1 00:00:00,090 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,720 continue to offer high quality educational resources for free. 5 00:00:10,720 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,280 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,280 --> 00:00:20,587 at ocw.mit.edu. 8 00:00:20,587 --> 00:00:22,920 MARK HARTMAN: And you guys can use these sheets of paper 9 00:00:22,920 --> 00:00:24,210 to actually write these things down 10 00:00:24,210 --> 00:00:26,251 until you feel comfortable enough remembering all 11 00:00:26,251 --> 00:00:28,170 the steps to do it yourself. 12 00:00:28,170 --> 00:00:31,259 So question-- and I'm going to give you 13 00:00:31,259 --> 00:00:32,259 this question this time. 14 00:00:32,259 --> 00:00:34,092 Sometimes you have to come up with your own. 15 00:00:34,092 --> 00:00:36,840 We want to find out how-- 16 00:00:43,730 --> 00:00:51,129 what is the linear height? 17 00:00:51,129 --> 00:00:52,420 Because what did we figure out? 18 00:00:52,420 --> 00:00:58,040 We did-- yeah, what is the linear height of the trash can? 19 00:01:02,539 --> 00:01:04,080 And again, this will be more exciting 20 00:01:04,080 --> 00:01:06,000 when, instead of measuring linear heights of trash cans, 21 00:01:06,000 --> 00:01:08,541 we're actually measuring linear heights of galaxies or things 22 00:01:08,541 --> 00:01:09,120 like that. 23 00:01:09,120 --> 00:01:16,620 What is the linear height of the trash can in the practice 24 00:01:16,620 --> 00:01:18,670 image? 25 00:01:18,670 --> 00:01:21,870 And you want to be as specific as you can, right? 26 00:01:21,870 --> 00:01:24,180 What is the linear height of what-- of the trash can? 27 00:01:24,180 --> 00:01:24,679 Where? 28 00:01:24,679 --> 00:01:26,190 In the practice image-- that's why 29 00:01:26,190 --> 00:01:29,110 we gave you this printout of the image that's right there. 30 00:01:29,110 --> 00:01:31,500 You've already labeled it, OK? 31 00:01:31,500 --> 00:01:33,260 So your question-- and actually I 32 00:01:33,260 --> 00:01:35,440 want you to physically write down the question, 33 00:01:35,440 --> 00:01:38,540 and nobody should be clicking on the computer. 34 00:01:38,540 --> 00:01:43,130 Jaylen, you shouldn't be using the computer at all right now. 35 00:01:43,130 --> 00:01:46,170 OK, so what is the linear height of the trash 36 00:01:46,170 --> 00:01:47,700 can in the practice image, that's 37 00:01:47,700 --> 00:01:49,320 what we're trying to get to. 38 00:01:49,320 --> 00:01:53,880 The next part is the diagram. 39 00:01:53,880 --> 00:01:54,970 Let's draw a diagram. 40 00:01:54,970 --> 00:01:57,420 We already have the image. 41 00:01:57,420 --> 00:02:00,510 You can draw, and you can even, like Peter said, 42 00:02:00,510 --> 00:02:01,260 you could draw-- 43 00:02:01,260 --> 00:02:06,240 you know, here's a little picture of the trash can. 44 00:02:06,240 --> 00:02:07,230 We measured this. 45 00:02:07,230 --> 00:02:12,960 This is the angular height. 46 00:02:12,960 --> 00:02:13,935 This is the image. 47 00:02:17,470 --> 00:02:19,600 But we also want to draw a diagram. 48 00:02:19,600 --> 00:02:22,160 So this is what the camera saw, and there 49 00:02:22,160 --> 00:02:25,367 is a ball here, and a couple of little other things, 50 00:02:25,367 --> 00:02:26,200 and the meter stick. 51 00:02:28,849 --> 00:02:30,390 Now that's what the image looks like. 52 00:02:30,390 --> 00:02:31,680 That's what we printed out. 53 00:02:31,680 --> 00:02:34,630 But to make it clear, we also want to draw the situation, 54 00:02:34,630 --> 00:02:36,120 remember, from the side. 55 00:02:36,120 --> 00:02:40,284 We want to know this is a three-dimensional situation, 56 00:02:40,284 --> 00:02:41,700 so we also want to say, well, this 57 00:02:41,700 --> 00:02:44,100 is what it looked like from the camera's point of view. 58 00:02:44,100 --> 00:02:47,640 This is a side view. 59 00:02:47,640 --> 00:02:51,150 Side view, right? 