1 00:00:00,000 --> 00:00:07,265 Here we are at a lecture hall at The Ohio State University. 2 00:00:07,265 --> 00:00:13,122 I'm Bart Snap. Today, we're going to see how we can use 3 00:00:13,122 --> 00:00:21,797 Calculus to study the path of an object through space, in particular this orange 4 00:00:21,797 --> 00:00:22,340 ball. 5 00:00:23,920 --> 00:00:34,860 6 00:00:34,860 --> 00:00:38,685 When the ball leaves my hand, it's going pretty fast. 7 00:00:38,685 --> 00:00:44,424 But as it approaches the ground, it's going faster and faster, and faster until 8 00:00:44,424 --> 00:00:48,470 it hits the ground, bounces up at nearly the same speed. 9 00:00:48,470 --> 00:00:53,914 And then it's going quickly, but slower, slower, slower, until, it reaches its 10 00:00:53,914 --> 00:00:58,770 pinnacle and then it stopped. And then it starts falling back down 11 00:00:58,770 --> 00:01:04,582 again, slowly at first, but then quicker, quicker and quicker, and then I, grab the 12 00:01:04,582 --> 00:01:10,400 ball. So how fast was the ball moving well lets 13 00:01:10,400 --> 00:01:17,346 see. The total distance the ball traveled was 14 00:01:17,346 --> 00:01:28,660 about, 3.14 meters and the total time the ball traveled for, was about 1.1 seconds. 15 00:01:28,660 --> 00:01:37,320 Dividing this, we find this is equal to 2.85 meters per second. 16 00:01:38,600 --> 00:01:44,321 This is the average speed of the ball. Now, of course, I took the ball and I 17 00:01:44,321 --> 00:01:50,583 threw it down with some force and then it came back up and hovered here and then 18 00:01:50,583 --> 00:01:55,841 came back down if you recall. So, what I'm trying to say is the speed 19 00:01:55,841 --> 00:02:01,871 of the ball was changing the entire time and this is the average speed and so 20 00:02:01,871 --> 00:02:06,433 sometimes the speed was faster and sometimes it was slower. 21 00:02:06,433 --> 00:02:11,063 Here. So, think about when I just threw the 22 00:02:11,063 --> 00:02:15,997 ball down. From leaving my hand to touching the 23 00:02:15,997 --> 00:02:23,450 ground, that was about 2. seconds. And the distance it traveled was about 24 00:02:23,450 --> 00:02:30,252 one meter. One meter divided by 2. seconds is 5 25 00:02:30,252 --> 00:02:37,084 meters per second. So we found the average speed to be 2.85 26 00:02:37,084 --> 00:02:41,836 meters per second. But that's not really the instantaneous 27 00:02:41,836 --> 00:02:44,937 speed of the ball. Because when I threw down the ball, the 28 00:02:44,937 --> 00:02:48,202 ball was traveling much faster. And then the ball bounced up. 29 00:02:48,202 --> 00:02:52,011 And then it started slowing down. And then it slowed down to almost no 30 00:02:52,011 --> 00:02:54,949 speed at all. And then it came back into my hand going 31 00:02:54,949 --> 00:02:58,900 a little faster. So how do we figure out how fast the 32 00:02:58,900 --> 00:03:02,910 balls going at say one of those other time intervals. 33 00:03:02,910 --> 00:03:08,737 We found the average speed over the course of the entire trajectory the full 34 00:03:08,737 --> 00:03:14,943 1.1 seconds to be 2.85 meters per second. However if we only consider the first 2. 35 00:03:14,943 --> 00:03:20,997 seconds we find the average speed to be five meters per second, this makes sense 36 00:03:20,997 --> 00:03:27,051 because if you watch the video you see the ball is moving rather quickly in the 37 00:03:27,051 --> 00:03:34,273 first 2. seconds of its trajectory. You probably asking yourself, how does 38 00:03:34,273 --> 00:03:40,459 Bart know that the ball traveled one meter during the first 2. seconds of its 39 00:03:40,459 --> 00:03:46,887 journey and 3.14 meters during the whole of 1.1 second journey down up and down? 40 00:03:46,887 --> 00:03:50,342 Well, we know this because we got the video. 