1 00:00:00,000 --> 00:00:00,390 2 00:00:00,390 --> 00:00:03,050 When I introduced you to the unit step function, I said, 3 00:00:03,049 --> 00:00:06,439 you know, this type of function, it's more exotic and 4 00:00:06,440 --> 00:00:10,240 a little unusual relative to what you've seen in just a 5 00:00:10,240 --> 00:00:12,900 traditional Calculus course, what you've seen in maybe your 6 00:00:12,900 --> 00:00:13,929 Algebra courses. 7 00:00:13,929 --> 00:00:15,990 But the reason why this was introduced is because a lot of 8 00:00:15,990 --> 00:00:18,589 physical systems kind of behave this way. 9 00:00:18,589 --> 00:00:21,699 That all of a sudden nothing happens for a long period of 10 00:00:21,699 --> 00:00:22,529 time and then bam! 11 00:00:22,530 --> 00:00:23,630 Something happens. 12 00:00:23,629 --> 00:00:24,449 And you go like that. 13 00:00:24,449 --> 00:00:26,439 And it doesn't happen exactly like this, but it can be 14 00:00:26,440 --> 00:00:29,330 approximated by the unit step function. 15 00:00:29,329 --> 00:00:34,390 Similarly, sometimes you have nothing happening for a long 16 00:00:34,390 --> 00:00:35,590 period of time. 17 00:00:35,590 --> 00:00:36,870 Nothing happens for a long period of 18 00:00:36,869 --> 00:00:39,349 time, and then whack! 19 00:00:39,350 --> 00:00:42,760 Something hits you really hard and then goes away, and then 20 00:00:42,759 --> 00:00:45,299 nothing happens for a very long period of time. 21 00:00:45,299 --> 00:00:46,919 And you'll learn this in the future, you can kind of view 22 00:00:46,920 --> 00:00:48,200 this is an impulse. 23 00:00:48,200 --> 00:00:49,679 And we'll talk about unit impulse 24 00:00:49,679 --> 00:00:50,600 functions and all of that. 25 00:00:50,600 --> 00:00:53,760 So wouldn't it be neat if we had some type of function that 26 00:00:53,759 --> 00:00:57,129 could model this type of behavior? 27 00:00:57,130 --> 00:01:00,359 And in our ideal function, what would happen is that 28 00:01:00,359 --> 00:01:04,909 nothing happens until we get to some point and then bam! 29 00:01:04,909 --> 00:01:07,149 It would get infinitely strong, but maybe it has a 30 00:01:07,150 --> 00:01:08,630 finite area. 31 00:01:08,629 --> 00:01:11,969 And then it would go back to zero and then go like that. 32 00:01:11,969 --> 00:01:16,659 So it'd be infinitely high right at 0 right there, and 33 00:01:16,659 --> 00:01:17,629 then it continues there. 34 00:01:17,629 --> 00:01:21,530 And let's say that the area under this, it becomes very-- 35 00:01:21,530 --> 00:01:23,879 to call this a function is actually kind of pushing it, 36 00:01:23,879 --> 00:01:26,649 and this is beyond the math of this video, but we'll call it 37 00:01:26,650 --> 00:01:27,990 a function in this video. 38 00:01:27,989 --> 00:01:32,299 But you say, well, what good is this function for? 39 00:01:32,299 --> 00:01:33,609 How can you even manipulate it? 40 00:01:33,609 --> 00:01:37,799 And I'm going to make one more definition of this function. 41 00:01:37,799 --> 00:01:40,069 Let's say we call this function represented by the 42 00:01:40,069 --> 00:01:43,089 delta, and that's what we do represent this function by. 43 00:01:43,090 --> 00:01:45,420 It's called the Dirac delta function. 44 00:01:45,420 --> 00:01:48,510 And we'll just informally say, look, when it's in infinity, 45 00:01:48,510 --> 00:01:52,480 it pops up to infinity when x equal to 0. 46 00:01:52,480 --> 00:01:56,525 And it's zero everywhere else when x is not equal to 0. 