1 00:00:01,154 --> 00:00:02,660 - [Narrator] So, I looked up the mass of a tennis ball. 2 00:00:02,660 --> 00:00:06,248 And it turns out a tennis ball is about 58 grams 3 00:00:06,248 --> 00:00:08,671 or point o five eight kilograms. 4 00:00:08,671 --> 00:00:10,410 And I wondered, if we shot that tennis ball 5 00:00:10,410 --> 00:00:13,663 to the right, straight toward a golf ball, 6 00:00:13,663 --> 00:00:15,380 and I looked up the mass of a golf ball. 7 00:00:15,380 --> 00:00:17,602 A golf ball's about 45 grams 8 00:00:17,602 --> 00:00:20,070 or point o four five kilograms. 9 00:00:20,070 --> 00:00:21,707 So if we shot these balls straight 10 00:00:21,707 --> 00:00:23,879 toward each other, at a certain speed, 11 00:00:23,879 --> 00:00:25,427 let's say the golf ball's moving 12 00:00:25,427 --> 00:00:27,435 around 50 meters per second. 13 00:00:27,435 --> 00:00:28,926 That's pretty fast. 14 00:00:28,926 --> 00:00:30,762 That's over 100 miles an hour. 15 00:00:30,762 --> 00:00:32,698 And we shoot the tennis ball to the right, 16 00:00:32,698 --> 00:00:34,999 at a speed of 40 meters per second, 17 00:00:34,999 --> 00:00:36,635 so that these balls collide. 18 00:00:36,635 --> 00:00:37,951 And they collide head on. 19 00:00:37,951 --> 00:00:39,744 In other words, I want them to collide 20 00:00:39,744 --> 00:00:41,694 and stay in this single direction. 21 00:00:41,694 --> 00:00:43,306 I don't want a glancing collision, 22 00:00:43,306 --> 00:00:45,128 where the golf ball goes flying 23 00:00:45,128 --> 00:00:46,307 up this way, or something like that. 24 00:00:46,307 --> 00:00:47,324 Let's not do that. 25 00:00:47,324 --> 00:00:48,804 So it's all gonna happen in one dimension. 26 00:00:48,804 --> 00:00:50,637 And my question is this, 27 00:00:50,637 --> 00:00:52,316 just given the initial velocities 28 00:00:52,316 --> 00:00:55,022 and the masses, can we figure out 29 00:00:55,022 --> 00:00:57,776 the final velocities of the golf ball 30 00:00:57,776 --> 00:00:59,098 and the tennis ball? 31 00:00:59,098 --> 00:01:00,014 Let's try it. 32 00:01:00,014 --> 00:01:01,768 So how can we start? 33 00:01:01,768 --> 00:01:03,406 When I'm doing a collision problem, 34 00:01:03,406 --> 00:01:05,990 I typically just start with conservation momentum. 35 00:01:05,990 --> 00:01:08,100 I just know, if it's gonna be a quick collision 36 00:01:08,100 --> 00:01:10,088 the momentum right before the collision 37 00:01:10,088 --> 00:01:12,047 should equal the momentum right 38 00:01:12,047 --> 00:01:12,880 after the collision. 39 00:01:12,880 --> 00:01:14,281 At least the total amounts. 40 00:01:14,281 --> 00:01:15,325 Why is that true? 41 00:01:15,325 --> 00:01:17,195 It's because this golf ball, 42 00:01:17,195 --> 00:01:19,080 the time that it's actually in contact 43 00:01:19,080 --> 00:01:21,656 with the tennis ball, it's gonna be so small 44 00:01:21,656 --> 00:01:24,034 that any external forces that might be there, 45 00:01:24,034 --> 00:01:26,090 like gravity, are gonna have so little time 46 00:01:26,090 --> 00:01:28,585 to act on the system, the external forces 47 00:01:28,585 --> 00:01:30,872 can't really impart a large amount 48 00:01:30,872 --> 00:01:32,641 of external impulse. 49 00:01:32,641 --> 00:01:34,562 And if there's no external impulse, 50 00:01:34,562 --> 00:01:36,697 the total momentum of our system, 51 00:01:36,697 --> 00:01:37,920 golf ball and tennis ball, 52 00:01:37,920 --> 00:01:39,285 has to stay constant. 53 00:01:39,285 --> 00:01:40,583 So, because these collisions happen, 54 00:01:40,583 --> 00:01:42,926 typically, over a very short time interval, 55 00:01:42,926 --> 00:01:45,292 we're just gonna say, the momentum right before total 56 00:01:45,292 --> 00:01:46,473 and the momentum right after total 57 00:01:46,473 --> 00:01:48,111 is gonna be the same. 58 00:01:48,111 --> 00:01:50,204 And that goes for basically any collision 59 00:01:50,204 --> 00:01:52,711 between two freely moving objects. 