For the purposes of these calculations it was assumed that the person traveling through the rain (“the traveler”) was the shape of a right cylinder of radius R and height h, with their base oriented parallel to the ground. The advantage of choosing a cylinder to model the traveler is that it is symmetrical in the X-Y plane and allows for easier calculation of how much rain hits the traveler from the side due to the wind. One could model the traveler as a rectangular box, or possibly several boxes of different sizes and orientations (corresponding to various body parts) so as to make a higher-order approximation, but this only complicates calculations further down the line when attempting to find optimal travel speeds.
The parameters considered in these calculations are:
All parameters are assumed to have fixed values as the traveler moves over a finite, straight-line, distance to their destination. All speeds are measured with respect to the ground and distances are measured from the traveler’s initial position. Take note of the symbols / abbreviations used to denote each parameter as they will be used in the equations presented below.
The amount of water absorbed by the traveler is calculated as:
Equation 1:
Where α is the percent of all raindrops hitting the travel which are absorbed (this is a user-set parameter), Time is the length of time it takes the traveler to complete their journey, and Flux is the total number of drops hitting the person from all sides per unit time.
The time spent out in the rain is merely the distance traveled divided by the traveler’s speed,
Flux is calculated as the product of the traveler’s area multiplied by the rate at which raindrops are attempting pass through that area. Since the traveler is modeled as a cylinder there are only two areas which need to be considered: the top and cross-sectional (‘side’) areas.
Top Area = πR^{2}
Side Area = 2 * h * R
The two components of the total Flux may be found as,
Where θ is the angle below the horizontal which the rain’s velocity is directed,
RainHSpeed is defined as the horizontal speed of the raindrops relative to the traveler,
The terminal downward velocity of the rain, v_{t}, may be found as,
It is assumed that the rain has been falling for such as long time that it has reached its terminal velocity in the downward direction and that the wind has been blowing it long enough such that the horizontal velocity of the rain is equal to the wind velocity.
Combining all of the terms which have been defined thus far back into Equation 1 we obtain what we will henceforth call the "Rain Equation" for a right-cylinder in the case of 2-dimentional motion and wind in the X-Y plane,
Above: The "Rain Equation" describing the amount of WaterAbsorbed (in mL) by the traveler over their journey.
The three constants, C_{1}, C_{2}, and C_{3}, seen in the Rain Equation are used in determining the downward terminal velocity of the raindrops as a function of their volume. The values of these constants were found experimentally (Reference 1) to be:
One can use the above Rain Equation to solve for the amount of rain absorbed given the specified parameters. Additionally, this equation may be used to find the optimal speed at which to travel so as to minimize the water absorbed by setting the first derivatives of the Rain Equation, with respect to the velocity terms, equal to zero. The derivative of the Rain Equation with respect to T_{x} is shown below (the derivative with respect to T_{y} is analogous).
Trying to solve for the speed at which the above equation is equal to zero, in the most general sense (i.e. not plugging in values for any of the parameters) is exceedingly tedious and should not be attempted by hand (though a computer may be used if one desires in order to generate the several, very long and complicated, solutions). Instead of solving for the optimal speed in the most general sense one can make a few assumptions which dramatically simplify the process.
Assuming there is no wind (W_{x} = W_{y} = 0), the derivative of the Rain Equation simplifies to,
Where the coordinate system can be redefined (which is valid since there is no wind) such that the motion is only along one axis, thus T_{x} = T, T_{y} = 0. One can see that the only time this equation is equal to zero is when T goes to infinity, confirming the assertion that one should always travel as fast as possible through the rain in order to minimize the amount of water absorbed.
Now consider the case when the wind is oriented perfectly perpendicular to the travel direction. We can define our coordinate system such that the motion is solely along the X-axis while the wind blows along the Y-axis (Ty = 0, Wx = 0). The derivative of the Rain Equation is thus,
The above equation has no Real zeros over the domain of interest (i.e. over feasible travel and wind speeds), as illustrated in the two graphs below.
Above: 3D plots illustrating the Rain Equation (Water Absorbed), Left, and the derivative of the Rain Equation, Right, as a function of both Travel Speed (T_{x}) and Wind Speed (W_{y}). Notice the monotonic decreasing trend of water absorbed as the travel speed increases for all wind speeds considered. Likewise, the derivative of the amount of water absorbed with respect to travel speed asymptotically approaches zero with increasing travel speed. The plateaus in the plots near the T_{x} = 0 axis are due to the fact that the amount of water absorbed goes to infinity as the traveler is nearly standing still. The graphing software cannot graph infinite (or other very very large numbers), so it shows a plateau in these locations.
Finally, consider the case where the wind is directed perfectly parallel to the direction of motion. Again, we can redefine the orientation of our coordinate system to make the equation simpler by assuming T_{x} = T, T_{y} = 0, W_{x} = W, and W_{y} = 0. Doing this, the equation simplifies to,
The above equation is equal to zero when,
When the wind speed blowing in the direction of motion, W, is less than this critical value (W*), or if it is blowing against the travel direction, then one should always travel as fast as possible to minimize the absorbed rain since the derivative of the Rain Equation is negative (indicating that the amount of water absorbed decreases with an increased speed). When W > W* the derivative of the Rain Equation becomes positive and one actually absorbs more water by increasing speed, therefore one should match the wind speed as closely as possible in order to minimize the amount of water absorbed. When W = W* it does not matter which of these options you choose, the amount of rain absorbed will be constant after surpassing the wind speed.
Above: Plots of the amount of water absorbed by the traveler as a function of travel speed. Three cases are considered: when W < W* (left), when W = W* (center), and when W > W* (right). One can see how the behavior of the Rain Equation is decidedly different for each of these cases as one’s travel speed exceeds the wind speed parallel to the direction of travel.
In order to solve for the optimal travel speed of more complicated cases Wolfram Alpha is used to find the minimum of the Rain Equation using the parameters specified by the user in the applet. The applet reads the parameters and generates a query string, similar to a Mathematica command, which is passed into Wolfram Alpha. Wolfram Alpha finds the minimum of the function, corresponding to the optimal travel speed (denoted as x in this case) and displays it as well as the amount of rain absorbed. Note that this feature, unlike the rest of the applet, assumes that all motion is along the X-axis and measures the wind direction with respect to this travel direction.
[1]: Atte Salmi, Lasse Elomaa, Vaisala Oyj,
"Measurement of the Terminal Velocity and Shape of Falling Raindrops at Vaisala Rain Laboratory".
Herb Bailey, "On Running in the Rain," The College Mathematics Journal, Vol. 33, 88-92, 2002.
Alessandro De Angelis, "Is it really worth running in the rain?," Eur. J. Phys. 8 201-202, 1987.
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