47 00:01:56,525 --> 00:01:58,040 And you say, how do I deal with that? 48 00:01:58,040 --> 00:01:59,840 How do I take the integral of that? 49 00:01:59,840 --> 00:02:02,219 And to help you with that, I'm going to make a definition. 50 00:02:02,219 --> 00:02:04,620 I'm going to tell you what the integral of this is. 51 00:02:04,620 --> 00:02:06,689 This is part of the definition of the function. 52 00:02:06,689 --> 00:02:09,469 I'm going to tell you that if I were to take the integral of 53 00:02:09,469 --> 00:02:12,509 this function from minus infinity to infinity, so 54 00:02:12,509 --> 00:02:16,750 essentially over the entire real number line, if I take 55 00:02:16,750 --> 00:02:22,009 the integral of this function, I'm defining it 56 00:02:22,009 --> 00:02:24,489 to be equal to 1. 57 00:02:24,490 --> 00:02:25,590 I'm defining this. 58 00:02:25,590 --> 00:02:27,599 Now, you might say, Sal, you didn't prove it to me. 59 00:02:27,599 --> 00:02:28,669 No, I'm defining it. 60 00:02:28,669 --> 00:02:31,959 I'm telling you that this delta of x is a function such 61 00:02:31,960 --> 00:02:34,640 that its integral is 1. 62 00:02:34,639 --> 00:02:38,169 So it has this infinitely narrow base that goes 63 00:02:38,169 --> 00:02:42,809 infinitely high, and the area under this-- I'm telling you-- 64 00:02:42,810 --> 00:02:44,090 is of area 1. 65 00:02:44,090 --> 00:02:47,379 And you're like, hey, Sal, that's a crazy function. 66 00:02:47,379 --> 00:02:50,490 I want a little bit better understanding of how someone 67 00:02:50,490 --> 00:02:52,110 can construct a function like this. 68 00:02:52,110 --> 00:02:56,520 So let's see if we can satisfy that a little bit more. 69 00:02:56,520 --> 00:02:57,880 But then once that's satisfied, then we're going to 70 00:02:57,879 --> 00:03:00,539 start taking the Laplace transform of this, and then 71 00:03:00,539 --> 00:03:03,239 we'll start manipulating it and whatnot. 72 00:03:03,240 --> 00:03:06,240 Let's see, let me complete this delta right here. 73 00:03:06,240 --> 00:03:08,420 Let's say that I constructed another function. 74 00:03:08,419 --> 00:03:14,109 Let's call it d sub tau And this is all just to satisfy 75 00:03:14,110 --> 00:03:17,640 this craving for maybe a better intuition for how this 76 00:03:17,639 --> 00:03:20,269 Dirac delta function can be constructed. 77 00:03:20,270 --> 00:03:24,930 And let's say my d sub tau of-- well, let me put it as a 78 00:03:24,930 --> 00:03:27,689 function of t because everything we're doing in the 79 00:03:27,689 --> 00:03:29,280 Laplace transform world, everything's been 80 00:03:29,280 --> 00:03:30,599 a function of t. 81 00:03:30,599 --> 00:03:40,000 So let's say that it equals 1 over 2 tau, and you'll see why 82 00:03:40,000 --> 00:03:42,289 I'm picking these numbers the way I am. 83 00:03:42,289 --> 00:03:48,804 1 over 2 tau when t is less then tau and 84 00:03:48,805 --> 00:03:51,260 greater than minus tau. 85 00:03:51,259 --> 00:03:53,639 And let's say it's 0 everywhere else. 86 00:03:53,639 --> 00:03:58,469 87 00:03:58,469 --> 00:04:01,889 So this type of equation, this is more reasonable. 88 00:04:01,889 --> 00:04:04,129 This will actually look like a combination of unit step 89 00:04:04,129 --> 00:04:06,120 functions, and we can actually define it as a combination of 90 00:04:06,120 --> 00:04:07,460 unit step functions. 