60 00:01:52,711 --> 00:01:54,261 You can just assume the total momentum's 61 00:01:54,261 --> 00:01:55,650 gonna be conserved. 62 00:01:55,650 --> 00:01:57,337 So what will that mean mathematically? 63 00:01:57,337 --> 00:01:59,838 Well it's gonna be that the total initial momentum, 64 00:01:59,838 --> 00:02:02,608 p is the letter we use for momentum, 65 00:02:02,608 --> 00:02:04,064 and the total, I'm gonna use Sigma 66 00:02:04,064 --> 00:02:05,208 to represent the total. 67 00:02:05,208 --> 00:02:07,521 This just means add up all the initial momentum, 68 00:02:07,521 --> 00:02:09,901 not just the momentum of one of the objects, 69 00:02:09,901 --> 00:02:12,056 but all the momentum of all the objects. 70 00:02:12,056 --> 00:02:13,671 And I'm even gonna put a vector sign up here 71 00:02:13,671 --> 00:02:15,867 because momentum's a vector. 72 00:02:15,867 --> 00:02:18,139 That's important, because momentum can be negative. 73 00:02:18,139 --> 00:02:20,050 So you can't forget the negative signs in here. 74 00:02:20,050 --> 00:02:21,470 And if momentum's conserved, 75 00:02:21,470 --> 00:02:23,564 then this, initial, total momentum 76 00:02:23,564 --> 00:02:26,579 should equal the final total momentum. 77 00:02:26,579 --> 00:02:27,916 This is what we mean when we say, 78 00:02:27,916 --> 00:02:29,775 "Momentum is conserved." 79 00:02:29,775 --> 00:02:31,556 so remember, the formula for momentum 80 00:02:31,556 --> 00:02:33,507 is mass times velocity. 81 00:02:33,507 --> 00:02:35,277 So the initial momentum of the tennis ball 82 00:02:35,277 --> 00:02:37,211 would be mass times velocity. 83 00:02:37,211 --> 00:02:38,982 So that would be zero point zero 84 00:02:38,982 --> 00:02:40,732 five eight kilograms. 85 00:02:41,643 --> 00:02:44,049 Times the velocity, initially, of the tennis ball 86 00:02:44,049 --> 00:02:45,991 is positive 40. 87 00:02:45,991 --> 00:02:47,474 And I'm gonna put a positive here 88 00:02:47,474 --> 00:02:49,957 to remind me that this is to the right. 89 00:02:49,957 --> 00:02:51,372 That's why I'm making it positive. 90 00:02:51,372 --> 00:02:53,334 The initial momentum of the golf ball 91 00:02:53,334 --> 00:02:55,156 would be also mass times velocity. 92 00:02:55,156 --> 00:02:57,312 So it'd be plus the mass of the golf ball 93 00:02:57,312 --> 00:03:00,106 is point o four five kilograms. 94 00:03:00,106 --> 00:03:01,744 And the initial velocity of the golf ball 95 00:03:01,744 --> 00:03:04,563 would be negative 50 meters per second. 96 00:03:04,563 --> 00:03:06,711 Because the golf ball is moving to the left. 97 00:03:06,711 --> 00:03:08,358 And we're gonna assume leftward is negative 98 00:03:08,358 --> 00:03:10,077 and rightward is positive. 99 00:03:10,077 --> 00:03:12,507 So if this is the total, initial momentum, 100 00:03:12,507 --> 00:03:14,548 and momentum's conserved, this should equal 101 00:03:14,548 --> 00:03:16,888 the total final momentum. 102 00:03:16,888 --> 00:03:19,163 So the final total momentum of the tennis ball 103 00:03:19,163 --> 00:03:23,506 is gonna be zero point zero five eight kilograms 104 00:03:23,506 --> 00:03:26,287 times v final of the tennis ball. 105 00:03:26,287 --> 00:03:28,536 I'm just gonna call that V-T, 106 00:03:28,536 --> 00:03:30,551 for v of the tennis ball, plus the final 107 00:03:30,551 --> 00:03:32,177 momentum of the golf ball's gonna be 108 00:03:32,177 --> 00:03:35,845 plus zero point zero four five kilograms 109 00:03:35,845 --> 00:03:38,652 times the final velocity of the golf ball's 110 00:03:38,652 --> 00:03:40,835 gonna be v, I'm gonna put V-G, 111 00:03:40,835 --> 00:03:42,357 for v of the golf ball. 112 00:03:42,357 --> 00:03:44,176 And that's gonna be the velocity after the collision. 