41 00:03:50,342 --> 00:03:55,645 Alright, here Bart is just about to release the ball from his hand. 42 00:03:55,645 --> 00:04:01,270 And 1, 2, 3, 4, 5, 6 frames later, the ball hits the ground. 43 00:04:01,270 --> 00:04:06,537 There's 30 frames being shot in every second so that means that it took 6 / 30 44 00:04:06,537 --> 00:04:09,797 or 2. seconds for the ball to hit the ground. 45 00:04:09,797 --> 00:04:14,751 Making a bunch more measurements, I can combine all this information in a graph. 46 00:04:14,751 --> 00:04:19,767 This is a graph of a function the input to this function is time so along the x 47 00:04:19,767 --> 00:04:24,533 axis I'm plotting time and seconds. The output to the function is the height 48 00:04:24,533 --> 00:04:28,170 the height of the ball at that particular moment in time. 49 00:04:28,170 --> 00:04:33,722 Now, on this graph, I can go back now and try to figure out how fast the ball is 50 00:04:33,722 --> 00:04:38,993 moving at say 4. seconds after Bart releases it and at 8. seconds after Bart 51 00:04:38,993 --> 00:04:42,648 releases it. So let's try to figure out how fast the 52 00:04:42,648 --> 00:04:47,217 ball is moving at 4. seconds. Here I've marked the position of the 53 00:04:47,217 --> 00:04:50,380 ball, .4 seconds after Bart releases the ball. 54 00:04:50,380 --> 00:04:55,272 Now we don't really have any way of figuring how fast the ball is moving at 55 00:04:55,272 --> 00:04:59,327 that particular moment. What I can do is figure out the average 56 00:04:59,327 --> 00:05:02,160 speed of the ball during some time interval. 57 00:05:02,160 --> 00:05:07,181 So as sort of a first guess to how fast this thing is moving at 4. seconds, I'm 58 00:05:07,181 --> 00:05:11,751 going to figure out the average speed of the ball between 4. seconds and 6. 59 00:05:11,751 --> 00:05:14,648 seconds. Alright, so I've got this handy table 60 00:05:14,648 --> 00:05:20,802 here, of function values at 4. seconds. The the ball was 101.1 centimeters above 61 00:05:20,802 --> 00:05:27,404 the ground and 6. seconds after Bart released the ball, the ball was 16.8 62 00:05:27,404 --> 00:05:32,555 centimeters above the ground. So I can put that information together to 63 00:05:32,555 --> 00:05:37,924 figure out the speed of the ball between 4. and 6. seconds. 64 00:05:37,924 --> 00:05:38,940 Alright. So 65 00:05:40,000 --> 00:05:49,634 4. seconds to say 6. seconds at 4. seconds on my chart the ball was a 101.1 66 00:05:49,634 --> 00:05:56,917 centimeters above the ground. At 6. seconds the ball was a 161.8 67 00:05:56,917 --> 00:06:04,983 centimeters above the ground. Now this time interval has a length of 2. 68 00:06:04,983 --> 00:06:11,567 seconds. And how far did the ball move during that 69 00:06:11,567 --> 00:06:16,789 time interval? Well 161.8 - 101.1 is 60.7 centimeters. 70 00:06:16,789 --> 00:06:23,751 So during the 2. seconds that elapsed from 4. seconds to 6. seconds after Bart 71 00:06:23,751 --> 00:06:30,988 released the ball, the ball traveled a distance of 60.7 centimeters, which means 72 00:06:30,988 --> 00:06:40,735 the speed, which is the distance traveled over time is 60.7 over 2. centimeters per 73 00:06:40,735 --> 00:06:51,225 second, which is 303.5 centimeters per second or 3.035 meters per second. 74 00:06:51,225 --> 00:06:59,990 Which is about seven miles per hour. Now I could do a little bit better, 75 00:06:59,990 --> 00:07:03,423 alright? Here I'm calculating the average speed of 76 00:07:03,423 --> 00:07:08,374 the ball between 4. and 6. seconds, but I'm trying to figure out how fast the 77 00:07:08,374 --> 00:07:13,391 ball is moving at this particular moment. So instead of just calculating the 78 00:07:13,391 --> 00:07:18,738 average speed during this time interval to be about seven miles per hour, I could 79 00:07:18,738 --> 00:07:21,709 do it over a shorter time interval, all right? 