 3 00:00:04,610 --> 00:00:07,265 when you don't know two of the velocities. 4 00:00:07,265 --> 00:00:09,728 In this case we don't know the final velocities. 5 00:00:09,728 --> 00:00:11,071 We know the initial velocity 6 00:00:11,071 --> 00:00:12,888 of the tennis ball and its mass. 7 00:00:12,888 --> 00:00:14,642 We know the initial velocity of the golf ball 8 00:00:14,642 --> 00:00:16,982 and its mass, but we don't know the final velocities 9 00:00:16,982 --> 00:00:18,961 of either ball, and the trick to make these calculations 10 00:00:18,961 --> 00:00:23,373 go faster for an elastic collision is to use this equation, 11 00:00:23,373 --> 00:00:26,727 which says the initial velocity of one of the objects 12 00:00:26,727 --> 00:00:29,796 before the collision, plus the final velocity 13 00:00:29,796 --> 00:00:33,846 of that same object after the collision should equal, 14 00:00:33,846 --> 00:00:36,543 if it's an elastic collision, it'll equal 15 00:00:36,543 --> 00:00:39,306 the initial velocity of the second object 16 00:00:39,306 --> 00:00:42,641 before the collision plus the final velocity 17 00:00:42,641 --> 00:00:45,230 of that second object after the collision. 18 00:00:45,230 --> 00:00:46,580 If you want to see where this comes from, 19 00:00:46,580 --> 00:00:48,719 we derived it in the previous video. 20 00:00:48,719 --> 00:00:50,206 But now that we know it, we can just use it 21 00:00:50,206 --> 00:00:52,674 for any elastic collision. 22 00:00:52,674 --> 00:00:54,381 So this expression right here only holds 23 00:00:54,381 --> 00:00:56,871 for elastic collisions. 24 00:00:56,871 --> 00:00:58,913 But you should love this formula right here, 25 00:00:58,913 --> 00:01:00,533 because what this does is it allows you to avoid 26 00:01:00,533 --> 00:01:03,423 having to use conservation of energy, and it avoids 27 00:01:03,423 --> 00:01:05,586 having to square terms and create 28 00:01:05,586 --> 00:01:07,687 horrible messes of algebra. 29 00:01:07,687 --> 00:01:10,021 This formula's gonna be much cleaner, much nicer. 30 00:01:10,021 --> 00:01:11,343 Let's see how to use it. 31 00:01:11,343 --> 00:01:13,340 Let's just say object one is the tennis ball 32 00:01:13,340 --> 00:01:15,458 and object two here is the golf ball. 33 00:01:15,458 --> 00:01:18,396 So in that case, the initial velocity of the tennis ball, 34 00:01:18,396 --> 00:01:21,298 that'd be 40, and it'd be positive 40. 35 00:01:21,298 --> 00:01:22,876 I'm gonna write positive just so I know 36 00:01:22,876 --> 00:01:25,433 that's to the right, and it should be positive. 37 00:01:25,433 --> 00:01:27,959 Plus the final velocity of the tennis ball, 38 00:01:27,959 --> 00:01:29,949 I'll write that as Vt. 39 00:01:29,949 --> 00:01:31,318 So instead of writing one, I'm gonna get confused 40 00:01:31,318 --> 00:01:33,665 which one was number one, so I'll write a Vt. 41 00:01:33,665 --> 00:01:34,744 That way I get to know 42 00:01:34,744 --> 00:01:36,797 which object's velocity I'm talking about. 43 00:01:36,797 --> 00:01:39,720 So Vt final, that's what Vt final's gonna mean, 44 00:01:39,720 --> 00:01:41,469 final velocity of the tennis ball. 45 00:01:41,469 --> 00:01:44,192 And that should equal the initial velocity 46 00:01:44,192 --> 00:01:47,468 of the second object, our second object is the golf ball. 47 00:01:47,468 --> 00:01:49,753 The initial velocity of the second object, 48 00:01:49,753 --> 00:01:51,427 of our golf ball, is not 50. 49 00:01:51,427 --> 00:01:53,795 It's negative 50, you've gotta be careful. 50 00:01:53,795 --> 00:01:56,160 These are velocities in this formula. 51 00:01:56,160 --> 00:01:58,364 So if you've got a velocity, that's directed 52 00:01:58,364 --> 00:02:00,957 in the negative direction, you better make it negative. 