(June 5, 1997, Gazette)
Have a burning question pertaining to science, engineering, medicine, the arts, music, humanities, social sciences, physical education, or any other field? Let us know; we'll do our best to find the answers for you, drawing upon the expertise of members of the university community.
Ken Hooper, an English student from Mount Pearl, has a problem with raindrops falling on his head. He writes:
"Here's one for your MUNsolved mysteries section (which I read frequently and enjoy). Recently, as I ran from the TSC to the QE II Library, I was getting soaked with rain. I began to wonder if running in the rain actually causes you to get hit with less raindrops, or if - because of your speed - you actually collide with more droplets of water and consequently get even more drenched than if you decided to walk. I look forward to resolving my quandary once and for all. Thanks for your assistance. All the best!"
Dr. John Whitehead, Physics and Physical Oceanography, gave Mr. Hooper's question serious consideration. "By running the in the rain you certainly spend less time in the rain than if you were to walk slowly," he observed. "On the other hand, the faster you run the larger your cross-sectional area, and hence the more water you will accumulate as you run. The question therefore arises that for a given set of conditions, is there an optimum speed at which to run?"
Dr. Whitehead said that if the simplest possible case were considered - that of a gentle rain falling vertically - it's easy to show that the optimal strategy for staying dry involves running as fast as you can.
"To see this, we note that the rate at which water will accumulate on you will be directly proportional to your speed - the faster you run the more rapidly the rain will accumulate on your front," he explained. "However, since the time taken for you to reach your destination will be inversely proportional to your velocity, the total amount of water that will accumulate on your front as you run through the rain will be essentially independent of your speed, and will depend only on the distance to your destination, the number of raindrops per unit volume, and their size. On the other hand, the rate at which rain falls on your head and shoulders is independent of your speed, and will be directly proportional to the time taken for you to get to where you're going. Since you exercise little control over the properties of the raindrops, to minimize the total amount of water that you accumulate you need to minimize the time you take to reach your destination - i.e., run as fast as possible."
Dr. Whitehead points out that since the "gentle rain falling vertically" is a largely hypothetical scenario in St. John's, it might be wiser for our purposes to also consider the effects of wind on the situation.In the case of a headwind, the optimal strategy is again to set forth post-haste. "In the case of a tailwind the optimal strategy is less obvious, since two possibilities present themselves," Dr. Whitehead noted. "The first is to run at the same speed as the wind (assuming this is within the bounds of your athletic capability), since this means no rain will fall on your front, only on your head and shoulders. The second possibility is to minimize the time spent in the rain by simply running as fast as you are able, regardless of the windspeed. Determining the best strategy, however, depends on the windspeed, the terminal velocity of the rain drops, and your shape (geometrical, not physical)."
Here's your formula for success:
"Denoting the vertical velocity of the rain by Vr, the wind velocity by Vw, then (roughly speaking) if Vw/Vr > D/h where D and h denote your diameter and height, the optimal strategy is to run at the speed of the wind. If Vw/Vr < D/h, the optimal strategy is to run as fast as you can regardless of the windspeed."
But it gets even more complicated if all possibilities are considered, such as the wind blowing at an angle to your path, or the angle at which you lean into the wind as you run. "While such complexity may make for a more interesting problem, the solution is unfortunately of little practical value given the vagaries of the Newfoundland climate, since it is unlikely that wind conditions would remain constant for the time taken by you to determine the optimal strategy," Dr. Whitehead said.
"Moreover, other constraints - such as one's ability to run at the speed of the wind over any substantial distance, possibly laden down with books, briefcase and perhaps a laptop computer - are probably of more importance in determining an optimal strategy for getting to where you're going on a rainy day."
Send your puzzler to MUNsolved Mysteries, Gazette, Arts and Administration Building, Memorial University of Newfoundland, St. John's, Nfld., A1C 5S7; e-mail firstname.lastname@example.org, or fax 709-737-8699. Please include your name and telephone number.