80 00:07:21,709 --> 00:07:25,280 Instead of 6. to 4,. I could go from 4. to say 5.. 81 00:07:25,280 --> 00:07:28,596 Here's half a second after Bart released the ball. 82 00:07:28,596 --> 00:07:33,703 And I could figure out the average speed of the ball during this part of its 83 00:07:33,703 --> 00:07:36,556 trajectory. Let's see how we calculate that. 84 00:07:36,556 --> 00:07:41,199 Well, it's the same kind of game. All right?.4 seconds after Bart released 85 00:07:41,199 --> 00:07:45,680 the ball, the ball was 101.1 centimeters above the ground. 86 00:07:45,680 --> 00:07:53,319 5. seconds after Barb released the ball, looking back at my table I find that the 87 00:07:53,319 --> 00:07:58,560 ball was 136.5 centimeters above the ground. 88 00:07:58,560 --> 00:08:04,986 136.5 centimeters. This time interval was 1. seconds long. 89 00:08:04,986 --> 00:08:11,412 And how far did the ball move during that time interval? 90 00:08:11,412 --> 00:08:18,289 Well, that's 35.4 centimeters. 136.5 minus 101.1 is 35.4. 91 00:08:18,289 --> 00:08:25,502 So, the ball moved 35.4 centimeters. During the point one seconds that elapsed 92 00:08:25,502 --> 00:08:30,189 point four seconds to 5. seconds after Bart released the ball. 93 00:08:30,189 --> 00:08:35,545 Speed is how far you've traveled over how long it took you so if I divide these 94 00:08:35,545 --> 00:08:41,103 this is the speed of the ball the average speed of the ball between point four and 95 00:08:41,103 --> 00:08:46,593 point five seconds and this works out to be 354 centimeters per second I mean I'm 96 00:08:46,593 --> 00:08:49,540 dividing by this very nice number point one. 97 00:08:49,540 --> 00:08:56,243 which is the same as 3.45 meters per second which is about eight miles per 98 00:08:56,243 --> 00:08:58,937 hour. And indeed, if you look back at this, 99 00:08:58,937 --> 00:09:01,467 this chart, between 4. and 6. seconds, yeah. 100 00:09:01,467 --> 00:09:05,136 Maybe the average speed was u, about seven miles per hour. 101 00:09:05,136 --> 00:09:09,311 Between 4. and 5. seconds the average speed was a little bit higher. 102 00:09:09,311 --> 00:09:14,435 You know, the average speed here worked out to be eight miles per hour instead of 103 00:09:14,435 --> 00:09:18,421 seven miles per hour. The average speed of the ball during this 104 00:09:18,421 --> 00:09:22,600 time interval is higher than during this whole time interval. 105 00:09:22,600 --> 00:09:26,126 We're still not there. We are trying to figure out how fast the 106 00:09:26,126 --> 00:09:31,052 ball is moving at this particular moment right not the average speed between 4. 107 00:09:31,052 --> 00:09:34,818 and 5. seconds. To get closer, right, we should take an 108 00:09:34,818 --> 00:09:39,031 even smaller time interval. Instead of 4. to 5. well, why not look 109 00:09:39,031 --> 00:09:44,003 back on our handy chart here and see well, here is where the ball is at 4. 110 00:09:44,003 --> 00:09:47,204 seconds. Here's where the ball is at 42. seconds. 111 00:09:47,204 --> 00:09:52,994 We could use this information to figure out the speed of the ball just during the 112 00:09:52,994 --> 00:09:58,306 very tiny time interval between 4. and 42. seconds after Bart releases the ball. 113 00:09:58,306 --> 00:10:00,281 Well, let's do that. All right. 114 00:10:00,281 --> 00:10:05,662 So again, .4 seconds after Bart releases the ball, the ball is 101.1 centimeters 115 00:10:05,662 --> 00:10:10,528 above the ground. .42 seconds after Bart releases the ball, 116 00:10:10,528 --> 00:10:17,674 the ball is 109 centimeters above the ground, that's what this chart is telling 117 00:10:17,674 --> 00:10:25,320 me .42 seconds after Bart releases the ball, 109 centimeters above the ground. 