How wet will you get by walking in the rain?
Will you stay drier by walking or running in the rain?
This Java applet is designed to calculate the amount of water absorbed by a person who is traveling in the rain. The person is modeled as a right cylinder with a specified height and radius for the purposes of calculations. The terminal downward velocity of the raindrops as a function of their volume is taken into account as well as the effects of wind speed and direction.
Below are 5 graphs generated using a MATLAB program similar to the above Java applet. Each graph plots the amount of water absorbed as a function of both the travel speed as well as one of 5 other factors (raindrop volume, rain density, distance traveled, wind speed parallel to the travel direction, and wind speed perpendicular to the travel direction).
The default values for user-set parameters are:
Raindrop Volume = 0.1 mL Rain Density = 75 Drops per second per square meter WindX = -4.0 m/s WindY = 0 m/s TravelY = 0 m/s Travel Distance = 50 m Person Height = 2 m Person Radius = 0.2 m Person Absorbancy = 100%
The effects of changing the values of some of the above parameters were studied; therefore, in the graphs below, the specified parameter is varied over a wide range of values as indicated. In each case, the travel speed along the X-direction is also varied between 1.0 m/s (~2.2 mph, a very leisurely walk) and 11.0 m/s (~24.6 mph, a little faster than the world record sprint speed).
Below is a brief summary of how changes in the 5 variables studied in the plots above affect the amount of water absorbed on the specified domain as well as a discussion on the optimal travel speeds for a few special cases.
Raindrop Volume
The amount of water absorbed is proportional to the raindrop volume. Raindrops of greater volume result in absorbing more water.
Rain Density
The amount of water absorbed is linearly proportional to the rain density. A greater number of raindrops per second per unit area results in absorbing more water.
Travel Distance
The amount of water absorbed is linearly proportional to the travel distance. Traveling a greater distance results in more water being absorbed.
Wind Speed Parallel to Travel Direction
The amount of water absorbed is linearly proportional to the relative wind speed (the difference between the travel speed and the speed of the wind parallel to the direction of travel). Higher magnitude wind speeds, relative to the travel speed, result in a greater amount of water absorbed.
Wind Speed Perpendicular to Travel Direction
The amount of water absorbed is proportional to the wind speed perpendicular to the travel direction. More water is absorbed at higher wind speeds perpendicular to the direction of travel.
Travel speed (and direction) is the only parameter which someone can really control in practicality, everything else is set by nature and the circumstances present at the time. One always has the ability to try to run faster, or slower, but one does not have control over how fast the wind is blowing or the volume of the raindrops. Optimizing the travel speed such that the amount of water absorbed is minimized is the question which one would like to be able to solve. However, this turns out to be a fairly non-trivial problem when one considers all of the many variables which come into play. By making a few simplifying assumptions, however, the optimal travel speed may be found for a few special sets of cases.
Cases Considered:
No wind
When there is no wind one should run as fast as possible through the rain in order to minimize the amount of water absorbed. Increasing one’s speed does mean one will absorb more rain on the side facing the direction of motion since now the rain is angled toward the traveler in their own reference frame. However, this increase in the rain absorbed on one’s front side is more than compensated by the decreased amount of time spent out in the rain, reducing the opportunity to absorb water on one’s head. Thus, traveling as fast as possible through the rain, with no wind, has the overall effect of minimizing the total amount of water absorbed. One will greatly benefit from running through the rain compared to leisurely walking, however as one’s speed increases the benefit from trying to travel faster decreases asymptotically to zero. Therefore running is advisable, but trying to run so fast as to set a world record may not be beneficial since the payoff is disproportionate to effort involved.
Only wind perpendicular to the travel direction
When the wind is only blowing perpendicular to the travel direction it is blowing rain onto one’s side, in addition to the rain absorbed on one’s front and head. As in the case of no wind, the amount of rain absorbed can be minimized by traveling as fast as possible. Running at a moderate speed is advisable since there are diminishing returns from trying to travel faster as one’s speed increases.
Only wind anti-parallel to direction of travel (Wind blowing directly against you)
When the wind is blowing directly against you (anti-parallel to the direction of travel) the way to minimize the total amount of water absorbed is, yet again, to travel as fast as possible (with diminishing returns at higher speeds), thus reducing the opportunity for wind to blow rain onto one’s front side.
Only wind parallel to direction of travel (Wind blowing in the direction of travel)
The situation when wind is blowing in the direction of travel is more complicated and interesting than those considered previously.
By traveling faster or slower than the wind speed one increases the amount of rain which one ‘walks into’ or has blows upon their backside. Thus by matching the wind speed one minimizes this factor. However, if the wind is blowing slowly and one matches the wind speed then one is spending a lot of time out in the rain and absorbing water on one’s head. Then there must be a wind speed at, or below, which one should travel as fast as possible, and above which one should always travel as fast as the wind is blowing.
This wind speed may be calculated as:
Where R is the radius of the person, h is their height, and v_{t} is the speed at which the raindrops are falling. This wind speed turns out to be about 1.5 m/s, or about 3.2 mph (about a normal walking speed) for the default parameter values. When the wind is blowing faster than this speed one should always travel at the same speed as the wind. When the wind is blowing slower than this speed then one should travel as fast as possible to minimize the water absorbed.
Within the above applet there is a button labeled “Optimal Tx”. The purpose of this button is to calculate the optimal travel speed in order to minimize the amount of water absorbed given the set of conditions specified in the applet window. When this button is clicked the applet reads the user-set parameters and passes them into Wolfram Alpha (a 3rd party website created by the makers of the scientific computing software, “Mathematica”) which is then used to calculate the optimal travel speed. Note: inside the Wolfram Alpha window which opens after clicking on the button, the travel speed is denoted by the variable x. Sometimes the value generated by Wolfram Alpha will be exceedingly large or even infinity. In this case one should infer that under the specified conditions, to minimize the amount of water absorbed, one should travel as fast as possible since, obviously, one cannot run at an infinite speed.
It should also be noted that, unlike the rest of the applet where both travel and wind directions are measured relative to a common reference point on the ground, Wolfram Alpha’s calculations measure the wind’s direction with respect to the travel direction (which is assumed to be solely along the +X-axis). Negative wind speeds in the x-direction indicate that the wind is blowing against the direction of travel, whereas positive velocities in the X-direction indicate the wind is blowing toward the direction of travel. Winds blowing in the Y-direction are blowing perpendicular to the direction of travel; the choice of positive and negative as left or right is purely arbitrary in this case.
The animation of the person walking in the rain in this applet is completely independent of the parameters which the user sets. The animation does not graphically display any relevant information; it is merely there to look good. It can be disabled by un-checking the box in the lower-left corner of the applet without any effect to the final results.
The amount of water absorbed by the person over their journey is shown directly below the animation panel. If the animation is enabled the amount of absorbed water which is displayed is progressively increased as the person travels across the animation panel until the final value is reached at the end of the trip. Without animation the final value is displayed immediately.
There is virtually no input protection provided on the user-set parameters, therefore it is possible to crash the applet and / or cause strange behavior. In general the user-input parameters should be real, finite, numeric entries. Also, the user-set parameters sometimes only make physical sense when they are positive values even though the applet may still run with invalid negative inputs (i.e. a negative raindrop volume is not physically valid). Just use common sense when providing input to the program.