Finding average speed or rate

SALMAN KHAN: I have some footage here of one of the most exciting moments in sports history. And to make it even more exciting, the commentator is speaking in German. And I'm assuming that this is OK under fair use, because I'm really using it for a math problem. But I want you to watch this video, and then I'll ask you a question about it. [CHEERING] COMMENTATOR: [SPEAKING GERMAN] SALMAN KHAN: So you see, it's exciting in any language that you might watch it. But my question to you is, how fast was Usain Bolt going? What was his average speed when he ran that 100 meters right there? And I encourage you to watch the video as many times as you need to do it. And now I'll give you a little bit of time to think about it, and then we will solve it. So we needed to figure out how fast was Usain Bolt going over the 100 meters. So we're really thinking about, in the case of this problem, average speed or average rate. And you might already be familiar with the notion that distance is equal to rate or speed-- I'll just write rate-- times time. And I could write times like that, but once we start doing algebra, the traditional multiplication symbol can seem very confusing because it looks just like the variable x. So instead, I will write times like this. So distance is equal to rate times time. And hopefully, this makes some intuitive sense for you. If your rate or your speed were 10 meters per second-- just as an example. That's not necessarily how fast he went. But if you went 10 meters per second, and if you were to do that for two seconds, then it should hopefully make intuitive sense that you went 20 meters. You went 10 meters per second for two seconds. And it also works out mathematically. 10 times 2 is equal to 20. And then you have seconds in the denominator and seconds up here in the numerator. I just wrote seconds here with an s. I wrote it out there. But they also cancel out, and you're just left with the units of meters. So you're just left with 20 meters. So hopefully this makes intuitive sense. With that out of the way, let's actually think about the problem at hand. What information do we actually have? So do we have the distance? So what is the distance in the video we just did? And I'll give you a second or two to think about it. Well, this race was the 100 meters. So the distance was 100 meters. Now, what else do we know? Do we know-- well, we're trying to figure out the rate. That's what we're going to figure out. What else do we know out of this equation right over here? Well, do we know the time? Do we know the time? What was the time that it took Usain Bolt to run the 100 meters? And I'll give you another few seconds to think about that. Well luckily, they were timing the whole thing. And they also showed that it's a world record. But this right over here is in seconds. It's how long it took Usain Bolt to run the 100 meters. It was 9.58 seconds. And I'll just write s for seconds. So given this information here, what you need to attempt to do is now give us our rate in terms of meters per second. I want you to think if you could figure out the rate in terms of meters per second. We know the distance, and we know the time. Well, let's substitute these values into this equation right over here. We know the distance is 100 meters. And it's equal to-- we don't know the rate, so I'll just write rate right over here. And let me write it in that same color. It's equal to rate times-- and what's the time? We do know the time. It's 9.58 seconds. And we care about rate. We care about solving for rate. So how can we do that? Well, if you look at this right hand side of the equation, I have 9.58 seconds times rate. If I were able to divide this right hand side by 9.58 seconds, I'll just have rate on the right hand side. And that's what I want to solve for. So you say, well, why don't I just divide the right hand side by 9.58 seconds? Because if I did that, the units cancel out, if we're doing dimensional analysis. Don't worry too much if that word doesn't make sense to you. But the units cancel out, and the 9.58 cancels out. But I can't just divide one side of an equation by a number. When we started off, this is equal to this up here. If I divide the right side by 9.58, in order for the equality to still be true, I need to divide the left side by the same thing. So I can't just divide the right side. I have to divide the left side in order for the equality to still be true. If I said one thing is equal to another thing, and I divide the other thing by something, in order for them to still be equal, I have to divide the first thing by that same amount. So I divide by 9.58 seconds. So on our right hand side-- and this was the whole point-- these two cancel out. And then on the left hand side, I'm left with 100 divided by 9.58. And my units are meters per second, which are the exact units that I want for rate, or for speed. And so let's get the calculator out to divide 100 by 9.58. So I've got 100 meters divided by 9.58 seconds gives me 10 point-- this says we've got about three significant digits here-- so let's say 10.4. So this gives us 10.4. And I'll write it in the rate color. 