113 00:03:44,176 --> 00:03:45,826 So can I solve now for the final velocity 114 00:03:45,826 --> 00:03:47,476 of the tennis ball and the golf ball? 115 00:03:47,476 --> 00:03:48,476 No, I can't. 116 00:03:49,323 --> 00:03:51,638 I've got one equation and I've got two unknowns 117 00:03:51,638 --> 00:03:53,143 sitting over here. 118 00:03:53,143 --> 00:03:54,966 So I'm not gonna be able to solve for either of them 119 00:03:54,966 --> 00:03:57,786 if I've got two variables in my single equation. 120 00:03:57,786 --> 00:03:59,762 In other words, I can add up this whole 121 00:03:59,762 --> 00:04:01,842 left hand side if I wanted to. 122 00:04:01,842 --> 00:04:03,001 If you add all this up you're gonna 123 00:04:03,001 --> 00:04:07,876 get zero point zero seven kilogram meters per second, 124 00:04:07,876 --> 00:04:09,826 is your total, initial momentum. 125 00:04:09,826 --> 00:04:10,878 Then if I solve this over here I'm gonna have 126 00:04:10,878 --> 00:04:13,139 equals two unknowns. 127 00:04:13,139 --> 00:04:15,154 I don't know V-T and I don't know 128 00:04:15,154 --> 00:04:17,418 the velocity of the golf ball either. 129 00:04:17,418 --> 00:04:19,954 So I need at least one more piece of information. 130 00:04:19,954 --> 00:04:21,433 I need to know, for instance, 131 00:04:21,433 --> 00:04:23,840 I knew one of these final velocities. 132 00:04:23,840 --> 00:04:25,881 I could easily solve for the other. 133 00:04:25,881 --> 00:04:27,961 So if the problem gave me the final velocity 134 00:04:27,961 --> 00:04:29,232 of the tennis ball. 135 00:04:29,232 --> 00:04:30,479 Well, I can plug that number into here 136 00:04:30,479 --> 00:04:32,677 and just solve, then for my final velocity 137 00:04:32,677 --> 00:04:33,862 of the golf ball. 138 00:04:33,862 --> 00:04:36,345 Or the problem could tell you that this collision, 139 00:04:36,345 --> 00:04:37,929 what type of collision is it? 140 00:04:37,929 --> 00:04:40,343 If it tells us that they stick together. 141 00:04:40,343 --> 00:04:41,528 Then I can assume that they both 142 00:04:41,528 --> 00:04:43,554 move off at the same velocity. 143 00:04:43,554 --> 00:04:46,467 That would be a perfectly inelastic collision. 144 00:04:46,467 --> 00:04:48,614 So if it was a perfectly inelastic collision, 145 00:04:48,614 --> 00:04:51,626 I'd just have equals one big mass over here. 146 00:04:51,626 --> 00:04:54,486 In other words, point o five eight kilograms. 147 00:04:54,486 --> 00:04:57,153 Plus the mass of the golf ball. 148 00:04:57,153 --> 00:04:59,479 Point o four five kilograms. 149 00:04:59,479 --> 00:05:02,002 It'd be one big mass because they'd stick together 150 00:05:02,002 --> 00:05:04,303 in a perfectly inelastic collision. 151 00:05:04,303 --> 00:05:06,579 Times just one final velocity, 152 00:05:06,579 --> 00:05:08,960 because they're both moving at the same velocity. 153 00:05:08,960 --> 00:05:10,714 So if you take this point o seven, 154 00:05:10,714 --> 00:05:13,143 divide by my total mass, that would give me 155 00:05:13,143 --> 00:05:16,949 the final velocity of the two balls combined. 156 00:05:16,949 --> 00:05:18,473 But that's unlikely. 157 00:05:18,473 --> 00:05:19,789 These balls aren't gonna stick together. 158 00:05:19,789 --> 00:05:21,400 I mean, a golf ball and a tennis ball, 159 00:05:21,400 --> 00:05:22,687 unless you've got some sort of adhesive 160 00:05:22,687 --> 00:05:24,169 on the front of them, 161 00:05:24,169 --> 00:05:25,635 I don't think these are gonna stick together, 162 00:05:25,635 --> 00:05:27,120 that seems unlikely. 163 00:05:27,120 --> 00:05:29,354 So let's assume that doesn't happen. 164 00:05:29,354 --> 00:05:30,916 But if you were told they stick together, 165 00:05:30,916 --> 00:05:32,357 in a collision, two masses, 166 00:05:32,357 --> 00:05:33,670 that's what you could do. 167 00:05:33,670 --> 00:05:35,010 It's much more likely, that if you're dealing 168 00:05:35,010 --> 00:05:37,195 with a golf ball and a tennis ball, 169 00:05:37,195 --> 00:05:39,484 that you're gonna be told that 170 00:05:39,484 --> 00:05:41,250 this collision was elastic. 