Aug 17 For a fourth (and last) offering for today, this is my math column last Friday in Mint. I wanted to dig a little into all the phenomenal performances I had watched during the Tokyo Olympics - or at least some of them - and find ways to put them in some perspective. How fast were those sprinters actually running? What did it take to pull off a world-record triple-jump? How fast was that javelin travelling as it soared to gold? And much more.
Didn't get to all of it, but at least some. Take a look: See how they run. And jump. And throw. https://www.livemint.com/opinion/columns/see-how-they-run-and-jump-and-throw-11628788555983.html (Friday Aug 13). Definitely let me know what you think. cheers, dilip PS: Faint musical reference. What? --- See how they run. And jump. And throw. The day after Neeraj Chopra's mark of 87.58 that won gold in Tokyo, an interesting tidbit landed in my inbox. No, not the clever quip about how he missed the Delhi University admission cutoff of 97.5 anyway. Instead, it was a quick diagram and calculation, scribbled on a piece of paper, working out how fast the javelin was travelling when it left Chopra's hand. Yes, it's possible to work that out, and it's a rather startling number. But before we get to that, take a look at some other numbers the Olympics generated. Start with the people usually described as the fastest in the world after their respective gold medals - the winners of the 100 metre dash, Italy's Marcell Jacobs and Jamaica's Elaine Thompson-Herah. Thompson-Herah finished in a phenomenal 10.61 seconds. Only one woman has ever run faster - Florence Griffith-Joyner (FloJo) who ran 10.49 seconds in 1988. Jacobs' time was 9.80 seconds - fast, but other men have run faster, all the way to Usain Bolt's 2009 world record of 9.58 seconds. But exactly how fast were these two running as they breasted the tape? The quick answer is to divide 100m by their times. That gives us their average speeds. Thompson-Herah ran at 33.93 kilometres per hour (kmh); Jacobs, 36.73kmh. Yet a little thought will tell you these numbers can't be right, especially in a race as short as the 100m. We do call those their average speeds. But it should be obvious that Jacobs and Thompson-Herah were not running 30+kmh through the whole 100m. After all, they started from zero, they accelerated for a while and only hit their all-out full speeds a few seconds into the race. If you watched the coverage of the races, you'll have seen some slick graphics that show this acceleration. Absent that, let's make the possibly reasonable assumption that the runners accelerated smoothly for the first 50m of the race, and ran flat out, at their peak speeds, for the last 50m. There are well-known and relatively simple equations that relate acceleration, speed, time and distance. I won't spell them out here. But use them with the numbers and assumptions above, and we find that Thompson-Herah reached 51.35kmh. Jacobs, 55kmh. There in Tokyo, the fastest woman and man in the world ran faster than some cars in our traffic-bound cities ever manage. Now you might argue with the 50m assumption above. Perhaps these remarkable athletes accelerated to their peak speeds more quickly, perhaps by the 30m mark? This would mean lower peak speeds. Still, the same equations apply, giving us 44.1kmh for Thompson-Herah, 47.8kmh for Jacobs. Still pretty fast, even for cars. If we ever get precise figures for how long - distance or time, it doesn't matter - it takes these athletes to reach their peak speeds, we can do these calculations again. I'd bet we'll find Thompson-Herah at somewhere nearing 40kmh, Jacobs 3-4kmh faster. In both cases, slightly slower than the numbers above. The reason for that is what a longer race tells us. (Without losing track of the point of this exercise, I'll stick to the women from here on.) Turn now to the 200m race. The great Thompson-Herah won that race too, completing a rare double-double - she won the same two races at the 2016 Rio Olympics as well. Her time in Tokyo was 21.53 seconds. Again, this was the second-fastest any woman has ever run 200m, and again, the fastest is FloJo, with her 21.34 seconds at the Seoul