10.4-- and the units are meters per second-- meters per second is equal to my rate. Now, the next question. So we got this in meters per second. But unfortunately, meters per second, they're not the-- when we drive a car, we don't see the speedometer in meters per second. We see either kilometers per hour or miles per hour. So the next task I have for you is to express this speed, or this rate-- and this is his average speed, or his average rate, over the 100 meters. But to think about this in terms of kilometers per hour. So try to figure out if you can rewrite this in kilometers per hour. Well, let's just take this step by step. So I'm going to write-- so let me just go down here, start over. So I started off with 10.4. And I'll write meters in blue, and seconds in magenta. Now, we want to get to kilometers per hour. Right now we're meters per second. So let's take baby steps. Let's first think about it in terms of kilometers per second. And I'll give you a second to think about what we would do this to turn this into kilometers per second. Well, the intuition here, if I'm going 10.4 meters per second, how many kilometers is 10.4 meters? Well, kilometers is a much larger unit of measurement. It's 1,000 times larger. So 10.4 meters will be a much smaller number of kilometers. And in particular, I'm going to divide by 1,000. Another way to think about it, if you want to focus on the units, we want to get rid of this in meters, and we want a kilometers. So we want a kilometers, and we want to get rid of these meters. So if we had meters in the numerator, we could divide by meters here. They would cancel out. But the intuitive way to think about it is we're going from a smaller unit, meters, to a larger unit, kilometers. So 10.4 meters are going to be a much smaller number of kilometers. But if we look at it this way, how many meters are in 1 kilometer? 1 kilometer is equal to 1,000 meters. This right over here, 1 kilometer over 1,000 meters, this is 1 over 1. We're not changing the fundamental value. We're essentially just multiplying it by one. But when we do this, what do we get? Well, the meters cancel out. We're left with kilometers and seconds. And the numbers, you get 10.4 divided by 1,000. 10.4 divided by 1,000 is going to give you-- so if you divide by 10, you're going to get 1.04. You divide by 100, you get 0.104. You divide by 1,000, you get 0.0104. So that's just 10.4 divided by 1,000. And then our units are kilometers per second. So that's the kilometers, and then I have my seconds right over here. So let me write the equal sign. Now, let's try to convert this to kilometers per hour. And I'll give you a little bit of time to think about that one. Well, hours, there's 3,600 seconds in an hour. So however many kilometers I do in a second, I'm going to do 3,600 times that in an hour. And the units will also work out. If I do this many in a second, so it's going to be times 3,600, there are 3,600 seconds in an hour. And another way to think about it is we want hours in the denominator. We had seconds. So if we multiply by seconds per hour, there are 3,600 seconds per hour, the seconds are going to cancel out, and we're going to be left with hours in the denominator. So seconds cancel out, and we're left with kilometers per hour. But now we have to multiply this number times 3,600. I'll get the calculator out for that. So we have 0.0104 times 3,600 gives us, I'll just say 37.4. So this is equal to 37.4 kilometers per hour. So that's his average speed in kilometers per hour. And now the last thing I want to do, for those of us in America, we'll convert into imperial units, or sometimes called English units, which are ironically not necessarily used in the UK. They tend to be used in America. So let's convert this into miles per hour. And the one thing I will tell you, just in case you don't know, is that 1.61 kilometers is equal to 1 mile. So I'll give you a little bit of time to convert this into miles per hour. Well, as you see from this, a mile is a slightly larger or reasonably larger unit than a kilometer. So if you're going 37.4 kilometers in a certain amount of time, you're going to go slightly smaller amount of miles in a certain amount of time. Or in particular, you're going to divide by 1.61. So let me rewrite it. If I have 37.4 kilometers per hour, we're going to a larger unit. We're going to miles. So we're going to divide by something larger than one. So we have one-- let me write it in blue-- 1 mile is equal to 1.61 kilometers. Or you could say there's one 1.61th mile per kilometer. It also, once again, works out with units. We want to get rid of the kilometers in the numerator. So we would want it in the denominator. We want a mile in the numerator. So that's why we have a mile in the numerator here. So let's once again multiply, or I guess in this case we're dividing by 1.61. And we get-- let's just divide our previous value by 1.61. And we get 23 point-- I'll just round up-- 23.3. This is equal to 23.3. And then we have miles per hour. Which is obviously very fast. He's the fastest human. But it's not maybe as fast as you might have imagined. In a car, 23.3 miles per hour doesn't seem so fast. And especially relative to the animal world, it's not particularly noteworthy. This is actually slightly slower than a charging elephant. Charging elephants have been clocked at 25 miles per hour.