91 00:04:07,460 --> 00:04:10,219 So if I draw, that's my x-axis. 92 00:04:10,219 --> 00:04:14,129 93 00:04:14,129 --> 00:04:19,680 And then if I put my y-axis right here. 94 00:04:19,680 --> 00:04:20,930 That's my y-axis. 95 00:04:20,930 --> 00:04:23,519 96 00:04:23,519 --> 00:04:25,709 Sorry, this is a t-axis. 97 00:04:25,709 --> 00:04:26,989 I have to get out of that habit. 98 00:04:26,990 --> 00:04:29,980 This is the t-axis, and, I mean, we could call it the 99 00:04:29,980 --> 00:04:33,430 y-axis or the f of t-axis, or whatever we want to call it. 100 00:04:33,430 --> 00:04:35,160 That's the dependent variable. 101 00:04:35,160 --> 00:04:36,910 So what's going to happen here? 102 00:04:36,910 --> 00:04:38,820 It's going to be zero everywhere until we get to 103 00:04:38,819 --> 00:04:40,870 minus t, and then at minus t, we're going to 104 00:04:40,870 --> 00:04:43,290 jump up to some level. 105 00:04:43,290 --> 00:04:44,819 Just let me put that point here. 106 00:04:44,819 --> 00:04:48,824 So this is minus tau, and this is plus tau. 107 00:04:48,824 --> 00:04:52,009 108 00:04:52,009 --> 00:04:53,699 So it's going to be zero everywhere, and then at minus 109 00:04:53,699 --> 00:04:58,039 tau, we jump to this level, and then we stay constant at 110 00:04:58,040 --> 00:05:01,670 that level until we get to plus tau. 111 00:05:01,670 --> 00:05:05,430 And that level, I'm saying is 1 over 2 tau. 112 00:05:05,430 --> 00:05:11,079 So this point right here on the dependent axis, this is 1 113 00:05:11,079 --> 00:05:13,079 over 2 tau. 114 00:05:13,079 --> 00:05:16,629 So why did I construct this function this way? 115 00:05:16,629 --> 00:05:17,709 Well, let's think about it. 116 00:05:17,709 --> 00:05:21,409 What happens if I take the integral? 117 00:05:21,410 --> 00:05:23,640 Let me write a nicer integral sign. 118 00:05:23,639 --> 00:05:29,789 If I took the integral from minus infinity to infinity of 119 00:05:29,790 --> 00:05:37,700 d sub tau of t dt, what is this going to be equal to? 120 00:05:37,699 --> 00:05:41,099 Well, if the integral is just the area under this curve, 121 00:05:41,100 --> 00:05:41,700 this is a pretty 122 00:05:41,699 --> 00:05:43,899 straightforward thing to calculate. 123 00:05:43,899 --> 00:05:46,439 You just look at this, and you say, well, first of all, it's 124 00:05:46,439 --> 00:05:48,480 zero everywhere else. 125 00:05:48,480 --> 00:05:52,379 It's zero everywhere else, and it's only the area right here. 126 00:05:52,379 --> 00:05:56,730 I mean, I could rewrite this integral as the integral from 127 00:05:56,730 --> 00:06:00,560 minus tau to tau-- and we don't care if infinity and 128 00:06:00,560 --> 00:06:02,870 minus infinity or positive infinity, because there's no 129 00:06:02,870 --> 00:06:06,280 area under any of those points-- of 1 130 00:06:06,279 --> 00:06:11,229 over 2 tau d tau. 131 00:06:11,230 --> 00:06:12,900 Sorry, 1 over 2 tau dt. 132 00:06:12,899 --> 00:06:14,459 So we could write it this way too, right? 133 00:06:14,459 --> 00:06:17,909 Because we can just take the boundaries from here to here, 134 00:06:17,910 --> 00:06:23,130 because we get nothing whether t goes to positive infinity or 135 00:06:23,129 --> 00:06:24,129 minus infinity. 136 00:06:24,129 --> 00:06:27,379 And then over that boundary, the function is a constant, 1 137 00:06:27,379 --> 00:06:29,839 over 2 tau, so we could just take this integral. 