53 00:02:00,957 --> 00:02:03,378 And if you solve in here, you might get a negative, 54 00:02:03,378 --> 00:02:05,905 so these are vector values up here. 55 00:02:05,905 --> 00:02:08,238 You gotta plug them in with the proper sign. 56 00:02:08,238 --> 00:02:09,865 So this initial velocity of the golf ball 57 00:02:09,865 --> 00:02:12,539 would be negative 50 meters per second, 58 00:02:12,539 --> 00:02:14,330 'cause we're gonna assume leftward is negative, 59 00:02:14,330 --> 00:02:15,753 and rightward is positive. 60 00:02:15,753 --> 00:02:18,521 Plus the final velocity of the second object, 61 00:02:18,521 --> 00:02:20,422 second object is our golf ball. 62 00:02:20,422 --> 00:02:22,934 I'll call this Vg instead of V2. 63 00:02:22,934 --> 00:02:26,007 Vg will be V of the golf ball and then f for final, 64 00:02:26,007 --> 00:02:27,820 i.e. after the collision. 65 00:02:27,820 --> 00:02:29,832 So we can solve this for Vt final. 66 00:02:29,832 --> 00:02:32,333 I can subtract 40 meters per second from both sides 67 00:02:32,333 --> 00:02:35,268 and I get that the final velocity of the tennis ball 68 00:02:35,268 --> 00:02:37,953 is gonna equal, I'll have this Vg final just sittin' 69 00:02:37,953 --> 00:02:40,172 over here, final velocity of the golf ball 70 00:02:40,172 --> 00:02:41,181 after the collision. 71 00:02:41,181 --> 00:02:43,443 And then negative 50 minus a 40 is gonna be 72 00:02:43,443 --> 00:02:46,640 negative 90 meters per second. 73 00:02:46,640 --> 00:02:48,871 So this formula alone was not enough, 74 00:02:48,871 --> 00:02:51,100 'cause I've still got two unknowns. 75 00:02:51,100 --> 00:02:52,601 I can't solve for either. 76 00:02:52,601 --> 00:02:53,920 I've gotta use another equation, 77 00:02:53,920 --> 00:02:55,754 and the other equation we're gonna use is 78 00:02:55,754 --> 00:02:57,367 conservation of momentum. 79 00:02:57,367 --> 00:02:58,944 'Cause during this collision, the momentum 80 00:02:58,944 --> 00:03:01,698 should be conserved, assuming the collision happens so fast, 81 00:03:01,698 --> 00:03:04,823 any net external impulse is negligible. 82 00:03:04,823 --> 00:03:07,058 So we can say that the total initial momentum 83 00:03:07,058 --> 00:03:09,195 is equal to the total final momentum. 84 00:03:09,195 --> 00:03:11,459 We basically do this for every single collision 85 00:03:11,459 --> 00:03:13,205 because we make that assumption, that the net 86 00:03:13,205 --> 00:03:16,741 external impulse during this collision is gonna be small. 87 00:03:16,741 --> 00:03:18,594 That means the momentum should be conserved. 88 00:03:18,594 --> 00:03:21,060 So the formula for momentum is mass times velocity. 89 00:03:21,060 --> 00:03:23,537 So the momentum of this tennis ball initially 90 00:03:23,537 --> 00:03:26,397 is the mass of the tennis ball, .058, times the 91 00:03:26,397 --> 00:03:28,794 initial velocity, would be positive 40. 92 00:03:28,794 --> 00:03:31,152 Positive because it's directed to the right, 93 00:03:31,152 --> 00:03:32,974 and I'm gonna consider rightwards positive. 94 00:03:32,974 --> 00:03:34,664 Plus the initial momentum of the golf ball 95 00:03:34,664 --> 00:03:38,725 would be .045 times the initial velocity of the golf ball 96 00:03:38,725 --> 00:03:40,618 and that'd be negative 50. 97 00:03:40,618 --> 00:03:43,216 Again, you have to be careful with the negative signs. 98 00:03:43,216 --> 00:03:45,961 Momentum is also a vector, so if these velocities 99 00:03:45,961 --> 00:03:48,171 are ever negative, you've got to plug them in 100 00:03:48,171 --> 00:03:49,550 with their negative sign. 101 00:03:49,550 --> 00:03:52,433 And that initial momentum should equal the final momentum. 102 00:03:52,433 --> 00:03:54,458 So the final momentum of the tennis ball 103 00:03:54,458 --> 00:03:58,189 is gonna be 0.058 times our final velocity 104 00:03:58,189 --> 00:03:59,131 of the tennis ball. 105 00:03:59,131 --> 00:04:01,196 And I'm gonna use the same nomenclature. 