WEBVTT 00:00:00.680 --> 00:00:01.880 I have some footage here. 00:00:01.880 --> 00:00:04.410 Of one of the most exciting moments in sports history. 00:00:04.410 --> 00:00:08.520 And to make it even more exciting, the commentator is speaking in German. 00:00:08.520 --> 00:00:10.930 And I'm assuming that this is okay under fair use 00:00:10.930 --> 00:00:14.320 because I'm really using it for a, a, a math problem. 00:00:14.320 --> 00:00:20.324 But I want you to watch this video and then I'll ask you a question about it. 00:00:20.324 --> 00:00:30.324 >> [FOREIGN] 00:00:40.838 --> 00:00:44.061 >> So you see it's exciting in any language that you might watch it. 00:00:44.061 --> 00:00:49.980 But my question to you is how fast was Ussain Bolt going? 00:00:49.980 --> 00:00:54.580 What was his average speed when he ran that 100 meters right there? 00:00:54.580 --> 00:00:56.140 And I encourage you to watch. 00:00:56.140 --> 00:00:58.725 The video as many times as you need to do it. 00:00:58.725 --> 00:01:00.705 And now I will give you a little bit of 00:01:00.705 --> 00:01:03.485 time to think about it, and then we will solve it. 00:01:03.485 --> 00:01:07.607 [BLANK_AUDIO] 00:01:07.607 --> 00:01:12.680 So we needed to figure out how fast was Ussain Bolt going over the 100 meters. 00:01:12.680 --> 00:01:14.510 So we're really thinking about, in the case 00:01:14.510 --> 00:01:17.620 of this problem, average speed, or average rate. 00:01:17.620 --> 00:01:20.150 And you might already be familiar with the notion. 00:01:20.150 --> 00:01:25.350 That distance, distance is equal to rate 00:01:25.350 --> 00:01:28.870 or speed, I'll just write rate, times time. 00:01:28.870 --> 00:01:33.120 I could write times like that, but once we start doing algebra, the traditional 00:01:33.120 --> 00:01:35.670 multiplication symbol can seem very confusing cuz 00:01:35.670 --> 00:01:37.910 it looks just like the variable x. 00:01:37.910 --> 00:01:40.440 So instead, I will write times like this. 00:01:40.440 --> 00:01:45.840 So distance is equal to rate The times is equal to rate times time. 00:01:45.840 --> 00:01:48.250 And hopefully this makes some intuitive sense for you. 00:01:48.250 --> 00:01:52.290 If your rate or your speed were ten meters per second just as an 00:01:52.290 --> 00:01:56.140 example, that's not necessarily how fast he went, but if you went ten meters 00:01:56.140 --> 00:02:02.700 per second and if you were to do that for For two seconds, for two seconds, 00:02:02.700 --> 00:02:07.190 seconds, then it should hopefully make intuitive sense that you went 20 meters. 00:02:07.190 --> 00:02:10.770 You went ten meters per seconds for two seconds. 00:02:10.770 --> 00:02:14.400 And it also works out mathematically, ten times two is equal to 20. 00:02:14.400 --> 00:02:16.390 And then you have seconds in the 00:02:16.390 --> 00:02:18.700 denominator, and seconds up here in the numerator. 00:02:18.700 --> 00:02:20.270 I just wrote seconds here with an s. 00:02:20.270 --> 00:02:21.520 I wrote it out there. 00:02:21.520 --> 00:02:25.790 But they also cancel out, and you're just left with the, the units of meters. 00:02:25.790 --> 00:02:28.740 So you're just left with 20 meters. 00:02:28.740 --> 00:02:31.050 So hopefully this makes intuitive sense. 00:02:31.050 --> 00:02:33.720 With that out of the way, let's actually think about, 00:02:33.720 --> 00:02:37.290 let's actually think about the problem, the problem at hand. 00:02:37.290 --> 00:02:40.100 What information do we actually have? 00:02:40.100 --> 00:02:42.920 So we, do we have the distance? 