1 00:00:00,060 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,030 Your support will help MIT OpenCourseWare 4 00:00:06,030 --> 00:00:10,120 continue to offer high quality educational resources for free. 5 00:00:10,120 --> 00:00:12,660 To make a donation or to view additional materials 6 00:00:12,660 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:20,496 at ocw.mit.edu 8 00:00:20,496 --> 00:00:22,930 MARK HARTMAN: Build a well-- 9 00:00:22,930 --> 00:00:37,880 let's just say build a model of how angular size-- 10 00:00:37,880 --> 00:00:40,070 or let's just say angular width-- 11 00:00:43,420 --> 00:00:54,880 linear width, and distance relate. 12 00:00:54,880 --> 00:00:55,873 All right? 13 00:00:55,873 --> 00:00:59,740 What we've spent doing most of today is just figuring out 14 00:00:59,740 --> 00:01:02,390 how do we measure angular width, right? 15 00:01:02,390 --> 00:01:03,940 We learned about pixels. 16 00:01:03,940 --> 00:01:07,780 We looked at how to convert between one unit and another. 17 00:01:07,780 --> 00:01:10,000 And then we looked at how to use scientific notation 18 00:01:10,000 --> 00:01:12,410 to make that work. 19 00:01:12,410 --> 00:01:14,860 So now let's put everything together. 20 00:01:14,860 --> 00:01:19,810 Because we want to be able to predict if we can get 21 00:01:19,810 --> 00:01:23,830 pictures of objects, either in the room or in outer space. 22 00:01:23,830 --> 00:01:27,100 We want to be able to predict how big they actually are, 23 00:01:27,100 --> 00:01:30,250 how wide, or how tall they actually 24 00:01:30,250 --> 00:01:33,280 are if we know how far away they are, 25 00:01:33,280 --> 00:01:35,660 or maybe the other way around. 26 00:01:35,660 --> 00:01:38,660 So I want everybody to flip back in their notebook. 27 00:01:38,660 --> 00:01:42,070 And I want somebody to come up, and I 28 00:01:42,070 --> 00:01:46,930 want you to reproduce from yesterday when 29 00:01:46,930 --> 00:01:50,690 we looked at our relationship of the distance 30 00:01:50,690 --> 00:01:51,565 away from the camera. 31 00:01:56,790 --> 00:02:05,392 And then we actually said the actual width of the object, 32 00:02:05,392 --> 00:02:06,880 all right? 33 00:02:06,880 --> 00:02:11,680 And that was our diameter of those different balls. 34 00:02:11,680 --> 00:02:13,626 But instead of saying actual width now-- 35 00:02:13,626 --> 00:02:15,250 that's kind of what we said yesterday-- 36 00:02:15,250 --> 00:02:18,550 we're going to say the linear width as opposed 37 00:02:18,550 --> 00:02:20,640 to the angular width. 38 00:02:20,640 --> 00:02:26,200 The angular width we measured from the pixels on the image 39 00:02:26,200 --> 00:02:27,790 So who would like to come up? 40 00:02:27,790 --> 00:02:30,340 Some of you may have the width out here and the distance 41 00:02:30,340 --> 00:02:31,840 up this way. 42 00:02:31,840 --> 00:02:33,700 But who would like to come up and just 43 00:02:33,700 --> 00:02:38,320 say in words what relationship they found for objects that 44 00:02:38,320 --> 00:02:41,350 have the same angular size? 45 00:02:41,350 --> 00:02:45,100 Because remember, how wide they look in the image 46 00:02:45,100 --> 00:02:47,110 is our way of saying angular size. 47 00:02:47,110 --> 00:02:49,040 That's what we looked at this morning. 