106 00:04:01,196 --> 00:04:02,661 I'm gonna use the same symbol over here 107 00:04:02,661 --> 00:04:04,461 that I used over here. 108 00:04:04,461 --> 00:04:06,131 This Vt final, final velocity of the tennis ball 109 00:04:06,131 --> 00:04:08,929 is the same as this Vt final, final velocity 110 00:04:08,929 --> 00:04:13,104 of the tennis ball, plus .045 times same thing, 111 00:04:13,104 --> 00:04:14,918 final velocity of the golf ball. 112 00:04:14,918 --> 00:04:19,084 I'll use the same symbol which is gonna be Vg final. 113 00:04:19,084 --> 00:04:22,224 I could multiply out this entire left hand side, 114 00:04:22,224 --> 00:04:26,391 and what I get is 0.07 kilogram meters per second. 115 00:04:27,331 --> 00:04:28,998 And that equals on the right hand side, 116 00:04:28,998 --> 00:04:31,332 this entire expression right here. 117 00:04:31,332 --> 00:04:32,479 I'll just copy that. 118 00:04:32,479 --> 00:04:34,948 So we've got two unknowns in this equation as well, 119 00:04:34,948 --> 00:04:36,750 so we can't solve this directly 120 00:04:36,750 --> 00:04:38,835 or either of the final velocities. 121 00:04:38,835 --> 00:04:42,099 But we do have two equations and two unknowns now. 122 00:04:42,099 --> 00:04:43,659 And whenever you have that situation, 123 00:04:43,659 --> 00:04:46,649 you can solve one of the equations for one of the variables 124 00:04:46,649 --> 00:04:50,207 and plug that expression into the other equation. 125 00:04:50,207 --> 00:04:52,358 In other words, I know that the final velocity 126 00:04:52,358 --> 00:04:55,020 of the tennis ball is equal to the final velocity 127 00:04:55,020 --> 00:04:56,883 of the golf ball minus 90, so I can take 128 00:04:56,883 --> 00:05:00,993 this entire term right here, since it's equal to Vt final, 129 00:05:00,993 --> 00:05:04,313 and just plug that in for Vt final. 130 00:05:04,313 --> 00:05:06,586 And what that would do for me is give me one expression, 131 00:05:06,586 --> 00:05:09,622 all in terms of the final velocity of the golf ball. 132 00:05:09,622 --> 00:05:10,521 So let's do that. 133 00:05:10,521 --> 00:05:15,327 We've still got 0.07 kilogram meters per second on the left. 134 00:05:15,327 --> 00:05:20,006 And that's gonna equal 0.058, that's still here, kilograms, 135 00:05:20,006 --> 00:05:21,914 times Vt final. 136 00:05:21,914 --> 00:05:24,257 I'm plugging in this entire expression for Vt final. 137 00:05:24,257 --> 00:05:28,190 So that gets multiplied by Vg final, the final velocity 138 00:05:28,190 --> 00:05:31,524 of the golf ball, minus 90 meters per second. 139 00:05:31,524 --> 00:05:33,926 That was the term I plugged in for Vt final, 140 00:05:33,926 --> 00:05:36,283 and it got multiplied by this mass here, 141 00:05:36,283 --> 00:05:37,805 so I can't forget about that mass. 142 00:05:37,805 --> 00:05:39,740 And then I still have to add this final momentum 143 00:05:39,740 --> 00:05:43,158 of the golf ball, .045 kilograms times the final velocity 144 00:05:43,158 --> 00:05:44,653 of the golf ball. 145 00:05:44,653 --> 00:05:46,069 So at this point you might be feeling ripped off. 146 00:05:46,069 --> 00:05:47,983 You might be like, easy way to do this? 147 00:05:47,983 --> 00:05:50,232 This isn't easy, this is hard. 148 00:05:50,232 --> 00:05:53,453 I've gotta plug one equation into another, and then solve? 149 00:05:53,453 --> 00:05:57,317 Well, this easy approach does not avoid having to plug 150 00:05:57,317 --> 00:05:59,526 one equation into the other, that's true. 151 00:05:59,526 --> 00:06:01,509 But the reason that it's easy is because 152 00:06:01,509 --> 00:06:03,874 the equations that we're plugging into each other 153 00:06:03,874 --> 00:06:07,059 are a whole lot simpler than the kinetic energy formula 154 00:06:07,059 --> 00:06:08,644 that you would have to use 155 00:06:08,644 --> 00:06:11,333 if you didn't know this expression here. 156 00:06:11,333 --> 00:06:13,760 Because we have this one, we do not have to plug 157 00:06:13,760 --> 00:06:16,705 conservation of momentum into conservation of energy. 