118 00:10:25,320 --> 00:10:31,051 So that means during the very tiny time interval 02. seconds that elapsed between 119 00:10:31,051 --> 00:10:37,111 4. seconds and 42. seconds after Bart releases the ball, the ball has traveled 120 00:10:37,111 --> 00:10:42,777 how far well 109 - 101.1 centimeters is just 7.9 centimeters. 121 00:10:42,777 --> 00:10:48,310 So in two hundredths of a second the ball has traveled 7.9 centimeters. 122 00:10:48,310 --> 00:10:57,579 To figure out the speed I again divide 7.9 / 02. is 395 centimeters per second. 123 00:10:57,579 --> 00:11:06,540 Which is 3.95 meters per second. Which is about 9 miles per hour. 124 00:11:06,540 --> 00:11:13,050 This is a much better approximation to the instantaneous speed of the ball at 4. 125 00:11:13,050 --> 00:11:15,684 seconds. Look, here's the graph again. 126 00:11:15,684 --> 00:11:21,682 Between point four and point six seconds, the ball is travelling maybe seven miles 127 00:11:21,682 --> 00:11:25,633 per hour on average. Between point four and point five 128 00:11:25,633 --> 00:11:31,719 seconds, the ball is travelling maybe. Eight miles per hour between 4. and 42. 129 00:11:31,719 --> 00:11:38,398 seconds we just calculated that the speed of the ball is about nine miles per hour. 130 00:11:38,398 --> 00:11:41,778 And that makes a whole lot of sense. Right? 131 00:11:41,778 --> 00:11:48,055 The speed of the ball from here to here is slower than the average speed from 132 00:11:48,055 --> 00:11:52,320 here to here. Which is slower than average speed from 133 00:11:52,320 --> 00:11:55,883 0.4 to 0.42 seconds. The ball's slowing down in its 134 00:11:55,883 --> 00:12:00,230 trajectory, so the average speed over these shorter time intervals is 135 00:12:00,230 --> 00:12:03,318 decreasing. So, let's figure out how fast the ball 136 00:12:03,318 --> 00:12:07,287 was moving at 8. seconds. That's when the ball was at the top of 137 00:12:07,287 --> 00:12:09,933 its trajectory. we can't really do that. 138 00:12:09,933 --> 00:12:14,911 All I can really do is figure out the average speed of the ball over some time 139 00:12:14,911 --> 00:12:19,320 interval but I've got a table of values of the function. 140 00:12:19,320 --> 00:12:22,969 And I know how high the ball was at 8. seconds after release. 141 00:12:22,969 --> 00:12:27,957 It was 182 centimeters, and I can compute its average speed over a very short time 142 00:12:27,957 --> 00:12:32,519 interval, like the time interval between 8. and 81. seconds after release, all 143 00:12:32,519 --> 00:12:35,378 right? And the ball didn't move very far during 144 00:12:35,378 --> 00:12:40,000 that time interval but of course that time interval also isn't very long, so 145 00:12:40,000 --> 00:12:44,745 it's not super clear how fast the ball might be moving on average during that 146 00:12:44,745 --> 00:12:47,604 time interval. We can do the calculation though. 147 00:12:47,604 --> 00:12:52,420 Let's do it now, so. 0.8 seconds after the ball was released, 148 00:12:52,420 --> 00:12:56,680 the ball was 182 centimeters above the ground. 149 00:12:56,680 --> 00:13:05,156 0.81 seconds after the ball was released, the ball was 181.9 centimeters above the 150 00:13:05,156 --> 00:13:09,279 ground. Now this time interval between 8. and 81. 151 00:13:09,279 --> 00:13:13,257 seconds has the duration of just one hundredth of a second. 152 00:13:13,257 --> 00:13:17,770 81. - 8. is 01.. That's a very short amount of time and 153 00:13:17,770 --> 00:13:22,360 during that short amount of time, how far do the ball move? 154 00:13:22,360 --> 00:13:26,281 Well, 181.9 - 182 centimeters, that's just.1 centimeters. 155 00:13:26,281 --> 00:13:30,424 And if we're being pedantic, it's negative 1. centimeters. 