00:02:42.920 --> 00:02:48.054 So, what is, what is the distance in the video we just did? 00:02:48.054 --> 00:02:51.123 And I'll give you a second or two to think about it. 00:02:51.123 --> 00:02:53.558 [BLANK_AUDIO] 00:02:53.558 --> 00:02:56.130 Well, this race was 100 meters. 00:02:56.130 --> 00:02:58.930 So, the distance was 100. 00:02:58.930 --> 00:03:01.120 100 meters. 00:03:01.120 --> 00:03:02.300 Now, what else do we know? 00:03:02.300 --> 00:03:05.540 Well, we're trying to figure out the rate, that's what we're gonna figure out. 00:03:05.540 --> 00:03:09.800 What else do we know out of, out of this, out of this equation over here? 00:03:09.800 --> 00:03:11.780 Well do we know the time? 00:03:11.780 --> 00:03:13.200 Do we know the time? 00:03:13.200 --> 00:03:18.270 What was the time that it took Ussain Bolt to run the 00:03:18.270 --> 00:03:21.039 100 meters, and I'll give you another few seconds to think about that. 00:03:23.250 --> 00:03:26.770 Well, luckily, they were timing the whole thing, and they've not, 00:03:26.770 --> 00:03:29.020 and they've also showed that it's a, it's a world record. 00:03:29.020 --> 00:03:30.860 But this right over here is in seconds, is how 00:03:30.860 --> 00:03:33.460 long it took Usain Bolt to run the 100 meters. 00:03:33.460 --> 00:03:40.780 It was 9.58, 9.58 seconds, and I'll just write s for seconds. 00:03:40.780 --> 00:03:44.300 So given this information here what you need to attempt to do 00:03:44.300 --> 00:03:48.730 is now give us our rate in terms of meters per second. 00:03:48.730 --> 00:03:51.880 I want you to think if you can figure out the rate in terms of meters per second. 00:03:51.880 --> 00:03:54.560 We know the distance and we know the time. 00:03:57.240 --> 00:04:00.890 Well, let's substitute these values into this equation right over here. 00:04:00.890 --> 00:04:02.599 We know the distance is 100 meters. 00:04:03.680 --> 00:04:06.050 The distance is 100 meters. 00:04:06.050 --> 00:04:10.860 And its equal to, we don't know the rate, so I'll just write rate right over here. 00:04:10.860 --> 00:04:13.670 Its, and I'm going to write in that same color. 00:04:13.670 --> 00:04:18.310 Its equal to rate, rate times, and what's the time? 00:04:18.310 --> 00:04:19.327 We do know the time. 00:04:19.327 --> 00:04:20.956 It's 9.58 seconds. 00:04:20.956 --> 00:04:25.392 9.58, 9.58 seconds. 00:04:25.392 --> 00:04:27.850 And we care about rate. 00:04:27.850 --> 00:04:30.070 We care about solving for rate. 00:04:30.070 --> 00:04:31.960 So how can we do that? 00:04:31.960 --> 00:04:33.750 Well, if you look at this right hand side 00:04:33.750 --> 00:04:37.520 of the equation, I have 9.58 seconds times rate. 00:04:37.520 --> 00:04:41.610 If I were able to divide this right hand side by 9.58 seconds, I'll just 00:04:41.610 --> 00:04:45.190 have rate on the right hand side, and that's what I want to solve for. 00:04:45.190 --> 00:04:46.840 So you say well wait why don't I just 00:04:46.840 --> 00:04:51.250 divide the right-hand side by 9.58 seconds, 9.58 seconds? 00:04:51.250 --> 00:04:56.870 Cuz if I did that the 9, the units cancel out if we're doing 00:04:56.870 --> 00:05:00.300 dimensional analysis, don't worry too much if that word doesn't make sense to you. 00:05:00.300 --> 00:05:03.180 But the units cancel out and the 9.58 cancel out. 00:05:03.180 --> 00:05:08.250 But I can't just divide one side of the equation by a number. 