PGS Mathematics: Rain Puddles and Speed

Here, Princeton University Press author John Adam presents two math problems you might encounter in your quest for fall foliage this month. The problems and solutions are similar to those found in his books A Mathematical Nature Walk and draw upon the skills you described in Guesstimation: Solving the World’s Problems on the Back of a Cocktail Napkin.

The season of Autumn can be one of great beauty, especially where the foliage changes to a bright variety of reds, oranges and yellows. Why do leaves change their colors in the fall? There are in fact several different reasons, but the most important is the increasing length of night and cooler temperatures at night. Other factors are the amount of rainfall and the overall weather patterns in the preceding months. Just like sunsets, the weather before each fall is different. Basically the production of chlorophyll slows and stops in the autumn months, causing the green color of the leaves to disappear, and the colors remaining are mixtures of brown, red, orange and yellow, depending on the types of tree. To have any real chance of seeing the wonderful fall foliage, you have to go to the right places at the right time. Going to the beach in summer or even the fall won’t do! And going to the Blue Ridge Mountains or New England in the depths of winter will not enable you to see the fall foliage either, pretty though the snow-covered trees may be!

But there are some other aspects of this season that are present at anytime of the year, and do not depend on the leaves changing. Those aspects involve trees, rain and, in this case, my left foot!

Consider this: you are in the hills of New Hampshire, or West Virginia, or perhaps you are somewhere on the Appalachian Trail enjoying the glorious fall colors, when suddenly a rain squall appears out of nowhere, or so it seems. You run to take cover in a deserted shelter a hundred yards away near the trail. Playfully (there being nothing else to do) you stick your foot outside the shelter, and of all things, photograph the rain falling around it! After ten minutes of intense rainfall, it stops as suddenly as it started. You wonder how fast that rain must have been falling to create the scene you now survey: large puddles all along the trail, drops dripping from every available leaf above you, and the temporary dark-brown stains on the trunks of rain-soaked silver birches.

As you set out on your way again, you start to notice the wave patterns formed when the drops falling from the branches above you hit the surface of puddles. Question: which of the following patterns represents this situation?

Answer: the one on the left. Raindrops falling on the surface of a puddle generate wave patterns that are dominated by the effects of surface tension. The speed of these waves is inversely proportional to the square root of the wavelength; thus shorter waves travel faster and move out first. Note the expanding region of calm associated with, and inside these waves. The other pattern is dominated by gravity, which produces longer waves with speeds directly proportional to the square root of the wavelength, so the longer waves travel faster and move out first.

Once home, you upload your pictures onto the computer. On noticing the picture you took back at the shelter, you realize that knowing the exposure time of the shot, you can estimate the speed of the rain. Here is the picture: can you find the speed of the rain?

This rather pedestrian photograph was taken from a sheltered area outside the Mt. Washington Resort, Bretton Woods, NH on September 30th, courtesy of Tropical Storm Nicole. Because of the heavy rain, Mt. Washington was nowhere to be seen! Nevertheless, it seemed like fun to “guesstimate” the speed of the raindrops, given that the exposure time for the shot was 1/200 second and an estimate of the width of my sneaker (you don’t need to know my shoe size to do this.). My foot and the raindrops shown were about the same distance from the camera. You can assume that the foreshortening of the rain streaks (due to the downward angle of the camera) is not significant.
Answer: From the photograph, The width of my sneaker is approximately 5 inches, so the raindrops travelled approximately 5/3 inches in 1/200th of a second, or 1000/3 inches/s, or 1000/36 ft/s. Therefore

Since 1 m/s 2.2 mph, this is about 8.6 m/s.]
For those who don’t have access to my foot, anything in the range 4 inches to 6 inches wide is a reasonable estimate. This would result in a range of speeds 15-23 mph (7-10 m/s).

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  1. This was a very interesting post. A photo enthusiast myself, I could really relate to how it correlates