48 00:02:49,040 --> 00:02:59,455 So this is for objects with the same angular size. 49 00:03:04,280 --> 00:03:05,370 This was the relationship. 50 00:03:05,370 --> 00:03:09,510 So who would like to come up and put up the graph that they saw, 51 00:03:09,510 --> 00:03:12,416 and then explain that pattern in words? 52 00:03:12,416 --> 00:03:15,300 It's a chance to get up in front of your peers 53 00:03:15,300 --> 00:03:17,180 and put your ideas out there. 54 00:03:27,010 --> 00:03:29,090 [? Asif, ?] you want to go for it? 55 00:03:29,090 --> 00:03:31,870 OK, so bring up your plot. 56 00:03:31,870 --> 00:03:33,670 And it doesn't have to be perfect, but just 57 00:03:33,670 --> 00:03:35,690 show us the relationship between those two, 58 00:03:35,690 --> 00:03:37,615 and then explain to us in words. 59 00:03:37,615 --> 00:03:39,695 [? ASIF: ?] [INAUDIBLE] when I first 60 00:03:39,695 --> 00:03:42,560 plotted my point for my first object, 61 00:03:42,560 --> 00:03:47,590 the distance of an object was [INAUDIBLE] from the camera. 62 00:03:47,590 --> 00:03:49,886 And the actual width of the object-- 63 00:03:49,886 --> 00:03:53,722 or the angular width of the object-- 64 00:03:53,722 --> 00:03:55,810 was [? 5. ?] 65 00:03:55,810 --> 00:03:56,860 MARK HARTMAN: Hang on. 66 00:03:56,860 --> 00:03:59,177 Is it the actual width or is it the angular width? 67 00:03:59,177 --> 00:04:00,260 [? ASIF: ?] Angular width. 68 00:04:00,260 --> 00:04:01,090 Angular width. 69 00:04:01,090 --> 00:04:04,149 [INAUDIBLE] Yeah, angular width of the object. 70 00:04:04,149 --> 00:04:06,190 MARK HARTMAN: OK, what does that say right there? 71 00:04:06,190 --> 00:04:07,618 ASIF: It's actual width. 72 00:04:07,618 --> 00:04:09,950 [INAUDIBLE] I meant to say angular. 73 00:04:09,950 --> 00:04:11,830 MARK HARTMAN: OK. 74 00:04:11,830 --> 00:04:13,462 Did we measure yesterday anything 75 00:04:13,462 --> 00:04:14,545 to do with angular widths? 76 00:04:14,545 --> 00:04:16,089 [? ASIF: ?] No, it's actual width. 77 00:04:16,089 --> 00:04:17,380 MARK HARTMAN: No, so it wasn't. 78 00:04:17,380 --> 00:04:19,709 We didn't measure angular width. 79 00:04:19,709 --> 00:04:21,459 Instead of calling it angular and actual-- 80 00:04:21,459 --> 00:04:23,542 because that's kind of confusing-- [? Asif, ?] can 81 00:04:23,542 --> 00:04:27,650 you change that so that it says linear width of the object? 82 00:04:27,650 --> 00:04:30,580 Because the linear width, you could measure with a ruler. 83 00:04:30,580 --> 00:04:32,770 An angular width, you have to have a camera, 84 00:04:32,770 --> 00:04:36,620 and you measure the number of pixels. 85 00:04:36,620 --> 00:04:37,620 [? ASIF: ?] [INAUDIBLE]? 86 00:04:37,620 --> 00:04:39,355 MARK HARTMAN: Yep, perfect. 87 00:04:39,355 --> 00:04:44,190 [? ASIF: ?] The linear width of the object was [? 5, ?] 88 00:04:44,190 --> 00:04:50,384 and as it proceeded, [INAUDIBLE] the second object was 89 00:04:50,384 --> 00:04:53,484 a [? 50 ?] mark. 90 00:04:53,484 --> 00:05:00,826 And the linear width of it was [? 6. ?] And as it proceeded 91 00:05:00,826 --> 00:05:05,396 to the last one, which was [INAUDIBLE] distant mark 92 00:05:05,396 --> 00:05:08,256 [INAUDIBLE] linear width was [? 9. ?] 