60 00:02:51,150 --> 00:02:52,970 Here's our detector. 61 00:02:52,970 --> 00:02:58,350 Here's our camera, and then here's the board. 62 00:02:58,350 --> 00:03:01,590 And the trash can, it's kind of like this. 63 00:03:01,590 --> 00:03:03,850 You can draw it three dimensional if you want. 64 00:03:03,850 --> 00:03:07,530 Well, if it's from the side, here's the board, 65 00:03:07,530 --> 00:03:13,410 here's the chalkboard, here's the trash can like that. 66 00:03:13,410 --> 00:03:20,695 And we know that this is the distance to detector. 67 00:03:25,000 --> 00:03:27,160 If you think it's too hard to draw the side view, 68 00:03:27,160 --> 00:03:29,470 you can also draw a little perspective view, 69 00:03:29,470 --> 00:03:31,570 and you can say here's my camera. 70 00:03:31,570 --> 00:03:35,890 And then along the wall, I had the trash can. 71 00:03:35,890 --> 00:03:37,230 I had the ball. 72 00:03:37,230 --> 00:03:39,100 I had there was another thing. 73 00:03:39,100 --> 00:03:41,635 And again, this is still the distance. 74 00:03:46,830 --> 00:03:49,490 So this is the distance to the detector. 75 00:03:49,490 --> 00:03:53,210 This, in the image, we said that was the angular 76 00:03:53,210 --> 00:03:55,400 height of the trash can. 77 00:03:55,400 --> 00:03:58,175 Now what is this if we could measure this quantity? 78 00:04:04,780 --> 00:04:06,636 Is it the angular height of the trash can 79 00:04:06,636 --> 00:04:07,885 if we're drawing this diagram? 80 00:04:16,390 --> 00:04:18,339 In this case, if this is our side view, 81 00:04:18,339 --> 00:04:20,440 this is the distance to the detector. 82 00:04:20,440 --> 00:04:28,620 This is the linear height of the trash can. 83 00:04:33,080 --> 00:04:35,450 You want to label everything that you can. 84 00:04:38,480 --> 00:04:41,000 So here was our image, and we know that, in an image, 85 00:04:41,000 --> 00:04:44,990 we can only measure angular heights, or angular widths, 86 00:04:44,990 --> 00:04:48,320 from our side view-- or this is a perspective view. 87 00:04:53,356 --> 00:04:54,855 That's the distance to the detector. 88 00:05:00,950 --> 00:05:08,110 This height right here is, again, the linear height 89 00:05:08,110 --> 00:05:08,950 of the trash can. 90 00:05:11,650 --> 00:05:15,880 I cannot overemphasize enough how important it is to try 91 00:05:15,880 --> 00:05:18,070 your best to draw these pictures, 92 00:05:18,070 --> 00:05:21,070 these little diagrams, because that's going to help you 93 00:05:21,070 --> 00:05:24,970 remember what did the situation look like? 94 00:05:24,970 --> 00:05:27,400 This is your kind of three-dimensional model 95 00:05:27,400 --> 00:05:30,100 of what you think is going on, the physical situation, 96 00:05:30,100 --> 00:05:33,850 that you want to apply your mathematical model to. 97 00:05:33,850 --> 00:05:34,720 OK? 98 00:05:34,720 --> 00:05:37,621 So there is a side view, there's the perspective view, 99 00:05:37,621 --> 00:05:38,370 there's the image. 100 00:05:38,370 --> 00:05:39,704 So now we've labeled everything. 101 00:05:39,704 --> 00:05:41,869 We know that we're going to worry about the distance 102 00:05:41,869 --> 00:05:42,724 to the detector. 103 00:05:42,724 --> 00:05:44,890 We know we're going to worry about the linear height 104 00:05:44,890 --> 00:05:45,473 of the object. 105 00:05:48,821 --> 00:05:49,320 Next part. 106 00:06:01,790 --> 00:06:06,950 Next known-- what do we know about this situation? 107 00:06:06,950 --> 00:06:08,840 We've just labeled this situation. 