138 00:06:29,839 --> 00:06:31,310 And either way we evaluate it. 139 00:06:31,310 --> 00:06:32,990 We don't even have to know calculus to know what this 140 00:06:32,990 --> 00:06:34,680 integral's going to evaluate to. 141 00:06:34,680 --> 00:06:41,340 This is just the area under this, which is just the base. 142 00:06:41,339 --> 00:06:42,129 What's the base? 143 00:06:42,129 --> 00:06:44,274 The base is 2 tau. 144 00:06:44,274 --> 00:06:47,939 145 00:06:47,939 --> 00:06:50,550 You have one tau here and then another tau there. 146 00:06:50,550 --> 00:06:54,240 So it's equal to 2 tau times your height. 147 00:06:54,240 --> 00:06:58,460 And your height, I just said, is 1 over 2 tau. 148 00:06:58,459 --> 00:07:01,879 So your area for this function, or for this 149 00:07:01,879 --> 00:07:03,560 integral, is going to be 1. 150 00:07:03,560 --> 00:07:04,680 You could evaluate this. 151 00:07:04,680 --> 00:07:07,160 You could get this is going to be equal to-- you take the 152 00:07:07,160 --> 00:07:10,630 antiderivative of 1 over 2 tau, you get-- I'll do this 153 00:07:10,629 --> 00:07:14,230 just to satiate your curiosity-- t over 2 tau, and 154 00:07:14,230 --> 00:07:18,750 you have to evaluate this from minus tau to tau. 155 00:07:18,750 --> 00:07:23,310 And when you would put tau in there, you get tau over 2 tau, 156 00:07:23,310 --> 00:07:29,550 and then minus minus tau over 2 tau, and then you get tau 157 00:07:29,550 --> 00:07:33,850 plus tau over 2 tau, that's 2 tau over 2 tau, 158 00:07:33,850 --> 00:07:35,110 which is equal to 1. 159 00:07:35,110 --> 00:07:37,120 Maybe I'm beating a dead horse. 160 00:07:37,120 --> 00:07:40,490 I think you're satisfied that the area under this is going 161 00:07:40,490 --> 00:07:43,300 to be 1, regardless of what tau was. 162 00:07:43,300 --> 00:07:45,650 I kept this abstract. 163 00:07:45,649 --> 00:07:48,509 Now, if I take smaller and smaller values of tau, what's 164 00:07:48,509 --> 00:07:49,449 going to happen? 165 00:07:49,449 --> 00:07:56,729 If my new tau is going to be here, let's say my new tau is 166 00:07:56,730 --> 00:08:00,357 going to be there, I'm just going to pick up my new tau 167 00:08:00,357 --> 00:08:03,320 there, then my 1 over 2 tau, the tau is 168 00:08:03,319 --> 00:08:04,560 now a smaller number. 169 00:08:04,560 --> 00:08:06,250 So when it's in the denominator, my 1 over 2 tau 170 00:08:06,250 --> 00:08:12,209 is going to be something like this, right? 171 00:08:12,209 --> 00:08:14,669 I mean, I'm just saying, if I pick smaller and smaller taus. 172 00:08:14,670 --> 00:08:18,379 So then if I pick an even smaller tau than that, then my 173 00:08:18,379 --> 00:08:20,899 height is going to be have to be higher. 174 00:08:20,899 --> 00:08:22,750 My 1 over 2 tau is going to have to even 175 00:08:22,750 --> 00:08:25,009 be higher than that. 176 00:08:25,009 --> 00:08:26,860 And so I think you see where I'm going with this. 177 00:08:26,860 --> 00:08:31,080 What happens as the limit as tau approaches zero? 178 00:08:31,079 --> 00:08:41,168 So what is the limit as tau approaches zero of my little d 179 00:08:41,168 --> 00:08:44,379 sub tau function? 180 00:08:44,379 --> 00:08:46,070 What's the limit of this? 181 00:08:46,070 --> 00:08:49,370 Well, these things are going to go infinitely close to 182 00:08:49,370 --> 00:08:50,529 zero, but this is the limit. 183 00:08:50,529 --> 00:08:52,459 They're never going to be quite at zero. 184 00:08:52,460 --> 00:08:56,050 And your height here is going to go infinitely high, but the 185 00:08:56,049 --> 00:08:59,399 whole time, I said no matter what my tau is, because it was 186 00:08:59,399 --> 00:09:01,340 defined very arbitrarily, was my area is 187 00:09:01,340 --> 00:09:03,129 always going to be 1. 