93 00:05:08,256 --> 00:05:18,920 As I plotted it, I noticed that each time the bigger object 94 00:05:18,920 --> 00:05:24,630 came and I used it, I had to move the object far more. 95 00:05:24,630 --> 00:05:28,290 And the linear width of the object was really [? hard. ?] 96 00:05:28,290 --> 00:05:31,990 So it just kept on increasing as the object got 97 00:05:31,990 --> 00:05:32,982 bigger and bigger. 98 00:05:32,982 --> 00:05:33,690 MARK HARTMAN: OK. 99 00:05:33,690 --> 00:05:36,300 That's a perfectly good explanation 100 00:05:36,300 --> 00:05:39,390 in words of what you saw from your experiment. 101 00:05:39,390 --> 00:05:41,835 Great, thanks, [? Asif. ?] Let's do a-- 102 00:05:41,835 --> 00:05:43,200 [APPLAUSE] 103 00:05:43,200 --> 00:05:44,880 So we're going to continue to try 104 00:05:44,880 --> 00:05:46,470 to have opportunities for you guys 105 00:05:46,470 --> 00:05:49,390 to stand up in front of everybody. 106 00:05:49,390 --> 00:05:52,080 But let's take a look at this. 107 00:05:52,080 --> 00:05:54,240 Because today we said anything that 108 00:05:54,240 --> 00:05:58,800 has the same angular size, which means the same angle coming out 109 00:05:58,800 --> 00:06:00,630 from the camera-- 110 00:06:00,630 --> 00:06:03,960 if you looked at those two lines, 111 00:06:03,960 --> 00:06:06,540 and you put an object that was very, very large 112 00:06:06,540 --> 00:06:09,420 but you put it really far away, that 113 00:06:09,420 --> 00:06:12,220 would be like saying let's put something even further. 114 00:06:12,220 --> 00:06:16,800 But let's predict how wide that object 115 00:06:16,800 --> 00:06:19,440 would have to be to be the same angular size. 116 00:06:19,440 --> 00:06:22,080 It would probably be up there. 117 00:06:22,080 --> 00:06:27,420 If we wanted to predict where we would need to put the linear-- 118 00:06:27,420 --> 00:06:33,300 if we had an object that was, let's say 2 centimeters wide, 119 00:06:33,300 --> 00:06:36,360 where do you think we'd need to put it? 120 00:06:36,360 --> 00:06:38,430 So maybe be like 20, right? 121 00:06:38,430 --> 00:06:40,570 It's going to be along this line. 122 00:06:40,570 --> 00:06:43,590 So let's say 2 and 20, right about there. 123 00:06:43,590 --> 00:06:45,465 Well, it's close enough. 124 00:06:45,465 --> 00:06:46,950 All right? 125 00:06:46,950 --> 00:06:50,280 So this understanding of this relationship 126 00:06:50,280 --> 00:06:54,510 between angular width, linear width, and distance-- 127 00:06:54,510 --> 00:06:56,970 we can create a mathematical model 128 00:06:56,970 --> 00:07:03,000 to help us predict where we should 129 00:07:03,000 --> 00:07:06,300 put an object of a certain linear width 130 00:07:06,300 --> 00:07:10,410 so that it always comes out to be the same angular size. 131 00:07:10,410 --> 00:07:13,260 Our mathematical model in this case-- 132 00:07:13,260 --> 00:07:16,140 we don't extend it down past zero-- 133 00:07:16,140 --> 00:07:18,492 says that if you have a really big object, 134 00:07:18,492 --> 00:07:19,950 you have to put it really far away. 135 00:07:19,950 --> 00:07:21,366 If you have a really small object, 136 00:07:21,366 --> 00:07:22,890 you can put it pretty close. 137 00:07:22,890 --> 00:07:25,290 If you had a zero-sized object, where 138 00:07:25,290 --> 00:07:29,316 would you need to put it in order to take up some space? 