171 00:05:41,250 --> 00:05:43,487 And remember, elastic means that 172 00:05:43,487 --> 00:05:45,502 the total kinetic energy in this collision 173 00:05:45,502 --> 00:05:47,868 is gonna be constant or conserved. 174 00:05:47,868 --> 00:05:50,180 You're not gonna lose any of that kinetic energy 175 00:05:50,180 --> 00:05:52,536 to any thermal energy or sound. 176 00:05:52,536 --> 00:05:54,612 And it turns out, just being told this, 177 00:05:54,612 --> 00:05:56,460 that the collision is elastic 178 00:05:56,460 --> 00:05:59,616 is enough to solve for these final velocities. 179 00:05:59,616 --> 00:06:01,557 And the reason is, this is implying 180 00:06:01,557 --> 00:06:03,569 the kinetic energy is conserved. 181 00:06:03,569 --> 00:06:05,025 So we can use that to our advantage. 182 00:06:05,025 --> 00:06:06,731 We can just say, "All right, not only 183 00:06:06,731 --> 00:06:09,345 "is momentum conserved now, but if we say it's elastic, 184 00:06:09,345 --> 00:06:11,032 "that means the total amount of 185 00:06:11,032 --> 00:06:12,723 "kinetic energy is conserved." 186 00:06:12,723 --> 00:06:15,024 so the initial, total, kinetic energy 187 00:06:15,024 --> 00:06:18,990 has to equal the final, total kinetic energy. 188 00:06:18,990 --> 00:06:22,031 And remember, kinetic energy is 1/2 M-V squared. 189 00:06:22,031 --> 00:06:24,242 So I can say that, all right, 1/2 point 190 00:06:24,242 --> 00:06:26,709 zero five eight kilograms, the mass 191 00:06:26,709 --> 00:06:27,725 of the tennis ball. 192 00:06:27,725 --> 00:06:31,355 Times it's initial velocity was 40 meters per second. 193 00:06:31,355 --> 00:06:32,404 You can't forget to square it, 194 00:06:32,404 --> 00:06:35,357 kinetic energy's 1/2 M-V squared. 195 00:06:35,357 --> 00:06:37,762 Then I do plus the initial kinetic energy 196 00:06:37,762 --> 00:06:39,856 of the golf ball's gonna be 1/2, 197 00:06:39,856 --> 00:06:43,561 mass of the golf ball was point o four five kilograms. 198 00:06:43,561 --> 00:06:46,202 We multiply by it's initial speed squared. 199 00:06:46,202 --> 00:06:48,360 And I'm just gonna do positive 50. 200 00:06:48,360 --> 00:06:50,428 Because we're gonna square this. 201 00:06:50,428 --> 00:06:52,300 This is just the speed in kinetic energy. 202 00:06:52,300 --> 00:06:54,637 It doesn't matter if you make it positive or negative. 203 00:06:54,637 --> 00:06:57,092 Over here, it definitely matters in momentum, 204 00:06:57,092 --> 00:06:59,114 whether you make it positive or negative. 205 00:06:59,114 --> 00:07:00,411 But since you're squaring it. 206 00:07:00,411 --> 00:07:02,140 And since kinetic energy's a scaler 207 00:07:02,140 --> 00:07:03,585 it can't be negative, doesn't matter 208 00:07:03,585 --> 00:07:05,170 whether you put the positive or negative in here. 209 00:07:05,170 --> 00:07:06,547 It's gonna go away when you square it. 210 00:07:06,547 --> 00:07:08,615 We can say that this total, initial kinetic energy 211 00:07:08,615 --> 00:07:11,279 should equal the total, final kinetic energy. 212 00:07:11,279 --> 00:07:13,490 So I can say that, this total amount here 213 00:07:13,490 --> 00:07:15,467 should equal, I'm just gonna put 214 00:07:15,467 --> 00:07:16,754 the equals sign down here, 215 00:07:16,754 --> 00:07:18,573 the final kinetic energy of the tennis ball 216 00:07:18,573 --> 00:07:22,405 would be 1/2 point o five eight kilograms. 217 00:07:22,405 --> 00:07:26,110 Times the final velocity of the tennis ball squared. 218 00:07:26,110 --> 00:07:27,530 And then I have to add to that 219 00:07:27,530 --> 00:07:29,375 the final kinetic energy of the golf ball. 220 00:07:29,375 --> 00:07:31,262 Which is gonna be 1/2. 221 00:07:31,262 --> 00:07:32,963 Mass of the golf ball is point 222 00:07:32,963 --> 00:07:35,175 o four five kilograms. 