Olympics in 1988. We can divide 200m by Thompson-Herah's 21.53 seconds to get her average speed through the race: 33.44kmh. But again, that doesn't account for her acceleration at the start. Assuming once more that she took the first 30m to reach her peak speed, we find that she ran at 38.5kmh over the remaining 170m. Do you see a puzzle here? Assuming the same 30m acceleration phase at the start, we have her running 44.1kmh for the 100m, but 38.5kmh for the 200m. If Thompson-Herah can touch 44.1kmh, why didn't she maintain that speed in the 200m race? If she had, she'd have obliterated not just FloJo's world record, but the men's world record - Usain Bolt, 19.19 seconds - as well. For she'd have finished the race in 18.78 seconds. But she didn't. One possible lesson: she reaches her fastest speed earlier than 30m into whatever race she's in. Still, since we don't know better, let's stick with that figure. Another possible lesson: her top speed might indeed be 44.1kmh, but while she can keep that up over 70m, she can't do it over the much longer 170m. You can sometimes see this in longer races, when an athlete goes out fast at the beginning, can't keep up that blistering pace and has faded by the end of the race. Which is why I've always thought the most compelling race in the Olympics is the 400m. It's long enough that a runner cannot sprint it like she would do the 100m or 200m, but not short enough for her to bide her time jogging along and make a late push for victory. In Tokyo, Shaunae Miller-Uibo of the Bahamas won gold in the 400m, in a time of 48.36 seconds. Again applying the 30m-to-accelerate metric, she ran 32kmh the rest of the way. In even longer races, we can dispense with the acceleration metric, because 30m is such a small fraction of the whole distance, and factoring it in doesn't make much of a difference. Thus: * Athing Mu, 800m gold in 1:55.21: 25kmh. * Faith Kipyegon, 1500m gold in 3:53.11: 23kmh. * Sifan Hassan, 5000m gold in 14:36.79: 20.5kmh. * Sifan Hassan, 10,000m gold in 29:55.32: 20kmh. * Peres Jepchirchir, marathon (42.195km) gold in 2:27:20: 17kmh. The longer the race, the lower the average speed, of course. But it's not as steep a drop-off as you might imagine. There are plenty more such numbers to extract from the Olympics. But let's turn to Neeraj Chopra and his javelin. Imagine throwing an object out in front of you. If you throw it parallel to the ground, it will drop there quickly enough. Thus to get any kind of distance into your throw, you need to throw it an angle to the ground, so that it gains height before dropping. But not too steep an angle, because it then goes higher rather than farther. If you throw it straight up, after all, it returns to bonk you on the head. So you will agree that the optimum angle, if you're looking to throw as far as possible, is somewhere in between. In fact, there's a relatively simple distance equation which tells us that the optimal angle is 45 degrees. Again, I won't spell it out. But given that Chopra's winning throw was 87.58m, we can use the equation to calculate the speed at which he threw the javelin. And what's that speed? 105kmh. Let that sink in for at least the few seconds it took his javelin to touch down. This is a nearly 3m long spear, weighing at least 800gm. What does it take to fling such a projectile significantly faster than the speed limit on the Bandra-Worli Sealink? It takes being an Olympic champion. So respect to Elaine Thompson-Herah, Marcell Jacobs, Shaunae Miller-Uibo, Athing Mu, Faith Kipyegon, Sifan Hassan, Peres Jepchirchir, Neeraj Chopra and every other champion at Tokyo. Your excellence leaves us far behind. But we want even more. -- My book with Joy Ma: "The Deoliwallahs" Twitter: @DeathEndsFun Death Ends Fun: http://dcubed.blogspot.com -- You received this message because you are subscribed to the Google Groups "Dilip's essays" group. To unsubscribe from this group and stop receiving emails from it, send an email to dilips-essays+unsubscr...@googlegroups.com. To view this discussion on the web, visit https://groups.google.com/d/msgid/dilips-essays/CAEiMe8paoijNoXn3vEMYrjKMFKg6dVZhxQpHeoBNBQiVvFC9VA%40mail.gmail.com.