188 00:09:03,129 --> 00:09:08,860 So you're going to end up with your Dirac delta function. 189 00:09:08,860 --> 00:09:09,590 Let me write it now. 190 00:09:09,590 --> 00:09:10,430 I was going to write an x again. 191 00:09:10,429 --> 00:09:14,629 Your Dirac delta function is a function of t, and because of 192 00:09:14,629 --> 00:09:22,129 this, if you ask what's the limit as tau approaches zero 193 00:09:22,129 --> 00:09:26,220 of the integral from minus infinity to infinity of d sub 194 00:09:26,220 --> 00:09:30,899 tau of t dt, well, this should still be 1, right? 195 00:09:30,899 --> 00:09:33,230 Because this thing right here, this evaluates to 1. 196 00:09:33,230 --> 00:09:36,629 So as you take the limit as tau approaches zero-- and I'm 197 00:09:36,629 --> 00:09:38,939 being very generous with my definitions 198 00:09:38,940 --> 00:09:40,020 of limits and whatnot. 199 00:09:40,019 --> 00:09:41,329 I'm not being very rigorous. 200 00:09:41,330 --> 00:09:43,665 But I think you can kind of understand the intuition of 201 00:09:43,664 --> 00:09:44,469 where I'm going. 202 00:09:44,470 --> 00:09:46,629 This is going to be equal to 1. 203 00:09:46,629 --> 00:09:51,049 And so by the same intuitive argument, you could say that 204 00:09:51,049 --> 00:09:55,689 the limit from minus infinity to infinity of our Dirac delta 205 00:09:55,690 --> 00:10:01,600 function of t dt is also going to be 1. 206 00:10:01,600 --> 00:10:05,000 And likewise, the Dirac delta function-- I mean, this thing 207 00:10:05,000 --> 00:10:09,370 pops up to infinity at t is equal to 0. 208 00:10:09,370 --> 00:10:14,830 This thing, if I were to draw my x-axis like that, and then 209 00:10:14,830 --> 00:10:17,980 right at t equals 0, my Dirac delta function 210 00:10:17,980 --> 00:10:19,180 pops up like that. 211 00:10:19,179 --> 00:10:20,969 And you normally draw it like that. 212 00:10:20,970 --> 00:10:23,730 And you normally draw it so it goes up to 1 to kind 213 00:10:23,730 --> 00:10:24,670 of depict its area. 214 00:10:24,669 --> 00:10:26,629 But you actually put an arrow there, and so this is your 215 00:10:26,629 --> 00:10:28,269 Dirac delta function. 216 00:10:28,269 --> 00:10:29,819 But what happens if you want to shift it? 217 00:10:29,820 --> 00:10:32,800 218 00:10:32,799 --> 00:10:36,859 How would I represent my-- let's say I want 219 00:10:36,860 --> 00:10:38,659 to do t minus 3? 220 00:10:38,659 --> 00:10:40,699 What would the graph of this be? 221 00:10:40,700 --> 00:10:43,450 Well, this would just be shifting it to the right by 3. 222 00:10:43,450 --> 00:10:46,100 For example, when t equals 3, this will become the Dirac 223 00:10:46,100 --> 00:10:47,159 delta of 0. 224 00:10:47,159 --> 00:10:51,629 So this graph will just look like this. 225 00:10:51,629 --> 00:10:54,549 This will be my x-axis. 226 00:10:54,549 --> 00:10:58,609 And let's say that this is my y-axis. 227 00:10:58,610 --> 00:11:00,330 Let me just make that 1. 228 00:11:00,330 --> 00:11:04,810 And let me just draw some points here, so it's 1, 2, 3 229 00:11:04,809 --> 00:11:06,359 That's t is equal to 3. 230 00:11:06,360 --> 00:11:07,389 Did I say that was the x-axis? 231 00:11:07,389 --> 00:11:08,929 That's my t-axis. 232 00:11:08,929 --> 00:11:09,934 This is t equal to 3. 