108 00:06:08,840 --> 00:06:11,460 Do we know any of these quantities? 109 00:06:11,460 --> 00:06:13,090 What do we know, [INAUDIBLE]? 110 00:06:13,090 --> 00:06:14,070 AUDIENCE: [INAUDIBLE] 111 00:06:14,070 --> 00:06:15,195 MARK HARTMAN: We know what? 112 00:06:15,195 --> 00:06:17,570 AUDIENCE: [INAUDIBLE] distance to the detector. 113 00:06:17,570 --> 00:06:19,695 MARK HARTMAN: We know the distance to the detector, 114 00:06:19,695 --> 00:06:27,020 so we're going to say distance to detector is equal to, 115 00:06:27,020 --> 00:06:29,185 what, 10 meters. 116 00:06:31,850 --> 00:06:33,170 OK, good. 117 00:06:33,170 --> 00:06:36,090 Do we know anything else? 118 00:06:36,090 --> 00:06:40,020 OK, we know the angular height of the object-- 119 00:06:40,020 --> 00:06:53,006 so angular height of the trash can equals what? 120 00:06:53,006 --> 00:06:53,942 AUDIENCE: 93. 121 00:06:53,942 --> 00:06:55,205 MARK HARTMAN: What was it? 122 00:06:55,205 --> 00:06:56,690 AUDIENCE: 93. 123 00:06:56,690 --> 00:06:58,270 MARK HARTMAN: 93 is just a number. 124 00:06:58,270 --> 00:07:00,317 I'm looking for a quantity. 125 00:07:00,317 --> 00:07:04,140 AUDIENCE: Then 93 phy. 126 00:07:04,140 --> 00:07:08,318 MARK HARTMAN: OK, but phy is short for? 127 00:07:08,318 --> 00:07:09,560 AUDIENCE: Physics-- 128 00:07:09,560 --> 00:07:11,270 MARK HARTMAN: Physical units. 129 00:07:11,270 --> 00:07:14,360 But we really mean pixels. 130 00:07:14,360 --> 00:07:16,880 That phy is pixels. 131 00:07:16,880 --> 00:07:21,090 So it's 93 pixels, OK? 132 00:07:23,990 --> 00:07:25,160 Do we know anything else? 133 00:07:30,110 --> 00:07:31,000 Not really, right? 134 00:07:31,000 --> 00:07:36,310 I mean, we know which camera we're using, but want-- 135 00:07:36,310 --> 00:07:37,580 what do we want? 136 00:07:37,580 --> 00:07:39,200 What do we find out? 137 00:07:39,200 --> 00:07:41,950 In this point, always look back at the original question. 138 00:07:41,950 --> 00:07:43,040 What do I want? 139 00:07:43,040 --> 00:07:45,890 What is the linear height of the trash can? 140 00:07:45,890 --> 00:07:48,500 So that's already also labeled in the diagram. 141 00:07:48,500 --> 00:07:54,700 You want the linear height of the trash can. 142 00:07:57,700 --> 00:07:59,560 All right. 143 00:07:59,560 --> 00:08:02,646 Do you want anything else? 144 00:08:02,646 --> 00:08:04,020 No, it's a very simple statement. 145 00:08:04,020 --> 00:08:07,690 What is the linear height of the trash can? 146 00:08:07,690 --> 00:08:18,230 Next part, relationships-- we want to look at the quantities 147 00:08:18,230 --> 00:08:20,960 that we know and the quantities that we want, and we 148 00:08:20,960 --> 00:08:23,330 want to think about is there a mathematical relationship 149 00:08:23,330 --> 00:08:25,550 or a mathematical model that describes 150 00:08:25,550 --> 00:08:28,010 how those are related. 151 00:08:28,010 --> 00:08:30,320 And there is. 152 00:08:30,320 --> 00:08:33,320 AUDIENCE: [INAUDIBLE] 153 00:08:35,600 --> 00:08:38,100 MARK HARTMAN: OK, so I actually want you to physically write 154 00:08:38,100 --> 00:08:41,590 that down, linear width. 155 00:08:41,590 --> 00:08:43,470 Well, in that case, we said width, 156 00:08:43,470 --> 00:08:45,240 but we could also say height, right? 157 00:08:45,240 --> 00:08:47,370 Linear height is just an up and down measurement. 158 00:08:47,370 --> 00:09:01,910 So linear height is equal to angular height in radians 159 00:09:01,910 --> 00:09:07,450 times distance to the detector. 160 00:09:10,360 --> 00:09:12,990 OK, so that's it. 161 00:09:12,990 --> 00:09:15,060 State, in words, what these equations 162 00:09:15,060 --> 00:09:18,290 mean in terms of a physical situation. 163 00:09:18,290 --> 00:09:20,170 [INAUDIBLE] already did that for us before. 