158 00:06:16,705 --> 00:06:19,161 That would square the term we put in, 159 00:06:19,161 --> 00:06:22,239 that would get nasty, the algebra would be a lot worse. 160 00:06:22,239 --> 00:06:24,920 These formulas that we're dealing with in this process 161 00:06:24,920 --> 00:06:26,440 only have velocity. 162 00:06:26,440 --> 00:06:28,160 None of these velocities are squared, 163 00:06:28,160 --> 00:06:30,175 so the algebra doesn't get nearly as bad. 164 00:06:30,175 --> 00:06:31,617 And we're actually almost done over here. 165 00:06:31,617 --> 00:06:33,504 Let me show you how close we are to finishing this thing. 166 00:06:33,504 --> 00:06:36,711 I just need to multiply through this term right here. 167 00:06:36,711 --> 00:06:37,807 So what am I going to get, 168 00:06:37,807 --> 00:06:39,757 this left hand side stays the same. 169 00:06:39,757 --> 00:06:42,759 And then when I multiply through .058, I'm gonna get 170 00:06:42,759 --> 00:06:46,759 0.058 times the final velocity of the golf ball, 171 00:06:47,723 --> 00:06:51,806 and then negative 90 times .058 is negative 5.22, 172 00:06:53,101 --> 00:06:55,623 and that would be units of kilogram meters per second. 173 00:06:55,623 --> 00:06:57,270 And I still just have this term right here, 174 00:06:57,270 --> 00:06:59,103 so I'll copy that one. 175 00:07:00,212 --> 00:07:01,556 And now here's a key step. 176 00:07:01,556 --> 00:07:03,890 The whole reason we plugged one formula into the other 177 00:07:03,890 --> 00:07:07,456 is so that we had the same unknown in that formula. 178 00:07:07,456 --> 00:07:10,314 Only one unknown, there's only one unknown variable in here, 179 00:07:10,314 --> 00:07:12,640 which is the final velocity of the golf ball. 180 00:07:12,640 --> 00:07:14,114 It's located in two spots, 181 00:07:14,114 --> 00:07:16,215 but at least it's the same variable. 182 00:07:16,215 --> 00:07:18,667 And what that allows us to do is to combine these terms. 183 00:07:18,667 --> 00:07:21,009 If on this right hand side I have this much 184 00:07:21,009 --> 00:07:22,843 of the final velocity of the golf ball, 185 00:07:22,843 --> 00:07:24,838 and that much of the final velocity of the golf ball, 186 00:07:24,838 --> 00:07:26,460 when I add them up, I can just 187 00:07:26,460 --> 00:07:28,827 add these two factors out front. 188 00:07:28,827 --> 00:07:31,471 In other words I'll have 0.07 equals, 189 00:07:31,471 --> 00:07:34,971 I can rewrite this as 0.058 kilograms plus 190 00:07:36,460 --> 00:07:40,627 0.045 kilograms times the final velocity of the golf ball. 191 00:07:42,682 --> 00:07:45,988 And then I still have this minus 5.22. 192 00:07:45,988 --> 00:07:48,121 So if that looked like mathematical witchcraft, 193 00:07:48,121 --> 00:07:51,014 all I did was I combined the terms that had Vg final. 194 00:07:51,014 --> 00:07:54,867 Cause if you had A times Vg final plug B times Vg final, 195 00:07:54,867 --> 00:07:58,300 that's the same as A plus B, times Vg final. 196 00:07:58,300 --> 00:07:59,662 When you multiply this through, 197 00:07:59,662 --> 00:08:01,821 you'll just get both of these terms back again. 198 00:08:01,821 --> 00:08:04,377 So we keep going, don't divide by this first, 199 00:08:04,377 --> 00:08:06,870 sometimes people try to divide by this whole term right now. 200 00:08:06,870 --> 00:08:08,913 You don't want to do that, you gotta go in the right order. 201 00:08:08,913 --> 00:08:12,044 I need to add this 5.22 to both sides 202 00:08:12,044 --> 00:08:13,428 to get rid of it first. 