## Prediction

Here is a physics problem from a book I was reading:

The year is 2100 and the pole vault record stands at 7.5m. Estimate the world record for the 100m sprint at this time.

This question is one of our favourites since it shows the power of physics to make the apparently most unlikely connections. Incidentally, both records would be held by women by this date – see Nature, Jan 2 1992.

The pole vault question is a classic, and a good one. The idea is that in pole vaulting you turn kinetic energy of running into potential energy of being high off the ground. If you know how much potential energy you had at the top of your pole vault, you know how much kinetic energy you had at top speed, and hence how fast you can run. Also, if you are Jacob Bronowski then pole vaulting is what separates man from the animals.

Solving the problem is a nice calculation. The potential energy per unit mass of the vaulter is her height $h$ times gravitational acceleration $g$ (PE = $gh$). The kinetic energy per unit mass is half the square of her velocity (KE = $1/2 v^2$). Setting these equal, we get

$gh = 1/2 v^2$

and solving for $v$

$v = \sqrt{2gh}$,

a result physics 1 students have seen many times when being asked about pennies dropping from the Empire State Building and such. The vaulter’s center of gravity starts out half a meter above the ground and rises to the height of the bar (or possibly just under). For $h = 7m$ and $g = 10m/s^2$ we get

$v = \sqrt{140m^2/s^2} = 12 m/s$

Using this as the speed of a sprinter, we get

$\frac{100m}{12m/s} = 8.3s$

Is this a good prediction? If the pole vault record really does advance to 7.5m, will the 100m time similarly improve?

Let’s do the calculation over using the current world records. If it doesn’t work for them, there’s no reason to believe it’ll start working in the future! The current world pole vault record is 6.14m and g = 9.8m/s^2, so we estimate the 100m record at 9.11s, or about 5% off from Usain Bolt’s record of 9.58s.

That’s pretty good. But would you be so impressed if you had done the same calculation in 1994? Sergey Bubka set the 6.14m pole vault record that year, and it has stood since. But in the same year Leroy Burrell set the 100m world record at 9.85s. Now we have about a 9% error. Still decent.

But hold on. I said this dude Sergey did the pole vaulting and Leroy did the running. Shouldn’t my calculation actually tell me how fast Sergey runs, not Leroy? Bubka was unusually fast for a vaulter, and could perhaps have run 10.1s. Now the error is 11%.

When you run 100m, though, you don’t instantly accelerate to a certain speed and stay there the whole time. You speed up to the 40m mark or so, and after 60m you start to slow down. Further, the run-up to a pole vault is much shorter than 100m, and you run more slowly than your best sprint speed because you’re carrying a giant pole. What the calculation is really going at is your speed right before takeoff. Bubka’s was about 9.9m/s right before planting the pole, compared to about 11m/s estimated with our cute physics problem. Ultimately, the error is more than 10%.

10% error is a lot if you want to make an estimate of the future world record, to place a bet on it, for example. And there’s a worse problem – our error goes the wrong way. Based on physics, we would expect the runner to be inefficient at converting all their kinetic energy to potential energy. Some gets lost in the bending of the pole, which isn’t perfectly elastic, and the vaulter still has some kinetic energy at the top of their flight because they move horizontally.

In general, we know that conversion of energy is not 100% efficient. So for a given speed, the vaulter should not quite make it to the calculated height, or conversely, for a given height, the vaulter should need more than the calculated speed. The authors point this out in their solution, estimating that the energy conversion might be 10% inefficient, so the real 100m time, give a pole vault height, might be 5% faster than predicted by the equations. This is backwards of reality, though. In the real world Sergey has less speed than what we calculated, not more. Evidently Bubka could do work after taking off from the ground, perhaps by pushing down on the pole.

The physics estimate for the 100m dash worked out pretty well, but only by accident. After all, if we believe this calculation, it predicts that Bubka should not have the world record for the pole vault – Usain Bolt should! In fact no one has ever held the world records for both pole vault and 100m (in the modern era, anyway. Maybe Tarzan did it.) We could repeat the calculation with the women’s world records (5.06m and 10.49s by Yelena Isinbayeva and Florence Griffith-Joyner) and again find that the estimated 100m time is too fast (10.0s). Or we could use data from the same person – Bryan Clay, the world’s best decathlete. Clay has run 10.36s and jumped 5.15m. His pole vault predict 9.95s for 100m.

