November 29, 2009

Running in the Rain

Many years ago, when I was a high school physics teacher, one of my favorite books was The Flying Circus of Physics by Jearl Walker. The book offered dozens of provocative questions about the everyday world to get you thinking about physics, whether you were cooking, flying, lazing at the beach, or getting caught in the rain.

This raises the question, "Should you run or walk when you cross the street in the rain without an umbrella?"

Walker noted that running would mean less time spent in the rain. However, running into some rain might make you wetter than if you had walked. He suggested doing a rough calculation, making the approximation that your body is a rectangular object. "Using such a model, can you tell if your answer (whether to run or walk) depends on whether the rain is falling vertically or at a slant?" Walker wrote.

The original edition of The Flying Circus of Physics included references but didn't provide any answers, so teachers and students were on their own. A later edition added answers, but often with the caution that the underlying physics hadn't yet been worked out satisfactorily.

Walker's solution to the "run or walk" question was to run as fast as possible if the rain were toward your front or overhead. If the rain were toward your back, you should run with a speed equal to the horizontal velocity of the rain (that is, along with the rain).

Many others have posed the same question and come up with roughly the same answer, as summarized in the following limerick by Matthew Wright from a 1995 New Scientist article.

When caught in the rain without mac,
Walk as fast as the wind at your back,
But when the wind's in your face
The optimal pace
Is fast as your legs will make track.

Some people have noted, however, that for a sufficiently weak tail-wind, it is actually better to run as quickly as possible rather than "as fast as the wind."

Moreover, shape matters, as Dan Kalman of American University (writing as Dank Hailman of Jamaicarain University) and Bruce Torrence of Randolph-Macon College (writing as Bruce Torrents of Raindrop-Macon College) point out in the article "Keeping Dry: The Mathematics of Running in the Rain," published in the October Mathematics Magazine.

Most previous analyses modeled the damp pedestrian as a rectangular solid. Kalman and Torrence considered an ellipsoidal traveler. Their results indicate that, under nearly all conditions, it is beneficial to run at top speed.

"Our recommendation, therefore, is to RUN in the rain unless you find yourself traveling in the perfect storm—where the tail-wind is half your top running speed, the cross-wind is minimal, and the rainfall is light," Kalman and Torrence conclude. "In such conditions, given the rounded features of the human body, it might make sense to dampen your pace (so to speak) from a run down to a speed that is just a bit faster than that of the tail-wind."

The authors offer the following advice:

When you find yourself caught in the rain,
while walking exposed on a plane,
for greatest protection
move in the direction
revealed by a fair weather vane.
Moving swiftly as the wind we'll concede,
for a box shape is just the right speed.
But a soul who's more rounded
will end up less drownded
if the wind's pace he aims to exceed.


Bailey, H. 2002. On running in the rain. College Mathematics Journal 33:88-92.

Deakin, M.A.B. 1972. Walking in the rain. Mathematics Magazine 45:246-253.

Hailman, D., and B. Torrents. 2009. Keeping dry: The mathematics of running in the rain. Mathematics Magazine 82(October):266-277.

Schwartz, B.L., and M.A.B. Deakin. Walking in the rain, reconsidered. Mathematics Magazine 46:272-276.

Walker, J. 1977. The Flying Circus of Physics (with Answers). Wiley.

November 8, 2009

Random Bias

When you're making an estimate, extraneous factors or irrelevant information can strongly bias your judgment, especially when the situation involves a lot of uncertainty.

Physicist and writer Leonard Mlodinow provided a vivid illustration of such bias during his recent colloquium talk on randomness, which he presented at the National Institute of Standards and Technology in Gaithersburg, Md. Mlodinow is currently a professor at Caltech. He is the author of The Drunkard’s Walk: How Randomness Rules Our Lives.

Your powers of estimation are easily influenced by minor things, random things that happen around you, Mlodinow argued.