223 00:07:35,175 --> 00:07:38,333 Times the final velocity of the golf ball squared. 224 00:07:38,333 --> 00:07:40,357 And you might be like, "How does this help us?" 225 00:07:40,357 --> 00:07:42,220 Look at how horrible this looks. 226 00:07:42,220 --> 00:07:43,650 These are squared. 227 00:07:43,650 --> 00:07:45,159 How's this gonna help me now. 228 00:07:45,159 --> 00:07:46,457 Well, now you can solve. 229 00:07:46,457 --> 00:07:48,072 Because I've got two equations. 230 00:07:48,072 --> 00:07:49,593 And I've got two unknowns. 231 00:07:49,593 --> 00:07:50,797 And the two unknowns over here 232 00:07:50,797 --> 00:07:52,789 are the same as the two unknowns over here. 233 00:07:52,789 --> 00:07:54,245 So whenever you have two equations 234 00:07:54,245 --> 00:07:55,871 and two unknowns, you can solve 235 00:07:55,871 --> 00:07:57,105 for one of your unknowns. 236 00:07:57,105 --> 00:07:59,316 You can actually solve for both of your unknowns. 237 00:07:59,316 --> 00:08:00,729 First you're gonna solve one of the equations 238 00:08:00,729 --> 00:08:03,474 and then substitute into the other. 239 00:08:03,474 --> 00:08:06,361 And we'll get one equation with one unknown. 240 00:08:06,361 --> 00:08:08,063 In other words, let me show you how that works. 241 00:08:08,063 --> 00:08:10,308 Let me clean up this side over here, 242 00:08:10,308 --> 00:08:12,277 this left hand side, which is kind of 243 00:08:12,277 --> 00:08:13,110 like the upper side right here. 244 00:08:13,110 --> 00:08:14,654 So if I add up all this initial, 245 00:08:14,654 --> 00:08:16,579 kinetic energy, over here. 246 00:08:16,579 --> 00:08:19,662 I get 102 point 65 joules of initial, 247 00:08:20,967 --> 00:08:22,616 total kinetic energy. 248 00:08:22,616 --> 00:08:24,906 But I've still got two unknowns in this equation. 249 00:08:24,906 --> 00:08:26,582 So what I'm gonna do is I'm gonna come over here. 250 00:08:26,582 --> 00:08:28,325 Let's just solve this for V-G. 251 00:08:28,325 --> 00:08:30,223 If I solve this for V-G, I'll subtract 252 00:08:30,223 --> 00:08:33,092 point o five eight V-T from both sides, 253 00:08:33,092 --> 00:08:34,925 point o four five V-G. 254 00:08:36,626 --> 00:08:37,513 And now I can divide both sides 255 00:08:37,513 --> 00:08:38,967 from point o four five. 256 00:08:38,967 --> 00:08:40,866 Over here, point o seven divided by 257 00:08:40,866 --> 00:08:44,886 point o four five, is equal to one point five six. 258 00:08:44,886 --> 00:08:46,316 And point o five eight divided by 259 00:08:46,316 --> 00:08:50,448 point o four five, is equal to one point two nine. 260 00:08:50,448 --> 00:08:53,129 And then this is multiplied by V-T. 261 00:08:53,129 --> 00:08:54,480 That's what's equal to V-G. 262 00:08:54,480 --> 00:08:55,885 So I have an expression for V-G. 263 00:08:55,885 --> 00:08:57,472 The final velocity of the golf ball 264 00:08:57,472 --> 00:08:59,834 is equal to this quantity right here. 265 00:08:59,834 --> 00:09:03,048 One point five six minus one point two nine V-T. 266 00:09:03,048 --> 00:09:04,777 So I'm gonna take this total expression, 267 00:09:04,777 --> 00:09:06,833 which is equal to V-G, 268 00:09:06,833 --> 00:09:09,523 and I'm gonna plug it in right over here. 269 00:09:09,523 --> 00:09:13,384 Which gives me 1/2 point o four five kilograms 270 00:09:13,384 --> 00:09:16,972 times the quantity, one point five six 271 00:09:16,972 --> 00:09:20,055 minus one point two nine V-T squared. 272 00:09:21,405 --> 00:09:24,317 Because this V-G was squared. 273 00:09:24,317 --> 00:09:26,358 And I'm just substituting the expression 274 00:09:26,358 --> 00:09:27,488 I have over here for V-G in 275 00:09:27,488 --> 00:09:29,478 for this quantity V-G. 276 00:09:29,478 --> 00:09:30,948 And I still have to multiply by the 1/2 277 00:09:30,948 --> 00:09:32,314 and the point o four five. 278 00:09:32,314 --> 00:09:33,665 And I still have all of this. 279 00:09:33,665 --> 00:09:36,694 So I still have 102 point 65 joules 280 00:09:36,694 --> 00:09:40,883 equals 1/2 point o five eight kilograms 281 00:09:40,883 --> 00:09:42,383 times V-T squared. 