Should You Run In the Rain

Suppose it’s raining outside and you need to go some short distance on foot, maybe to a building 50 feet away. Should you run or should you walk, in order to stay drier? Opinions can vary and may include “always run,” “always walk,” “it doesn’t matter,” or “it depends.” Of course the choice of the better approach always depends at least a little on the specific situation, but maybe we can arrive at some kind of general answer for the question.

How can this be modeled? There are many details of reality that can make this really complicated, such as the current degree of wetness or dealing with individual drops of rain. Fortunately, we can make the modeling task easier with some simplifying assumptions.

• Forget how wet our hypothetical pedestrian is at a given moment or whether he gets into any puddles. Instead I would like to focus only on how much rain the person encounters in that short trip on foot.
• Though the pedestrian’s arms, legs, and body will be moving relative to each other, it’s easier to assume he is moving only straight forward at a constant speed with no rotation or zigzagging. His motion would then resemble the stiff motion of a game token when being scooted to a new location on a game board.
• While dealing with the exact shape of a person would promote higher accuracy, people are really too complicated in terms of shapes and curves for quickly modeling as a 3-D object. Here we only care about how much rain is encountered from the front or the top. Otherwise we don’t care about the angle at which the rain is encountered. So we can treat the pedestrian as a rectangular block, like the shape of a child’s wooden building block or a cereal box. This block has a well-defined top face (flat surface) and front face.
• As there are too many rain drops to deal with individually, we can treat the falling rain as being spread out evenly through the 3-D space the pedestrian will be traveling in. Think of it like an evenly spaced apple orchard or wallpaper pattern. We might further assume that the density of rain in this space (the amount of rain per unit volume) is constant too and does not change over time. The rain rolling off of the person’s surface will be ignored for now and will only be counted once, when first encountered.
• In fact, let’s assume the speed and direction of the rain are also constant, and it always falls straight down. One consequence is that the amount of rain entering an otherwise empty region is equal to the amount leaving the region. A more useful consequence is that we can ignore the rain that might normally touch a person from the back, left, or right or even from below.
• Although it’s the rain that is failing, we can picture the rain as stationary and treat the pedestrian as moving upward through the rain. So now we can look at moving through a stationary rain field in the horizontal direction and the vertical direction.

With all this in mind, we can now look at calculating the amount of rain encountered for the duration of the trip. As mentioned above, we are modeling the pedestrian as a rectangular block moving upward and horizontally through stationary rain (evenly distributed in space). The space carved out by that vertical motion alone is itself a rectangular block, and the volume of that block is proportional to the amount of rain encountered vertically (from the top). The top area of this vertical travel-defined block is the same as the pedestrian’s top area. The height of the travel block is determined by the distance the pedestrian moves upward during the drip, which can be determined by the rain speed and the time of the trip. The well-known formula of rate*time = distance can be used for that height.

Similarly, the space carved out by the horizontal motion is proportional to the rain encountered horizontally (from the front). The front area of the horizontal travel block is the front area of the pedestrian. The length of the block in the direction of horizontal travel is the distance the pedestrian is walking or running through the rain.

Why are we looking at volume? Because we can. Though our goal is to look at the amount of rain encountered, it depends on the volumes of these blocks. Since we only need to compare the cases for running and walking and not get actual numbers for rain quantity, we can just compare the total volumes described above for each case and get the same answer. It’s like dividing both sides of an equation or inequality by the same number, the rain density.

So all we need to do is calculate the total volumes for running and walking and compare them. Now we actually need some numbers to calculate with. How fast does rain fall? The Weather Guys at the university of Wisconsin have said that rain can fall as fast as 20 MPH (http://wxguys.ssec.wisc.edu/2013/09/10/how-fast-do-raindrops-fall/). Other estimates I saw were in the range of 15 to 20 MPH. I got the impression that 20 MPH is for unusually heavy rain, and 15 MPH might be more typical.