00:05:08.250 --> 00:05:10.870 When we started off, this is equal to this up here. 00:05:10.870 --> 00:05:14.780 If I divide it, if I divide the right side by 9.58, in order for the 00:05:14.780 --> 00:05:19.710 equality To still be true, I needed to divide to left side by the same thing. 00:05:19.710 --> 00:05:22.220 So, I can't just divide the right side, I have to divide 00:05:22.220 --> 00:05:25.510 the left side in order for the quality to still be true. 00:05:25.510 --> 00:05:27.410 If I said, one thing is equal to another 00:05:27.410 --> 00:05:29.510 thing, and I divide the other thing by something, 00:05:29.510 --> 00:05:30.820 in order for them to still be equal I 00:05:30.820 --> 00:05:33.290 have to divide the first thing by the same amount. 00:05:33.290 --> 00:05:36.230 So I divide by 9.58 seconds. 00:05:36.230 --> 00:05:38.540 So on our right hand side and this was the whole point. 00:05:38.540 --> 00:05:42.100 These two cancel out and then on the left-hand 00:05:42.100 --> 00:05:44.680 side, I'm left with a hundred divided by 9.58. 00:05:44.680 --> 00:05:47.800 And my units are meters per second which are the 00:05:47.800 --> 00:05:51.610 exact units that I want for rate or for speed. 00:05:51.610 --> 00:05:56.620 And so let's get the calculator out to divide 100 by 9.58. 00:05:56.620 --> 00:06:01.170 So I've got 100 meters, divided 00:06:02.920 --> 00:06:07.390 by 9.58 seconds, gives me 10 point, let's see. 00:06:07.390 --> 00:06:09.930 We've got about three significant digits here. 00:06:09.930 --> 00:06:11.920 So let's say, 10.4. 00:06:11.920 --> 00:06:16.375 So this gives us 10.4 and I'll right in the rate color. 00:06:16.375 --> 00:06:22.680 10, 10.4 and the units are meters per second, meters, 00:06:22.680 --> 00:06:30.050 meters per second is equal to, is equal to my rate. 00:06:30.050 --> 00:06:32.290 Now, the next question, so we've got this in meters per second 00:06:32.290 --> 00:06:35.620 but unfortunately meters per second is, they're not the, you know, when 00:06:35.620 --> 00:06:38.690 we drive a car, we don't see the, the speedometer in meters 00:06:38.690 --> 00:06:42.690 per second, we see either kilometers per hour or miles per hour. 00:06:42.690 --> 00:06:47.820 So, the next task I, I have for you, is to express this speed, or this rate. 00:06:47.820 --> 00:06:51.140 And this is his average speed, or his average rate, over the hundred meters. 00:06:51.140 --> 00:06:56.630 But to think about this in terms of kilometers per, kilometers per hour. 00:06:56.630 --> 00:07:01.350 So try to figure out if you can, if you can rewrite this in kilometers per hour. 00:07:04.435 --> 00:07:06.450 Well let's just take this step by step. 00:07:06.450 --> 00:07:09.810 So, I'm gonna write, so let me just go down here, start over. 00:07:09.810 --> 00:07:15.300 So, I'm just, I started off with 10.4, and I'll write meters in blue, meters in 00:07:15.300 --> 00:07:20.780 blue, and seconds, seconds in magenta. 00:07:20.780 --> 00:07:24.330 Now, we wanna get the kilometers per hour, I know our meters per second. 00:07:24.330 --> 00:07:25.540 So let's take baby steps. 00:07:25.540 --> 00:07:30.420 Let's first think about it in terms of kilometers per second. 00:07:30.420 --> 00:07:32.740 And I'll give you second to think about what we 00:07:32.740 --> 00:07:35.739 would do to this, to turn this into kilometers per second. 00:07:37.840 --> 00:07:40.400 Well, the intuition here, if I'm going 10.4 meters 00:07:40.400 --> 00:07:45.280 per second, how many, how many kilometers is 10.4 meters? 00:07:45.280 --> 00:07:50.320 Well, kilometers is a much larger unit of measurement, it's 1000 times larger. 00:07:50.320 --> 00:07:53.770 So, 10.4 meters will be a much smaller number of kilometers. 