To demonstrate, Mlodinow divided his audience into two groups. Each member of each group independently wrote down on a slip of paper a numerical answer to the question: How many countries are there in Africa? The slips were collected and the results tallied.

Answers from the first group averaged about 47; answers from the second group, about 27. Why the difference?

Each group also answered an introductory yes-or-no question that the other group did not see. The first group was initially asked: Are there more than 180 countries in Africa? The second group was initially asked: Are there more than 5 countries in Africa?

Most members of the audience of scientists and engineers probably had no clear idea what the correct answer is. They had to make a guess and apparently were influenced by the number that they saw in the first question. The results would undoubtedly have been different with, say, an audience of Africa scholars, who would likely be much more familiar with the continent.

Africa actually has about 50 countries, depending on how you count disputed territories and whether you include offshore island nations.

At the NIST talk, I was in the first group, and I knew that Africa had a lot of countries. I also recalled that the United Nations had nearly 200 members, so I based my estimate on what fraction of the total would be in Africa. I came up with 70, which was too high and was probably influenced by my seeing 180 in the first question.

Mlodinow has tried the same experiment with a variety of audiences, always obtaining a striking difference between the two groups. When he presented the talk last year to an audience at Google, for example, the estimates averaged about 65 and 30. (At Microsoft, the estimates had been 50 and 24.)

Mlodinow called this effect "anchoring bias." When making estimates, "be careful before you trust them," he warned.

One practical lesson, however, is that in negotiations it pays to ask high, whether you're fishing for a higher allowance or suing someone for damages. In the face of uncertainty, this posited sum may very well influence the size of the final outcome.

We take in, filter, and interpret a lot of data every day, Mlodinow noted. When the situation involves uncertainty or randomness, we often make mistakes or draw improper conclusions.

November 3, 2009

Cosmic Reflection

The universe began with a bang 13.7 billion years ago, so it was no surprise that a terse bang accompanied a video illustration of the event. This sonic boom was itself just one fleeing moment in a narrated, 40-minute symphony, titled Cosmic Reflection, which debuted on Nov. 2, 2009, at the Kennedy Center in Washington, D.C.

Composed by Nolan Gasser, the symphony was his tribute to the Fermi Gamma-Ray Space Telescope in its first year in orbit. In words, pictures, and music, the composition told the story of the universe, offering a striking lesson in modern astrophysics and cosmology, from the big bang to dark energy and cold death.

Written by Classical Archives founder Pierre R. Schwob and physicist Lawrence M. Krauss, the narration presented the known facts and theoretical speculations simply and clearly—offering a model of accessible exposition. The poetry was in the music, played crisply and impressively by the Boston University Symphony Orchestra, and in the video images provided by the NASA Goddard Space Flight Center: twisting tendrils and flickering rhythms; swinging galaxies and swirling tones; flashing stars and melodic fragments; cataclysmic collapses and percussive effects.

The large audience, blending the sensibilities of the arts, science, and engineering, gave the performance a standing ovation.

While experiencing the event, I couldn't help wondering what a narrated, illustrated symphony devoted to a mathematical theme would be like. Of course, astronomy itself has been a significant driving force in the development of mathematics. Isaac Newton, for one, wrote his immortal Principia Mathematica in answer to the question of what sorts of orbits would occur under an inverse-square force law. I wrote about these connections in my book Newton’s Clock: Chaos in the Solar System. And mathematics has been an invaluable tool for astrophysicists and cosmologists peering into the deepest reaches of the universe. So, mathematics could lay a claim to Cosmic Reflection, too.

What about the story of the prime numbers and the 150-year-old Riemann Hypothesis, a mathematical question for which the Clay Mathematics Institute has offered a $1 million prize for its solution? But that story isn’t complete; there is no proof yet, and no certainty that anyone will ever find a proof. And what images would you use?

Despite the pervasiveness of mathematical thought and its crucial role in underpinning scientific research, I am at a loss. What story, told in words, images, and music, could match the incredible epic that astronomers and physicists have forged from their observations over many millennia of spots of light in the sky?