139 00:07:29,316 --> 00:07:31,164 AUDIENCE: [INAUDIBLE] 140 00:07:31,164 --> 00:07:32,550 AUDIENCE: At the zero? 141 00:07:32,550 --> 00:07:33,120 [INAUDIBLE] 142 00:07:33,120 --> 00:07:35,640 MARK HARTMAN: Yeah, you'd have to put it really, really 143 00:07:35,640 --> 00:07:39,690 close so that it would take up the same angular size. 144 00:07:39,690 --> 00:07:43,410 Well, things don't work particularly well all the way 145 00:07:43,410 --> 00:07:44,580 down to zero. 146 00:07:44,580 --> 00:07:46,180 But what do we see here? 147 00:07:46,180 --> 00:07:47,820 What kind of a relationship is this? 148 00:07:51,640 --> 00:07:52,140 Again? 149 00:07:52,140 --> 00:07:53,390 AUDIENCE: Linear. 150 00:07:53,390 --> 00:07:55,640 MARK HARTMAN: It's what we call a linear relationship. 151 00:07:55,640 --> 00:07:58,840 We are going to call this in the [? CAI. ?] 152 00:07:58,840 --> 00:08:01,520 And I want you guys to sketch this up, or maybe even 153 00:08:01,520 --> 00:08:03,110 on your graph. 154 00:08:03,110 --> 00:08:10,850 This is a direct relationship, all right? 155 00:08:10,850 --> 00:08:18,230 That means as one thing goes up, another thing goes up. 156 00:08:18,230 --> 00:08:21,560 Can you put this on the whiteboards? 157 00:08:21,560 --> 00:08:23,610 Put it on the projector? 158 00:08:23,610 --> 00:08:38,470 So a direct relationship means as one quantity goes up, 159 00:08:38,470 --> 00:08:39,429 another goes up. 160 00:08:42,280 --> 00:08:45,700 In this case, as our linear width of the object 161 00:08:45,700 --> 00:08:49,240 gets bigger, the distance that we 162 00:08:49,240 --> 00:08:51,660 have to put it from the camera also gets bigger. 163 00:08:56,330 --> 00:08:59,920 We can represent this as an equation. 164 00:08:59,920 --> 00:09:02,710 How many people have seen an equation like y 165 00:09:02,710 --> 00:09:07,540 equals mx plus b before? 166 00:09:07,540 --> 00:09:08,215 OK? 167 00:09:08,215 --> 00:09:12,940 Has anybody not seen y equals mx plus b? 168 00:09:12,940 --> 00:09:13,720 OK? 169 00:09:13,720 --> 00:09:16,030 So in first-year algebra, a lot of the times 170 00:09:16,030 --> 00:09:18,670 if you have two points, you can fit a line between them. 171 00:09:18,670 --> 00:09:21,010 If you're given a slope and a y-intercept, 172 00:09:21,010 --> 00:09:23,290 you can find the line between there. 173 00:09:23,290 --> 00:09:27,125 In this case, our y value is the linear width. 174 00:09:31,048 --> 00:09:31,960 All right? 175 00:09:31,960 --> 00:09:35,560 And you'll find whenever I write equations, 176 00:09:35,560 --> 00:09:38,300 I'm always going to write out the whole word. 177 00:09:38,300 --> 00:09:40,540 So here you want to say the linear width is 178 00:09:40,540 --> 00:09:43,990 equal to the slope, all right? 179 00:09:43,990 --> 00:09:46,210 So our slope here-- 180 00:09:46,210 --> 00:09:48,310 we don't know what that is just yet. 181 00:09:48,310 --> 00:09:51,010 So we're just going to call it m, 182 00:09:51,010 --> 00:09:53,480 because just like in this form. 183 00:09:53,480 --> 00:09:56,530 Now what is our x variable as we increase 184 00:09:56,530 --> 00:09:57,895 the distance from the camera? 185 00:10:03,650 --> 00:10:05,290 Say distance from detector. 