00:07:53.770 --> 00:07:57.250 And in particular, I'm gonna divide by 1000. 00:07:57.250 --> 00:07:59.410 Another way to think about it, if you wanna focus on the 00:07:59.410 --> 00:08:03.210 units, we wanna get rid of this meters and we want a kilometers. 00:08:03.210 --> 00:08:07.540 So, we wanted kilometers, and we wanna get rid of these meters. 00:08:07.540 --> 00:08:09.400 So, if, if we had meters in the numerator, we 00:08:09.400 --> 00:08:12.510 could divide by meters here, and they would cancel out. 00:08:12.510 --> 00:08:14.160 But, the intuitive way to think about it is, we're 00:08:14.160 --> 00:08:17.100 going from a smaller unit, meter, to a larger unit, kilometers. 00:08:17.100 --> 00:08:21.720 So, 10.4 meters is going to be a much smaller number of kilometers. 00:08:21.720 --> 00:08:25.550 But if we look at it this way, how many, how many meters are in one kilometer? 00:08:25.550 --> 00:08:29.430 One kilometer is equal to 1,000 meters. 00:08:29.430 --> 00:08:31.750 This right over here, 1 kilometer over 1,000 meters. 00:08:31.750 --> 00:08:35.539 This is over 1 over 1, we're sent, we're not changing the fundamental value. 00:08:35.539 --> 00:08:38.039 We're essentially just multiplying it by one. 00:08:38.039 --> 00:08:40.860 But when we do this, when we do this what do we get? 00:08:40.860 --> 00:08:43.820 Well, the meters cancel out, we're left with kilometers and 00:08:43.820 --> 00:08:48.361 seconds, and then the numbers you get 10.4 divided by 1,000. 00:08:48.361 --> 00:08:53.445 10.4 divided by 1,000 is going to you, so if you divide by 10, 00:08:53.445 --> 00:08:58.531 you're gonna get 1.04, you divide by 100, you get 0.104. 00:08:58.531 --> 00:09:07.288 You divide by 1000, you get 0.0104, so that's just 10.4 divided by 1000. 00:09:07.288 --> 00:09:12.999 And then our units are kilometers, kilometers per second. 00:09:12.999 --> 00:09:18.860 So that's the kilometers And then I have my seconds right over here. 00:09:18.860 --> 00:09:21.040 Now, so let me write the equal sign. 00:09:21.040 --> 00:09:25.080 Now let's try to convert this to kilometers per hour. 00:09:25.080 --> 00:09:27.720 And I'll give you a little bit of time to think about that one. 00:09:29.720 --> 00:09:35.330 Well, hours there's, there's 3,600 seconds in an hour, so however many kilometers 00:09:35.330 --> 00:09:39.490 I do in a second, I'm gonna do 3,600 times that in an hour. 00:09:39.490 --> 00:09:41.780 And the units will also work out. 00:09:41.780 --> 00:09:43.290 If I'm going, I'm, right now. 00:09:43.290 --> 00:09:47.490 Well, if I do this many in a second, so it's gonna be times 3600. 00:09:47.490 --> 00:09:50.080 There are 3600 seconds in an hour. 00:09:50.080 --> 00:09:52.160 3600 seconds in an hour. 00:09:53.270 --> 00:09:55.400 And another way to think about it is 00:09:55.400 --> 00:09:58.240 we want hours in the denominator, we had seconds. 00:09:58.240 --> 00:10:00.300 So if we multiply by seconds per hour. 00:10:00.300 --> 00:10:02.430 There are 3600 seconds per hour. 00:10:02.430 --> 00:10:03.930 The seconds are going to cancel out and we 00:10:03.930 --> 00:10:06.400 are going to be left with hours in the denominator. 00:10:06.400 --> 00:10:09.500 So seconds cancel out and we're left with kilometers per 00:10:09.500 --> 00:10:12.750 hour but now we have to multiply this number times 3600. 