203 00:08:13,428 --> 00:08:15,869 So when I do that, when I add 5.22 to both sides 204 00:08:15,869 --> 00:08:18,410 it'll cancel this term, and on the left hand side 205 00:08:18,410 --> 00:08:21,827 I'll get 5.29 kilogram meters per second. 206 00:08:23,058 --> 00:08:25,729 And that's gonna equal, if I add these two terms together, 207 00:08:25,729 --> 00:08:29,896 if I just add .058 and .045, I get 0.103 kilograms 208 00:08:30,860 --> 00:08:33,208 times the final velocity of the golf ball. 209 00:08:33,208 --> 00:08:36,341 And let me move these initial velocities down so we can see. 210 00:08:36,342 --> 00:08:39,714 At this point, if I divide both sides by .103, 211 00:08:39,714 --> 00:08:43,881 I'll get 51.36 meters per second on the left hand side. 212 00:08:45,100 --> 00:08:48,200 And that equals the final velocity of the golf ball. 213 00:08:48,200 --> 00:08:51,137 So we did it, we found one of the final velocities 214 00:08:51,137 --> 00:08:52,258 of these objects. 215 00:08:52,258 --> 00:08:54,076 We found the final velocity of the golf ball. 216 00:08:54,076 --> 00:08:56,551 But what about the final velocity of the tennis ball? 217 00:08:56,551 --> 00:08:58,004 How do we figure out what that is? 218 00:08:58,004 --> 00:08:59,776 'Cause that's also an unknown in this problem. 219 00:08:59,776 --> 00:09:00,967 Well that one's really easy. 220 00:09:00,967 --> 00:09:03,046 Now that we know the final velocity of the golf ball, 221 00:09:03,046 --> 00:09:06,312 I can just take that value, plug it right back into here, 222 00:09:06,312 --> 00:09:07,879 this expression that we plugged 223 00:09:07,879 --> 00:09:09,704 into conservation of momentum. 224 00:09:09,704 --> 00:09:11,009 And I can figure out what the velocity 225 00:09:11,009 --> 00:09:12,353 of the tennis ball is. 226 00:09:12,353 --> 00:09:14,142 So in other words, the velocity of the tennis ball 227 00:09:14,142 --> 00:09:17,056 I know, finally, should equal the final velocity 228 00:09:17,056 --> 00:09:20,625 of the golf ball, which was 51.36 meters per second, 229 00:09:20,625 --> 00:09:23,099 and then minus 90 meters per second, 230 00:09:23,099 --> 00:09:25,319 and I'll get the final velocity of the tennis ball 231 00:09:25,319 --> 00:09:28,402 was negative 38.64 meters per second. 232 00:09:29,718 --> 00:09:31,536 And it came out to be negative, that means that 233 00:09:31,536 --> 00:09:34,219 this tennis ball got deflected backwards. 234 00:09:34,219 --> 00:09:37,567 It was heading leftward, 38.64 meters per second 235 00:09:37,567 --> 00:09:39,032 after the collision. 236 00:09:39,032 --> 00:09:41,904 So recapping, we used this nice formula 237 00:09:41,904 --> 00:09:45,059 to get one equation that involved the velocities 238 00:09:45,059 --> 00:09:46,617 that we didn't know for an elastic collision, 239 00:09:46,617 --> 00:09:48,824 which you can only use for an elastic collision. 240 00:09:48,824 --> 00:09:50,580 If you want to see where this equation comes from, 241 00:09:50,580 --> 00:09:52,376 we derived it in the last video. 242 00:09:52,376 --> 00:09:54,177 We solved for one of the velocities and plugged it 243 00:09:54,177 --> 00:09:56,433 into conservation of momentum. 244 00:09:56,433 --> 00:10:00,277 We combined the like unknowns, and solved algebraically 245 00:10:00,277 --> 00:10:02,527 to get one of the final unknown velocities. 246 00:10:02,527 --> 00:10:05,234 Then we plugged that back into the first equation 247 00:10:05,234 --> 00:10:07,449 to get the other unknown velocity. 248 00:10:07,449 --> 00:10:10,008 And this is easier than the alternative, 249 00:10:10,008 --> 00:10:12,559 because the alternative involves kinetic energies, 250 00:10:12,559 --> 00:10:15,047 which means when you take one of these expressions, 251 00:10:15,047 --> 00:10:17,095 you'd have to square them, and the algebra 252 00:10:17,095 --> 00:00:00,000 would be significantly more difficult.