233 00:11:09,934 --> 00:11:12,759 And what I'm going to do here is the Dirac delta function is 234 00:11:12,759 --> 00:11:14,125 going to be zero everywhere. 235 00:11:14,125 --> 00:11:18,019 236 00:11:18,019 --> 00:11:19,870 But then right at 3, it goes infinitely high. 237 00:11:19,870 --> 00:11:22,240 And obviously, we don't have enough paper to draw an 238 00:11:22,240 --> 00:11:24,230 infinitely high spike right there. 239 00:11:24,230 --> 00:11:26,440 So what we do is we draw an arrow. 240 00:11:26,440 --> 00:11:27,660 We draw an arrow there. 241 00:11:27,659 --> 00:11:30,889 And the arrow, we usually draw the magnitude of the area 242 00:11:30,889 --> 00:11:32,480 under that spike. 243 00:11:32,480 --> 00:11:34,110 So we do it like this. 244 00:11:34,110 --> 00:11:35,279 And let me be clear. 245 00:11:35,279 --> 00:11:37,600 This is not saying that the function just goes to 1 and 246 00:11:37,600 --> 00:11:38,720 then spikes back down. 247 00:11:38,720 --> 00:11:41,080 This tells me that the area under the 248 00:11:41,080 --> 00:11:42,400 function is equal to 1. 249 00:11:42,399 --> 00:11:44,809 This spike would have to be infinitely high to have any 250 00:11:44,809 --> 00:11:50,250 area, considering it has an infinitely small base, so the 251 00:11:50,250 --> 00:11:55,470 area under this impulse function or under this Dirac 252 00:11:55,470 --> 00:11:57,250 delta function. 253 00:11:57,250 --> 00:12:01,940 Now, this one right here is t minus 3, but your area under 254 00:12:01,940 --> 00:12:03,740 this is still going to be 1. 255 00:12:03,740 --> 00:12:05,414 And that's why I made the arrow go to 1. 256 00:12:05,414 --> 00:12:07,959 257 00:12:07,960 --> 00:12:10,050 Let's say I wanted to graph-- let me do it in another color. 258 00:12:10,049 --> 00:12:14,750 Let's say I wanted to graph 2 times the Dirac 259 00:12:14,750 --> 00:12:17,519 delta of t minus 2. 260 00:12:17,519 --> 00:12:19,480 How would I graph this? 261 00:12:19,480 --> 00:12:21,830 Well, I would go to t minus 2. 262 00:12:21,830 --> 00:12:26,009 When t is equal to 2, you get the Dirac delta of zero, so 263 00:12:26,009 --> 00:12:27,490 that's where you would have your spike. 264 00:12:27,490 --> 00:12:31,549 And we're multiplying it by 2, so you would do a spike twice 265 00:12:31,549 --> 00:12:32,819 as high like this. 266 00:12:32,820 --> 00:12:35,250 Now, both of these go to infinity, but this goes twice 267 00:12:35,250 --> 00:12:36,460 as high to infinity. 268 00:12:36,460 --> 00:12:38,889 And I know this is all being a little ridiculous now. 269 00:12:38,889 --> 00:12:41,939 But the idea here is that the area under this curve should 270 00:12:41,940 --> 00:12:43,590 be twice the area under this curve. 271 00:12:43,590 --> 00:12:46,320 And that's why we make the arrow go to 2 to say that the 272 00:12:46,320 --> 00:12:49,020 area under this arrow is 2. 273 00:12:49,019 --> 00:12:51,199 The spike would have to go infinitely high. 274 00:12:51,200 --> 00:12:54,060 So this is all a little abstract, but this is a useful 275 00:12:54,059 --> 00:12:58,769 way to model things that are kind of very jarring. 276 00:12:58,769 --> 00:13:00,784 Obviously, nothing actually behaves like this, but there 277 00:13:00,784 --> 00:13:06,159 are a lot of phenomena in physics or the real world that 278 00:13:06,159 --> 00:13:08,819 have this spiky behavior. 279 00:13:08,820 --> 00:13:10,460 Instead of trying to say, oh, what does that spike 280 00:13:10,460 --> 00:13:11,180 exactly look like? 281 00:13:11,179 --> 00:13:13,359 We say, hey, that's a Dirac delta function. 