How fast might someone walk in the rain? The Verywell Fit web site says that a brisk walking pace is about 3 MPH (https://www.verywellfit.com/how-fast-is-brisk-walking-3436887). As for running, many years ago I recall running a quarter mile in less than 75 seconds. Multiplying that time by 4 gets 300 seconds per mile, which is 5 minutes per mile or 12 MPH.

The two missing inputs are the top and horizontal area for our pedestrian. I tried estimating these numbers for myself when doing the calculations in a spreadsheet. I’m not a big person, and I estimated my top area as 1.4 square feet, slightly tweaked for easier math. My front area estimate was tweaked to be a multiple of that, 8.4 sq. ft.

The results are telling:

Top Area (ft^2)	1.4
Front Area (ft^2)	8.4
Rain Speed (MPH)	15
Hor Dist in Rain (mi)	0.01
Hor Dist in Rain (ft)	52.8
	Walk	Run		Ratio (r/w)
Hor Speed (MPH)	3	12		0.25
Time in Rain (h)	0.00333	0.00083		4
Time in Rain (sec)	12	3		4
Vert Dist in Rain (mi)	0.05	0.0125		4
Vert Dist in Rain (ft)	264	66		4
Hor Vol	443.52	443.52		1
Vert Vol	369.6	92.4		4
Total Vol:	813.12	535.92		1.52

The far right column shows a ratio of the total rain space encountered for walking to running. So according to the model worked out above, a person may get around 52% wetter by walking. The implication is that our pedestrian should in fact run in the rain, if the situation matches the parameters used above.

The nice thing about a spreadsheet is that you can experiment with different inputs. Here are a few results from that.

• The ratio between the top and front areas is more important than the individual area values.
• Adjusting the top area to 1 and the front area to 20 still favors running, with the other numbers at their original values.
• I continued extreme inputs with top area 1, front area 20, rain speed 1, walk speed 3, and run speed 4, and it still favored running, but just barely (total vol. ratio of 1.004).
• Setting the top area to 0 (using all other original values) makes it a tie, so running might not matter if the pedestrian is carrying an umbrella.
• Setting the front area to 0 instead, causes a greater advantage for running.
• Setting the top and front area to 4 (with all other original values), again shows more advantage for running, with a ratio of about 2.36.
• Setting the rain speed alone to 1 (with all other original values), only causes a slight advantage for running, with a ratio of about 1.04. So maybe running or walking matters less when it’s just misting or drizzling or sprinkling out.

There are many ways to improve this model, like modifying it to account for any of the following:

• any change in the top or front area of the pedestrian due to (e.g.) his arms and legs swinging
• any change in speed or direction of any part of the pedestrian; basically any motion that is not constant speed in a straight line along the travel path
• how much the rain that hits the pedestrian actually contributes to wetness, as opposed to just bouncing off
• the rain impact speed and how it affects wetness
• how wet the pedestrian is at a given time
• the actual distribution of the rain in space and time
• the actual speed and direction of the rain, including due to wind
• what happens with the rain that runs down the pedestrian’s front or other semi-vertical surface (does it contribute to wetness or fall off?)
• how resistant the pedestrian and his clothes are to absorbing the rain that is encountered (a rain coat would certainly help)
• the pedestrian’s perspiration (especially for warmer weather and longer travel distances)
• situations when you would not want to run, like when doing so would doing so would be dangerous to you or any cargo you are carrying or when you need to wait for someone who is walking regardless
• many other things…

Yet in spite of the numerous inaccuracies, I think the model is good enough to make some decisions. I myself am now convinced that if I need to travel on foot through the rain without a rain coat or umbrella and not carrying anything, I would prefer to run, to get less wet. This model might even be useful in other situations like hail, (small) meteor showers, a sprinkler, etc. Obviously a model does not have to be perfectly accurate in order to help make decisions, or even intended for the situation at hand.

(c) Copyright 2018 by Mike Ferrell