In each case I’ve examined, the pole vault-projected 100m time is too fast for the real 100m time, and consistently around 5% too fast. In that case, maybe the pole vault record in 2100 is a good predictor for the 100m time in 2100, but not because of physics. Those two records just tend to track each other and progress simultaneously (with a fair amount of noise). We could go ahead and observe that the pole vault and 100m records tend to be related by some quadratic equation, and then use that equation to predict the future 100m time given the future pole vault. Toss out the physics, just note that it works and it’s good enough. Good idea or bad?

The problem is that we have no reason to believe that should continue to work, just because it has so far. Just because we can find some pattern in our past data doesn’t mean it will persist. Want an example? We already have one, provided in the original statement of the problem. The authors wrote, “Incidentally, both records would be held by women by this date,” a phrasing that makes them sound certain of it.

I’ve been around track and field for almost a decade. I can tell you, with utter assurance, that men are better at it than women, and they still will be in 2100. Men and women are physiologically different, and compared to an elite man, an elite woman, despite training with comparable intensity and dedication, is nowhere near as good.

So what led the authors to be so confident that women will hold the world records by 2100? The Nature reference is to a fantastically stupid letter titled “Will women soon outrun men?”, which predicted that female marathoners might become as good as male marathoners by 1998. They didn’t.

The discrepancy came up because the authors made a prediction by drawing a curve through the data about the past, and assuming the curve would continue the same way in the future. That was wrong. Women’s track and field, and especially women’s marathoning, is less mature than men’s. The women’s marathon was only added to the Olympics in 1984 (and steeplechasing in 2008). Few women were running marathons before then, and few were training for them seriously the way men do. Now, as more women join track and field and run professionally, the records can drop quickly. Men approached physiological barriers before women, so their improvement has been slower recently. This adds up so that when you draw a line through the times of the last 50 years for women and men, you predict that women will soon be faster.

Lines don’t keep going indefinitely into the future, though, even if you publish them in really fancy journals.

So now let’s go back to predicting the future 100m time based on the pole vault height. If the pole vault height gets 10% better, do you believe the 100m time will get 5% faster as our curve-fitting would likely predict? Unlike the estimates about men’s and women’s performances, it’s worth a close look. The difference is we have a physical model to back it up. We have an idea where it comes from. Those same equations we used at the beginning of the post predict that the pole vault will advance twice as much as the 100m dash.

We can’t predict the 100m time accurately with physics alone, because when we tried we were off by a lot. We can’t predict it accurately with curve-fitting alone, because it’s blind. When we combine them we may have a pretty decent predictor. What do you think?

References:
I got the world records from Peter Larsson’s web page Track and Field all-time Performances Homepage

Bryan Clay’s stats are from Wikipedia: Bryan Clay

The story about Bubka being especially fast is an anecdote I heard from a pole vault coach, who said Bubka was fast enough to be on the Russian national 4x100m team.

For an example of the velocity profile of elite sprinters in a 100m race, see Eriksen et. al. “Velocity Dispersions in a Cluster of Stars: How Fast Could Usain Bolt Have Run?” I think the title is a joke. Based on video analysis of Bolt’s 9.69s 100m world record in the 2008 Beijing Olympics, the authors predicted Bolt could run 9.61 +- .04s or 9.55 +- .04s if he had not started celebrating before finishing, depending on assumptions of their model. Bolt has now run 9.58s.

Bubka’s speed before takeoff is anecdotal from Wikipedia: Sergey Bubka.

The Nature article is not available unless you pay or have access from your university or something like that. It’s here but I haven’t read it because plenty of people have written similar things and they’re all worth your time to avoid.

The book of physics problems is “Physics to a Degree” by Thomas and Raine.

The video clip is from “Lower than Angels”, the first episode of Jakob Bronowski’s documentary “The Ascent of Man”.

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