156 00:13:30,424 --> 00:13:34,049 All right? The ball fell between 8. and 81. seconds, 157 00:13:34,049 --> 00:13:40,116 so this number is recording not only how far it moved but also the direction that 158 00:13:40,116 --> 00:13:43,815 it moved in. It's really displacement instead of a 159 00:13:43,815 --> 00:13:47,219 distance. Anyhow, .01 centimeters divided by 01. 160 00:13:47,219 --> 00:13:50,548 seconds, that will give me the velocity, right? 161 00:13:50,548 --> 00:13:56,978 Displacement over time. So if I divide these, this ratio here is 162 00:13:56,978 --> 00:14:02,515 10 centimeters per second or -10 if I am keeping track of the direction its moving 163 00:14:02,515 --> 00:14:05,075 in. It's falling down at a speed on average 164 00:14:05,075 --> 00:14:08,410 of ten centimeters per second during this time interval. 165 00:14:08,410 --> 00:14:14,442 That's -.1 meters per second. Which is about.2 miles per hour or -.2 166 00:14:14,442 --> 00:14:19,142 miles hour if I'm keeping track of the direction it's going. 167 00:14:19,142 --> 00:14:22,902 Anyway, .2 miles per hour is a really slow speed, 168 00:14:22,902 --> 00:14:26,193 right? The ball is not moving very much on 169 00:14:26,193 --> 00:14:31,335 average between 8. and 81. seconds. In light of this, it might make sense to 170 00:14:31,335 --> 00:14:36,961 say that the instantaneous speed of the ball at.4 seconds is nine miles per hour. 171 00:14:36,961 --> 00:14:40,226 Now, why? Well, the average speed between 4. and 6. 172 00:14:40,226 --> 00:14:45,644 seconds is maybe 7 miles per hour. The average speed from 4. and 5. is about 173 00:14:45,644 --> 00:14:49,534 eight miles per hour. The average speed between 4. and 42. 174 00:14:49,534 --> 00:14:53,980 seconds is about 9 miles per hour. You know and based on this, 175 00:14:53,980 --> 00:14:58,671 it seems like if we took a really short time interval, just after 4. seconds and 176 00:14:58,671 --> 00:15:03,482 tried to calculate how fast the ball was going on average during that very small 177 00:15:03,482 --> 00:15:07,996 time interval, you might conclude that the average speed during a very small 178 00:15:07,996 --> 00:15:12,829 time interval is about 9 miles per hour. It's in that sense that we're going to 179 00:15:12,829 --> 00:15:17,797 say that the instantaneous speed of the ball at point four seconds is 9 miles per 180 00:15:17,797 --> 00:15:20,772 hour. When you play the same game 8. seconds 181 00:15:20,772 --> 00:15:25,290 into the balls journey, alright. When it's just to the top of its 182 00:15:25,290 --> 00:15:28,396 trajectory. So the average speed of the ball, between 183 00:15:28,396 --> 00:15:33,972 8. and 81. seconds is exceedingly slow, and you can see that in the video, 184 00:15:33,972 --> 00:15:37,361 alright. The ball is barely moving, at the top of 185 00:15:37,361 --> 00:15:42,014 its trajectory. What's the instantaneous speed of the 186 00:15:42,014 --> 00:15:45,414 ball at the top of its trajectory? It's zero, right? 187 00:15:45,414 --> 00:15:49,177 I mean yes. The average speed over a time interval 188 00:15:49,177 --> 00:15:52,263 between 8. and 8000001. seconds isn't zero. 189 00:15:52,263 --> 00:15:58,435 But if you look at an average speed over an exceedingly small time interval, those 190 00:15:58,435 --> 00:16:04,381 average speeds over shorter and shorter time intervals are as close to zero as 191 00:16:04,381 --> 00:16:07,317 you like. That's the sense in which the 192 00:16:07,317 --> 00:16:13,112 instantaneous velocity at the top of the trajectory, the limit of the average 193 00:16:13,112 --> 00:16:16,960 velocities over small time intervals, is zero. 194 00:16:16,960 --> 00:16:25,883 Isn't calculus amazing? We're using the idea of limits to compute 195 00:16:25,883 --> 00:16:33,296 instantaneous speed. Using a little bit of math, we can 196 00:16:33,296 --> 00:16:41,532 understand the world around us. That's the power of calculus. 197 00:16:41,532 --> 00:16:42,631 [MUSIC]