186 00:10:08,560 --> 00:10:13,900 Now in this case, what is our y intercept? 187 00:10:13,900 --> 00:10:17,470 If we have x equals zero, what is the value of y? 188 00:10:17,470 --> 00:10:18,040 Zero. 189 00:10:18,040 --> 00:10:22,130 So in this case, we don't have any b value. 190 00:10:22,130 --> 00:10:27,070 So our mathematical relationship or our mathematical model 191 00:10:27,070 --> 00:10:30,700 is going to say in the case where an object is 192 00:10:30,700 --> 00:10:34,690 the same angular size, if I know the linear width, that 193 00:10:34,690 --> 00:10:37,000 is equal to some number-- 194 00:10:37,000 --> 00:10:38,650 the slope of that line-- 195 00:10:38,650 --> 00:10:41,260 times the distance from the detector. 196 00:10:41,260 --> 00:10:43,750 As the distance gets bigger, this linear width 197 00:10:43,750 --> 00:10:46,870 has to get bigger. 198 00:10:46,870 --> 00:10:52,210 This number m, the slope, is the angular size 199 00:10:52,210 --> 00:10:56,065 of that object in radians. 200 00:10:56,065 --> 00:10:57,440 So we're going to check this out. 201 00:10:57,440 --> 00:11:01,720 So I'm going to say linear width is 202 00:11:01,720 --> 00:11:16,100 equal to angular width in radians 203 00:11:16,100 --> 00:11:22,585 times the distance from the detector. 204 00:11:25,570 --> 00:11:27,325 Now does this make sense? 205 00:11:27,325 --> 00:11:28,200 Let's think about it. 206 00:11:28,200 --> 00:11:30,595 I just kind of identified this angular width 207 00:11:30,595 --> 00:11:33,880 in radians width the slope. 208 00:11:33,880 --> 00:11:36,527 In each case, when we looked at these dots-- 209 00:11:36,527 --> 00:11:38,860 this morning, when we looked at the different balls that 210 00:11:38,860 --> 00:11:41,170 were set up different distances away, 211 00:11:41,170 --> 00:11:45,790 the angle between the two sides, the two lines of sights, 212 00:11:45,790 --> 00:11:48,040 was always the same, right? 213 00:11:48,040 --> 00:11:51,850 So the angular width was the same. 214 00:11:51,850 --> 00:11:56,930 In the case of our mathematical model, at each of these points, 215 00:11:56,930 --> 00:12:00,007 the slope is always the same. 216 00:12:00,007 --> 00:12:01,090 So it kind of makes sense. 217 00:12:04,010 --> 00:12:13,990 So we're going to say our mathematical model 218 00:12:13,990 --> 00:12:29,580 for the relationship between angular width, linear width, 219 00:12:29,580 --> 00:12:31,830 and distance-- 220 00:12:31,830 --> 00:12:42,430 I'm just going to abbreviate angular width, linear width, 221 00:12:42,430 --> 00:12:43,370 and distance-- 222 00:12:48,450 --> 00:12:49,080 is this. 223 00:12:49,080 --> 00:12:49,940 All right? 224 00:12:49,940 --> 00:12:50,815 We're just going to-- 225 00:12:57,960 --> 00:13:01,175 Now if this mathematical model is any good-- 226 00:13:03,680 --> 00:13:07,015 just like any model, it has to make some predictions. 227 00:13:07,015 --> 00:13:10,740 It has to make some good predictions. 228 00:13:10,740 --> 00:13:14,370 So what I want you to do with your group is 229 00:13:14,370 --> 00:13:19,400 I want you to sit and think about how could I 230 00:13:19,400 --> 00:13:25,670 use this mathematical model to predict 231 00:13:25,670 --> 00:13:32,050 the size of an object in that picture that we took?