282 00:09:43,375 --> 00:09:45,667 Plus this quantity right here. 283 00:09:45,667 --> 00:09:48,007 Which is what I substituted in the V-G for. 284 00:09:48,007 --> 00:09:49,150 And it's getting a little messy. 285 00:09:49,150 --> 00:09:50,787 But at least I now have one equation 286 00:09:50,787 --> 00:09:52,516 with just one unknown. 287 00:09:52,516 --> 00:09:54,297 I just have V-T in here. 288 00:09:54,297 --> 00:09:57,409 So if I do the math I have 102 point 65 joules 289 00:09:57,409 --> 00:09:59,941 equals, if I just take point o five eight 290 00:09:59,941 --> 00:10:02,231 divided by two, I'm gonna get point o two nine 291 00:10:02,231 --> 00:10:04,310 and V-T squared. 292 00:10:04,310 --> 00:10:05,260 I'm gonna leave off the units. 293 00:10:05,260 --> 00:10:06,416 Things are gonna get messy. 294 00:10:06,416 --> 00:10:08,376 And then if I take point o four five divided by two 295 00:10:08,376 --> 00:10:11,397 I'll get point o two two five. 296 00:10:11,397 --> 00:10:13,476 And now I've gotta square this quantity. 297 00:10:13,476 --> 00:10:16,883 So if you remember, if you ever have a minus b squared, 298 00:10:16,883 --> 00:10:19,195 the result of that is gonna be 299 00:10:19,195 --> 00:10:23,199 a squared, which is one point five six squared. 300 00:10:23,199 --> 00:10:27,152 Minus two, times the quantity of the first one, 301 00:10:27,152 --> 00:10:29,361 one point five six times the quantity 302 00:10:29,361 --> 00:10:33,534 of the second one, which is one point two nine V-T. 303 00:10:33,534 --> 00:10:35,482 And then, plus, the final element here squared 304 00:10:35,482 --> 00:10:36,732 this b squared. 305 00:10:37,722 --> 00:10:40,931 Which is gonna be one point two nine squared 306 00:10:40,931 --> 00:10:44,559 times the velocity of the tennis ball squared. 307 00:10:44,559 --> 00:10:46,063 That may have made no sense at all. 308 00:10:46,063 --> 00:10:47,159 If so, what I'm really doing 309 00:10:47,159 --> 00:10:48,174 is I'm saying that if you ever have 310 00:10:48,174 --> 00:10:51,253 a minus b squared, that's just equal 311 00:10:51,253 --> 00:10:54,753 to a squared minus two a b plus b squared. 312 00:10:57,223 --> 00:10:58,089 And that's what I did. 313 00:10:58,089 --> 00:10:59,419 Here's my a. 314 00:10:59,419 --> 00:11:00,717 I did a squared, one point five six squared. 315 00:11:00,717 --> 00:11:04,513 And I did minus two times this first one 316 00:11:04,513 --> 00:11:07,266 times the second one, with the V-T in there. 317 00:11:07,266 --> 00:11:11,782 Plus b squared is gonna be plus this final term squared. 318 00:11:11,782 --> 00:11:15,069 Is one point two nine squared times V-T squared. 319 00:11:15,069 --> 00:11:16,371 So I've got this big mess now. 320 00:11:16,371 --> 00:11:17,615 I just need to clean it up. 321 00:11:17,615 --> 00:11:20,803 The left hand side is still 102 point 65. 322 00:11:20,803 --> 00:11:22,718 I've still got this point o two 323 00:11:22,718 --> 00:11:24,616 nine V-T squared sitting here. 324 00:11:24,616 --> 00:11:26,759 But I need to multiply this point o two two five 325 00:11:26,759 --> 00:11:28,891 throughout this whole quantity. 326 00:11:28,891 --> 00:11:30,620 Because it's multiplying this whole quantity. 327 00:11:30,620 --> 00:11:34,787 So if I do that, I've got plus point o five four eight. 328 00:11:35,849 --> 00:11:36,682 That's what? 329 00:11:36,682 --> 00:11:37,992 Point o two two five times one point 330 00:11:37,992 --> 00:11:39,018 five six squared is. 331 00:11:39,018 --> 00:11:42,601 Then I'll get minus point o nine o six V-T. 332 00:11:43,946 --> 00:11:45,833 That's what point o two two five times 333 00:11:45,833 --> 00:11:47,972 this whole quantity is. 334 00:11:47,972 --> 00:11:49,704 And then, finally, I'll get plus 335 00:11:49,704 --> 00:11:52,787 point o three seven four V-T squared. 