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.


Phil Hodge said...

Thanks, Prof. Ivars. I went out and tested it today, and have two additional variables to add. 1) It also depends how wet you are. Once you have reached saturation, or at least saturation on the relative windward side, further wetting becomes irrelevant. Therefore, the avoidance of discomfort is not a linear function. 2) It would seem prudent to factor traction, also. Running too fast for conditions could cause a slip, and a resultant huge increase in the wetness and discomfort.

dank hailman said...

Thanks for blogging about our article, Prof. Peterson. FYI there is a companion article in the November Math Horizons with prettier pictures and less math, for those who want a gentler overview of the analysis.