Did Usain Bolt REALLY run 100m in 9.63 seconds?

What's going on in London? Well, I don't know. Some small event known as the Olympic Games is going on in London at the moment. And uh last night in particular, there was 100 meter final and Usain Bolt winning it in 9.63 seconds. Well, it was a little bit less really if you uh if you include some of the relativistic effects from uh from Einstein's theory of relativity. So, uh I guess that's what we're going to talk about. All right. So, do you want me to draw a picture of the race? Yeah. Okay. I'll do my best. So, of course, you've got here we are in the the Olympic stadium in London. There's my racetrack. Okay, it's 100 meters long. The guys started down here, right? And then they they legged it along the track. So, let's draw a little picture. So, this is uh let's say this is Tyson Gay or something, right? And then somewhere out in front of him, we've got uh we've got Usain Bolt. So, Usain Battle front and a bunch of the other guys sort of on route. Okay, so Usain Bolt went past the finishing line in 9.63. 63 seconds according to the uh stadium clock. So let's make that clear the stadium clock. Now what I want you to imagine is that Usain Bolt this is this is Usain here has got a little watch on. Okay. So this is his watch. So this is Usain Bolt's clock. So what does Usain Bolt's clock or what did Usain Bolt's clock read at the moment that he crossed the uh the finishing line? Now you might think well well it's 9.63 seconds but it's not. It's actually a little bit less than that. It's 9.63 seconds less about more or less 5 millionth of a nancond. So he actually clocked according to his clock 5 minutes of a nancond less than the stadium clock less than this 9.63 seconds. Okay. So a nancond is 10^ theus 9 seconds. So if you do it in the American billions that's a billionth of a second and then it's a millionth of that billionth n point n that's three n we now got 12 but I want five four five six seven eight nine so Usain Boltz if you actually looked at his watch it would have read this it would have read 9 62 999 999995. Well, this is all because of special relativity, Einstein's theory of special relativity. So, what what's the key thing about Einstein's theory? It's a lot of people say it's that, oh, you can't travel faster than light. And and that's true, but that's not the most important thing about it. The most important thing about it is that everybody agrees that light travels at about 300 million meters/s in a vacuum. That's the most important thing. Everybody agrees on that, no matter whether they're moving relative to it or not. So, this really changes how you think about velocities and how you think about time in a in a really fundamental way. And sort of to illustrate that, imagine, you know, you've got you're going along a motorway and you're going in a car at 70 mph and the guy's going in the opposite direction at 70 mph relative to the road as well. Then you would say, well, what's the relative speed of you to the other car? And you'd say, well, what would you say it is? 140. 140. That's what most people would answer. But actually, that's not quite right. That's almost right. It's a little bit less than 140. to see that it it can't quite be the right answer. Consider what would happen if the other car was going along at the speed of light. Okay, then you would be saying, okay, what's our relative speed? Well, is it the speed of light plus 70 mph? Well, no, it can't be because everybody agrees that something going at the speed of light goes at the speed of light. So, you've clearly got to change what you think about velocities, what you think about time. Okay? So what actually happens is that you know when somebody's moving relative to something else is that their clock slows down and it slows down by by a particular factor. So if we were to work out what Usain Bolt's time is T-bolt and we were to compare it to T stadium, this is the time measured by Usain Bolt on his clock. This is the time measured by the stadium clock which is this one. Okay? And they differ by a factor. They're not the same. They differ by this factor. It's square<unk> of 1 minus v ^ 2 over c^ 2. Now v here is Usain Bolt's velocity relative speed relative to the track. Okay? And c of course is the speed of light. So this is a very small number this v^² over c^² but it's not zero. And these would only be equal if it were zero. So what we can do is if we want to sort of to get this this approximation that we've got here, well we do a little sort of bit of mathematical trick here. We do what's called a tailor expansion on this thing because this is this is something that's very small. This v^2 over c^² so we can do a tailor expansion and we can approximate this by 1 minus a half v ^ 2 over c^ 2 time t stadium. Okay, I'm going to have to look it up now. Okay. So, I'm going to put that in. So, that's 2 9 7 92 458. That's probably not exact either, but you're going to have to make do with that. 458 5.77 * 10 - 15. When I did this, I did it sort of essentially very quickly approximating things. But, uh, actually, maybe I should have done it a bit more accurately. And there should be a 78 there. So, this is the number that Usainb actually actually ran the time that he he clocked on on his watch, which is 9.62 999 9. How many nines have I got? 422 seconds. And that's the time that he actually would have clocked on on his clock. Now, there's actually something even more remarkable about this when you actually think about the implication. You could ask the question, so so okay, so he's ran it in a shorter time, it would seem. Does that mean he ran faster in some way? Is is that the implication? And no, it doesn't mean that because the laws of relativity don't allow that. Basically, you know, Usain Bolt's running at roughly 10 m/s relative to the track. And just by the laws of relativity, the track is moving at about 10 m/s relative to Usain Bolt, which is completely completely equivalent. So there is no change in in their speeds. So what's changing then? Well, actually the answer is even more remarkable. The answer is that the from Usain Bolt's perspective that track is moving and then because that track is moving it actually shrinks a little bit from his perspective by about 50 minutes of a nanometer. So he clocks a shorter time basically because the track shrank a little bit for him. So you could say he didn't complete 100 meters, which is uh just take his medal off him. And each one goes outside one ring and inside the other. But as you go around