336 00:11:54,828 --> 00:11:56,583 That's what point o two two five is 337 00:11:56,583 --> 00:11:59,036 times this quantity right here. 338 00:11:59,036 --> 00:12:00,286 So let's identify the V-T's. 339 00:12:00,286 --> 00:12:03,680 I got a V-T right here, just single V-T. 340 00:12:03,680 --> 00:12:06,333 And then I've got a V-T squared, right here. 341 00:12:06,333 --> 00:12:09,125 So I can combine this V-T squared term, 342 00:12:09,125 --> 00:12:11,325 with this V-T squared term. 343 00:12:11,325 --> 00:12:14,658 And I'll get point o six six V-T squared 344 00:12:15,600 --> 00:12:18,017 minus point o nine o six V-T, 345 00:12:19,603 --> 00:12:22,020 plus point o five four eight. 346 00:12:22,856 --> 00:12:24,607 Now we're getting close, I promise. 347 00:12:24,607 --> 00:12:28,768 If we subtract, there's 102 point 65 from both sides. 348 00:12:28,768 --> 00:12:32,579 We'll have zero equals this whole quantity again. 349 00:12:32,579 --> 00:12:34,255 And then point o five four eight 350 00:12:34,255 --> 00:12:35,838 minus 102 point 65. 351 00:12:36,741 --> 00:12:40,823 Is gonna be negative 102 point five nine five. 352 00:12:40,823 --> 00:12:42,163 And what this is right here, 353 00:12:42,163 --> 00:12:44,438 is the Quadratic Equation. 354 00:12:44,438 --> 00:12:45,932 And you can't solve this by just trying to 355 00:12:45,932 --> 00:12:48,142 isolate V-T on one side. 356 00:12:48,142 --> 00:12:49,339 It's never gonna work that way. 357 00:12:49,339 --> 00:12:51,822 You've got to use the Quadratic Formula. 358 00:12:51,822 --> 00:12:53,187 So in the Quadratic Formula, 359 00:12:53,187 --> 00:12:55,747 this term here, the point o six would be a. 360 00:12:55,747 --> 00:12:59,610 And this negative point o nine o six would be b. 361 00:12:59,610 --> 00:13:03,232 And this negative 102 point five nine five 362 00:13:03,232 --> 00:13:04,545 would be the c. 363 00:13:04,545 --> 00:13:06,262 You could either do this the long way by hand. 364 00:13:06,262 --> 00:13:08,835 Or you could just use a Quadratic Formula Solver. 365 00:13:08,835 --> 00:13:10,254 They're available online. 366 00:13:10,254 --> 00:13:11,658 They might be on your calculator. 367 00:13:11,658 --> 00:13:12,960 That's what I'm gonna do. 368 00:13:12,960 --> 00:13:14,220 I'm gonna do this on my calculator. 369 00:13:14,220 --> 00:13:15,310 So the two answers I'm getting 370 00:13:15,310 --> 00:13:16,469 out of this would be, 371 00:13:16,469 --> 00:13:20,444 V-T either equals, I'm getting 40 as one answer, 372 00:13:20,444 --> 00:13:21,575 meters per second. 373 00:13:21,575 --> 00:13:25,437 Or I'm getting negative 39 meters per second. 374 00:13:25,437 --> 00:13:26,827 And so which one is it? 375 00:13:26,827 --> 00:13:28,635 Is it gonna be 40 or negative 39? 376 00:13:28,635 --> 00:13:30,685 Well, we can figure out which one it is. 377 00:13:30,685 --> 00:13:32,797 Look at this V-T here, 40? 378 00:13:32,797 --> 00:13:34,955 That's the initial velocity it had already. 379 00:13:34,955 --> 00:13:37,476 We want the final velocity. 380 00:13:37,476 --> 00:13:39,725 Why is it giving us the initial velocity again? 381 00:13:39,725 --> 00:13:41,972 Because it turns out, one way to conserve 382 00:13:41,972 --> 00:13:45,311 momentum and energy, is for these objects 383 00:13:45,311 --> 00:13:46,694 to just miss each other. 384 00:13:46,694 --> 00:13:48,577 And fly right past each other. 385 00:13:48,577 --> 00:13:50,867 If the golf ball doesn't actually collide 386 00:13:50,867 --> 00:13:52,788 with the tennis ball. 387 00:13:52,788 --> 00:13:54,662 And the tennis ball just keeps going forward, 388 00:13:54,662 --> 00:13:56,798 they just both maintain whatever velocity 389 00:13:56,798 --> 00:13:57,983 they had initially. 