00:10:12.750 --> 00:10:14.910 I'll get the calculator out for that. 00:10:14.910 --> 00:10:16.800 I'll get the calculator out for that. 00:10:16.800 --> 00:10:21.190 So we have, we have 0.0104, 0.0104 times 3600, 3600 gives us 37., 00:10:21.190 --> 00:10:26.650 I'll just stay 37.4 so this is equal to 00:10:26.650 --> 00:10:34.166 37.4, 37.4 00:10:34.166 --> 00:10:38.856 kilometers. 00:10:39.980 --> 00:10:45.080 Kilometers, kilometers per hour. 00:10:45.080 --> 00:10:48.390 Kilometers per hour. 00:10:48.390 --> 00:10:52.000 So that's his average speed in kilometers per hour. 00:10:52.000 --> 00:10:55.320 And then, the last thing I wanna do, for those of you, for those of us in America. 00:10:55.320 --> 00:10:58.620 We'll convert into imperial units or sometimes called English units. 00:10:58.620 --> 00:10:59.840 Which are ironically. 00:10:59.840 --> 00:11:03.400 Not necessarily used in UK, they tend to be used in America. 00:11:03.400 --> 00:11:07.360 So lets convert this into miles per hour and the one thing I will tell you just in 00:11:07.360 --> 00:11:11.450 case you don't know is that their 1.61 kilometers 00:11:11.450 --> 00:11:15.210 is equal to 1 is equal to 1 mile. 00:11:15.210 --> 00:11:18.692 So I'll give you little bit of time to convert this into miles per hour. 00:11:18.692 --> 00:11:20.968 [BLANK_AUDIO] 00:11:20.968 --> 00:11:23.085 Well as you see from this, a mile is 00:11:23.085 --> 00:11:27.023 a slightly larger or reasonably larger unit than a kilometer. 00:11:27.023 --> 00:11:30.641 So, if you're going 37.4 kilometers in a certain amount of time, you're 00:11:30.641 --> 00:11:33.890 gonna go slightly smaller amount of miles in a certain amount of time. 00:11:33.890 --> 00:11:37.040 Or in particular, you're gonna divide by 1.61. 00:11:37.040 --> 00:11:38.490 So let me rewrite it. 00:11:38.490 --> 00:11:44.740 If I have 37.4 kilometers per hour, kilometers 00:11:44.740 --> 00:11:47.850 per hour, we're going to a larger unit. 00:11:47.850 --> 00:11:49.040 We're going to miles. 00:11:49.040 --> 00:11:51.800 So we're gonna divide by something larger than one. 00:11:51.800 --> 00:11:55.680 So we have one, we have one, let me right it in blue. 00:11:55.680 --> 00:12:04.578 One mile, one mile is equal to, is equal to 1.61 kilometers. 00:12:04.578 --> 00:12:07.920 Or you could say there's 1, 1.61 mile per kilometer. 00:12:07.920 --> 00:12:10.042 And also once again works out with units. 00:12:10.042 --> 00:12:13.070 We wanna get rid of the kilometers in the numerator, so we would 00:12:13.070 --> 00:12:15.410 want it in the denominator and we want a mile in the numerator. 00:12:15.410 --> 00:12:17.700 So that's why we have a mile in the numerator here. 00:12:17.700 --> 00:12:20.530 So let's once again multiply, or I guess in this 00:12:20.530 --> 00:12:24.550 case we're dividing by 1.61 and we get, so we get. 00:12:24.550 --> 00:12:29.540 Let's just divide our previous value by 1.61, 1.61, 00:12:29.540 --> 00:12:34.122 and we get 23, I'll just round up, 23.3. 00:12:34.122 --> 00:12:39.365 This is equal to 23.3, 23. 00:12:39.365 --> 00:12:48.510 23.3 and then we have miles, miles, miles per hour. 00:12:48.510 --> 00:12:54.570 20 miles per hour, which is obviously very fast, he's the fastest 00:12:54.570 --> 00:12:57.620 human But it's not maybe as fast as you might have imagined. 00:12:57.620 --> 00:13:00.530 You know, in the car 23.3 miles per hour doesn't seem so 00:13:00.530 --> 00:13:02.570 fast, and especially relative to the 00:13:02.570 --> 00:13:05.660 animal world, it's not particularly noteworthy. 00:13:05.660 --> 00:13:08.900 This is actually slightly slower than a charging elephant. 00:13:08.900 --> 00:13:12.870 Charging elephants have been clocked at 25 miles per hour.