282 00:13:13,360 --> 00:13:17,419 And we'll dictate its impulse by something like this. 283 00:13:17,419 --> 00:13:20,009 And just to give you a little bit of motivation behind this, 284 00:13:20,009 --> 00:13:23,200 and I was going to go here in the last video, but then I 285 00:13:23,200 --> 00:13:25,360 kind of decided not to. 286 00:13:25,360 --> 00:13:26,870 But I'm just going to show it, because I've been doing a lot 287 00:13:26,870 --> 00:13:28,370 of differential equations and I've been giving you no 288 00:13:28,370 --> 00:13:31,066 motivation for how this applies in the real world. 289 00:13:31,066 --> 00:13:35,720 But you can imagine, if I have just a wall, and then I have a 290 00:13:35,720 --> 00:13:42,740 spring attached to some mass right there, and let's say 291 00:13:42,740 --> 00:13:45,549 that this is a natural state of the spring, so that the 292 00:13:45,549 --> 00:13:48,789 spring would want to be here, so it's been stretched a 293 00:13:48,789 --> 00:13:53,339 distance y from its kind of natural where it wants to go. 294 00:13:53,340 --> 00:13:59,160 And let's say I have some external force right here. 295 00:13:59,159 --> 00:14:01,329 Let's say I have some external force right here on the 296 00:14:01,330 --> 00:14:03,600 spring, and, of course, let's say it's ice on ice. 297 00:14:03,600 --> 00:14:05,509 There's no friction in all of this. 298 00:14:05,509 --> 00:14:07,939 And I just want to show you that I can represent the 299 00:14:07,940 --> 00:14:10,860 behavior of this system with the differential equation. 300 00:14:10,860 --> 00:14:13,680 And actually things like the unit step functions, the Dirac 301 00:14:13,679 --> 00:14:17,519 delta function, actually start to become useful in this type 302 00:14:17,519 --> 00:14:18,659 of environment. 303 00:14:18,659 --> 00:14:24,610 So we know that F is equal to mass times acceleration. 304 00:14:24,610 --> 00:14:26,240 That's basic physics right there. 305 00:14:26,240 --> 00:14:28,940 Now, what are all of the forces on 306 00:14:28,940 --> 00:14:33,510 this mass right here? 307 00:14:33,509 --> 00:14:37,490 Well, you have this force right here. 308 00:14:37,490 --> 00:14:39,620 And I'll say this is a positive rightward direction, 309 00:14:39,620 --> 00:14:42,779 so it's that force, and then you have a minus force from 310 00:14:42,779 --> 00:14:44,649 the spring. 311 00:14:44,649 --> 00:14:47,100 The force from the spring is Hooke's Law. 312 00:14:47,100 --> 00:14:51,909 It's proportional to how far it's been stretched from its 313 00:14:51,909 --> 00:14:54,620 kind of natural point, so its force in that direction is 314 00:14:54,620 --> 00:14:57,879 going to be ky, or you could call it minus ky, because it's 315 00:14:57,879 --> 00:14:59,850 going in the opposite direction of what we've 316 00:14:59,850 --> 00:15:01,250 already said is a positive direction. 317 00:15:01,250 --> 00:15:06,970 So the net forces on this is F minus ky, and that's equal to 318 00:15:06,970 --> 00:15:13,470 the mass of our object times its acceleration. 319 00:15:13,470 --> 00:15:14,519 Now, what's its acceleration? 320 00:15:14,519 --> 00:15:20,509 If its position is y, so if y is equal to position, if we 321 00:15:20,509 --> 00:15:24,069 take the derivative of y with respect to t, y prime, which 322 00:15:24,070 --> 00:15:29,570 we could also say dy dt, this is going to be its velocity. 