390 00:13:57,983 --> 00:13:59,541 And that would correspond to this. 391 00:13:59,541 --> 00:14:01,647 This is a collision that missed. 392 00:14:01,647 --> 00:14:03,428 So we know that this collision 393 00:14:03,428 --> 00:14:05,767 was not the one we're looking for. 394 00:14:05,767 --> 00:14:06,771 That's if they didn't collide. 395 00:14:06,771 --> 00:14:08,786 We're looking for this velocity right here. 396 00:14:08,786 --> 00:14:10,137 So, after the collision, this tennis ball 397 00:14:10,137 --> 00:14:12,333 gets knocked backward, with negative 398 00:14:12,333 --> 00:14:14,748 39 meters per second of velocity. 399 00:14:14,748 --> 00:14:15,868 So how do we find the velocity 400 00:14:15,868 --> 00:14:17,325 of the golf ball after the collision? 401 00:14:17,325 --> 00:14:18,898 Well I've got the velocity of the tennis ball. 402 00:14:18,898 --> 00:14:22,186 Now all I have to do is bring that right back into here. 403 00:14:22,186 --> 00:14:24,057 And I can get what the velocity of the golf ball was. 404 00:14:24,057 --> 00:14:25,905 The velocity of the golf ball's 405 00:14:25,905 --> 00:14:27,441 now just gonna be one point five six 406 00:14:27,441 --> 00:14:29,520 minus one point two nine. 407 00:14:29,520 --> 00:14:32,270 Times this quantity, negative 39. 408 00:14:33,395 --> 00:14:34,421 And if I'm gonna multiply this out, 409 00:14:34,421 --> 00:14:37,682 I'm getting about 52 meters per second. 410 00:14:37,682 --> 00:14:39,503 Positive 52 meters per second, 411 00:14:39,503 --> 00:14:40,945 for the velocity of the golf ball. 412 00:14:40,945 --> 00:14:42,507 That means this golf ball got 413 00:14:42,507 --> 00:14:44,440 knocked back to the right, 414 00:14:44,440 --> 00:14:45,781 Because it's a positive velocity. 415 00:14:45,781 --> 00:14:47,224 And it got knocked out at a speed 416 00:14:47,224 --> 00:14:48,849 of 52 meters per second. 417 00:14:48,849 --> 00:14:50,432 So, recapping what we did, 418 00:14:50,432 --> 00:14:52,608 we were given the initial velocities and the masses. 419 00:14:52,608 --> 00:14:54,650 We tried to use Conservation of Momentum 420 00:14:54,650 --> 00:14:55,729 and that was fine. 421 00:14:55,729 --> 00:14:57,378 Except we had two unknowns. 422 00:14:57,378 --> 00:14:59,264 So we had to write down another equation. 423 00:14:59,264 --> 00:15:01,027 If we're told this collision is elastic, 424 00:15:01,027 --> 00:15:03,279 we know that total kinetic energy's conserved. 425 00:15:03,279 --> 00:15:05,384 We wrote down that equation 426 00:15:05,384 --> 00:15:08,052 but it also has two unknowns. 427 00:15:08,052 --> 00:15:09,086 So we solved the momentum equation 428 00:15:09,086 --> 00:15:12,092 for one of the variables, V-G. 429 00:15:12,092 --> 00:15:15,134 We substituted that expression into over here, 430 00:15:15,134 --> 00:15:17,253 for the V-G in this kinetic energy. 431 00:15:17,253 --> 00:15:19,191 We squared it, we had only one equation, 432 00:15:19,191 --> 00:15:20,896 with one unknown. 433 00:15:20,896 --> 00:15:22,987 But unfortunately, it gave us a Quadratic Equation. 434 00:15:22,987 --> 00:15:26,186 So we used the Quadratic Formula to solve. 435 00:15:26,186 --> 00:15:28,658 One of the velocities corresponded to 436 00:15:28,658 --> 00:15:30,892 the same as the initial velocity 437 00:15:30,892 --> 00:15:32,386 the object had in the first place. 438 00:15:32,386 --> 00:15:33,371 We don't want that one. 439 00:15:33,371 --> 00:15:34,295 Because that would mean that they 440 00:15:34,295 --> 00:15:35,420 didn't collide at all. 441 00:15:35,420 --> 00:15:37,352 We take the second one, 442 00:15:37,352 --> 00:15:38,850 if we wanna find the velocity of the first object. 443 00:15:38,850 --> 00:15:40,514 Then we take that, plug that back into 444 00:15:40,514 --> 00:15:42,877 this expression here. 445 00:15:42,877 --> 00:00:00,000 We get the velocity of the other object.