164 00:09:20,170 --> 00:09:24,900 He said, for objects farther from the detector, that 165 00:09:24,900 --> 00:09:28,170 must mean, if their angular height is the same, 166 00:09:28,170 --> 00:09:32,225 they must be a larger linear height. 167 00:09:32,225 --> 00:09:33,600 So you don't always have to write 168 00:09:33,600 --> 00:09:35,940 that down if you don't want to, but think 169 00:09:35,940 --> 00:09:37,920 about what this equation means. 170 00:09:37,920 --> 00:09:40,950 It's not just, you know, some numbers to put together. 171 00:09:40,950 --> 00:09:43,510 And you notice that I'm not writing letters or anything. 172 00:09:43,510 --> 00:09:47,500 I'm just using the actual phrase for right now. 173 00:09:47,500 --> 00:09:49,335 Now the next thing is convert units. 174 00:09:54,110 --> 00:09:57,650 This is what we learned how to do today, the unit conversion. 175 00:09:57,650 --> 00:10:00,154 We've been a lot of stuff today, so this is probably 176 00:10:00,154 --> 00:10:01,820 going to be one of the most intense days 177 00:10:01,820 --> 00:10:02,653 in the whole summer. 178 00:10:02,653 --> 00:10:04,730 From here on out, hopefully we won't 179 00:10:04,730 --> 00:10:07,700 be doing quite as much standing at the board. 180 00:10:07,700 --> 00:10:08,870 So convert the units. 181 00:10:08,870 --> 00:10:10,664 You guys have already done that, but if you 182 00:10:10,664 --> 00:10:12,080 were doing this for the first time 183 00:10:12,080 --> 00:10:15,470 then you'd say, oh, angular height in pixels 184 00:10:15,470 --> 00:10:19,050 but my relationship calls for angular height in radians. 185 00:10:19,050 --> 00:10:21,710 So I'm going to have to change my angular 186 00:10:21,710 --> 00:10:23,090 height from pixels to radians. 187 00:10:23,090 --> 00:10:26,126 And you would actually do that calculation right here. 188 00:10:26,126 --> 00:10:27,500 Now we've already done it before, 189 00:10:27,500 --> 00:10:39,244 but we can now say angular height of trash can equals-- 190 00:10:39,244 --> 00:10:39,910 and what was it? 191 00:10:43,300 --> 00:10:51,730 7.34 times 10 to the minus 2 radians, right? 192 00:10:51,730 --> 00:10:52,957 Now we converted the units. 193 00:10:52,957 --> 00:10:54,790 And now it says, now that you know all this, 194 00:10:54,790 --> 00:10:58,460 turn over and start calculating on the back. 195 00:10:58,460 --> 00:11:01,780 And it says, solve the equation for the quantity you want. 196 00:11:01,780 --> 00:11:06,010 Right now, it's already solved for linear height. 197 00:11:06,010 --> 00:11:09,580 Later on, we'll have to work with solving the equation 198 00:11:09,580 --> 00:11:11,980 for the right quantity. 199 00:11:11,980 --> 00:11:14,680 Plug in the numbers with units and simplify. 200 00:11:14,680 --> 00:11:17,200 Make only one math step per line and rewrite 201 00:11:17,200 --> 00:11:20,320 the entire relationship on every line. 202 00:11:20,320 --> 00:11:22,780 So what I want you guys to do now with your group-- 203 00:11:22,780 --> 00:11:26,350 now that you filled out the front of this, 204 00:11:26,350 --> 00:11:30,070 I want you to rewrite this relationship on the back 205 00:11:30,070 --> 00:11:34,660 at the top, and then plug in those numbers that you've got, 206 00:11:34,660 --> 00:11:40,990 calculate what you end up getting for the-- 207 00:11:40,990 --> 00:11:44,710 well, actually let's just do this right here. 208 00:11:44,710 --> 00:11:47,800 So we'd say linear height-- 209 00:11:47,800 --> 00:11:51,070 I want to go through this with you once-- 210 00:11:51,070 --> 00:12:01,105 equals angular height times distance to detector. 