323 00:15:29,570 --> 00:15:32,210 And then if we take the derivative of that, y prime 324 00:15:32,210 --> 00:15:36,200 prime, which is equal to d squared y with respect to dt 325 00:15:36,200 --> 00:15:37,940 squared, this is equal to acceleration. 326 00:15:37,940 --> 00:15:41,080 327 00:15:41,080 --> 00:15:48,340 So instead of writing a, we could right y prime prime. 328 00:15:48,340 --> 00:15:51,110 And so, if we just put this on the other side of the 329 00:15:51,110 --> 00:15:54,009 equation, what do we get? 330 00:15:54,009 --> 00:15:58,269 We get the force-- this force, not just this force; this was 331 00:15:58,269 --> 00:16:02,539 just F equals ma-- but this force is equal to the mass of 332 00:16:02,539 --> 00:16:05,610 our object, times the acceleration of the object 333 00:16:05,610 --> 00:16:09,399 plus whatever the spring constant is for the spring 334 00:16:09,399 --> 00:16:14,139 plus k times our position, times y. 335 00:16:14,139 --> 00:16:17,580 So if you had no outside force, if this was zero you'd 336 00:16:17,580 --> 00:16:20,370 have a homogeneous differential equation. 337 00:16:20,370 --> 00:16:23,149 And in that case, the spring would just start 338 00:16:23,149 --> 00:16:24,559 moving on its own. 339 00:16:24,559 --> 00:16:27,589 But now this F, all of a sudden, it's kind of a 340 00:16:27,590 --> 00:16:31,610 non-homogeneous term, it's what the outside force you're 341 00:16:31,610 --> 00:16:33,899 applying to this mass. 342 00:16:33,899 --> 00:16:36,779 So if this outside force was some type of Dirac delta 343 00:16:36,779 --> 00:16:42,309 function-- so let's say it's t minus 2 is equal to our mass 344 00:16:42,309 --> 00:16:50,189 times y prime prime plus our spring constant times y, this 345 00:16:50,190 --> 00:16:54,180 is saying that at time is equal to 2 seconds, we're just 346 00:16:54,179 --> 00:16:56,769 going to jar this thing to the right. 347 00:16:56,769 --> 00:16:58,960 And it's going to have an-- and I'll talk more about it-- 348 00:16:58,960 --> 00:17:00,360 it's going to have an impulse of 2. 349 00:17:00,360 --> 00:17:04,279 It's force times time is going to be-- or its impulse is 350 00:17:04,279 --> 00:17:05,910 going to have 1. 351 00:17:05,910 --> 00:17:07,560 And I don't want to get too much into the physics here, 352 00:17:07,559 --> 00:17:11,629 but its impulse or its change in momentum, is going to be of 353 00:17:11,630 --> 00:17:14,430 magnitude 1, depending on what our units are. 354 00:17:14,430 --> 00:17:16,890 But anyway, I just wanted to take a slight diversion, 355 00:17:16,890 --> 00:17:20,329 because you might think Sal is introducing me to these weird, 356 00:17:20,328 --> 00:17:21,348 exotic functions. 357 00:17:21,348 --> 00:17:23,568 What are they ever going to be good for? 358 00:17:23,568 --> 00:17:27,368 But this is good for the idea that sometimes you just jar 359 00:17:27,368 --> 00:17:29,750 this thing by some magnitude and then let go. 360 00:17:29,750 --> 00:17:33,789 And you do it kind of infinitely fast, but you do it 361 00:17:33,789 --> 00:17:38,539 enough to change the momentum of this in a well-defined way. 362 00:17:38,539 --> 00:17:41,139 Anyway, in the next video, we'll continue with the Dirac 363 00:17:41,140 --> 00:17:41,870 delta function. 364 00:17:41,869 --> 00:17:44,829 We'll figure out its Laplace transform and see what it does 365 00:17:44,829 --> 00:17:47,649 to the Laplace transforms of other functions.