211 00:12:04,080 --> 00:12:06,580 All right, so we plug those values in. 212 00:12:06,580 --> 00:12:14,950 Our angular height, 7.34, times 10 to the minus 2 radians, 213 00:12:14,950 --> 00:12:17,300 times what's the distance to the detector? 214 00:12:17,300 --> 00:12:17,905 10 meters. 215 00:12:21,830 --> 00:12:28,690 So then that gives us 7.34 times 10 to the minus 2 216 00:12:28,690 --> 00:12:33,325 plus 1 radian meters. 217 00:12:36,370 --> 00:12:37,780 Nice. 218 00:12:37,780 --> 00:12:44,200 So 7.34 times 10 to the minus 2 plus 1 219 00:12:44,200 --> 00:12:50,700 is minus 1 radian meters. 220 00:12:50,700 --> 00:12:53,260 OK, here's where I have to just say 221 00:12:53,260 --> 00:12:56,740 a radian, because it's a unit of angle, 222 00:12:56,740 --> 00:13:01,550 doesn't quite work out to be the same way as other units. 223 00:13:01,550 --> 00:13:05,050 So when you have a radian times another quantity, 224 00:13:05,050 --> 00:13:08,800 you just end up with that quantity. 225 00:13:08,800 --> 00:13:11,770 Because our answer, we want it in linear height. 226 00:13:11,770 --> 00:13:15,370 A linear height is going to be measured in meters. 227 00:13:15,370 --> 00:13:18,160 So when you're working with radians, 228 00:13:18,160 --> 00:13:21,610 you can just drop those units out 229 00:13:21,610 --> 00:13:24,310 so long as it makes sense with what you're trying to end up 230 00:13:24,310 --> 00:13:25,492 with, the linear height. 231 00:13:25,492 --> 00:13:26,950 And the linear height's going to be 232 00:13:26,950 --> 00:13:40,880 measured in meters, so then if I rewrite this linear height as-- 233 00:13:40,880 --> 00:13:45,190 now if I multiply these out, 7.34 times 10 to the minus 1, 234 00:13:45,190 --> 00:13:49,590 that means I move the decimal point over 0.1, 235 00:13:49,590 --> 00:13:53,000 or I move the decimal point over one place. 236 00:13:53,000 --> 00:14:00,860 So I've got 0.734 meters is my prediction 237 00:14:00,860 --> 00:14:05,330 for the linear height of the trash can. 238 00:14:05,330 --> 00:14:09,530 Predictions are not good unless they actually 239 00:14:09,530 --> 00:14:14,210 are somewhat close to reality, so what I want to do 240 00:14:14,210 --> 00:14:18,785 is ask [INAUDIBLE] if he could grab a meter stick. 241 00:14:18,785 --> 00:14:24,140 And I want you to measure how tall that trash can is. 242 00:14:38,517 --> 00:14:40,100 So measure from the bottom all the way 243 00:14:40,100 --> 00:14:41,210 to the top of the ridge. 244 00:14:41,210 --> 00:14:45,710 I think most people measured to the top of the curved part. 245 00:14:45,710 --> 00:14:47,450 There you go. 246 00:14:47,450 --> 00:14:49,358 What does it come out? 247 00:14:49,358 --> 00:14:50,640 AUDIENCE: It's about 81. 248 00:14:50,640 --> 00:14:50,960 MARK HARTMAN: How much? 249 00:14:50,960 --> 00:14:51,710 AUDIENCE: 81. 250 00:14:51,710 --> 00:14:54,560 MARK HARTMAN: About 81. 251 00:14:54,560 --> 00:14:56,344 So 81 what? 252 00:14:56,344 --> 00:14:57,260 AUDIENCE: Centimeters. 253 00:14:57,260 --> 00:14:58,490 MARK HARTMAN: 81 centimeters. 254 00:15:02,980 --> 00:15:08,890 So 81 centimeters is about 0.81 meters, 255 00:15:08,890 --> 00:15:10,670 and I'll let you guys do the factor label 256 00:15:10,670 --> 00:15:11,570 conversion on that. 257 00:15:11,570 --> 00:15:14,310 That's actually going to be one of your problems to work on. 258 00:15:14,310 --> 00:15:17,300 So it's pretty close, but why is it not exact? 259 00:15:20,378 --> 00:15:21,770 AUDIENCE: Rounded off. 260 00:15:21,770 --> 00:15:23,840 MARK HARTMAN: OK, maybe we rounded off too much. 261 00:15:23,840 --> 00:15:26,660 Maybe our measurement of the-- 262 00:15:26,660 --> 00:15:29,000 it was hard to measure exactly where the top and bottom 263 00:15:29,000 --> 00:15:32,350 of that trash can were, right?