The Driest Way to Get Soaked
You may have noticed while driving through a rainstorm that when you are stopped at an intersection your windshield stays quite dry, displaying only a few drops of precipitation. However, as you accelerate and get up to the college student speeds of triple digits, your windshield is covered with water and your wipers cannot hope to keep up (but then at such high speeds, seeing stuff would only scare you and make you lose control anyway, right?). Would this suggest that you stay dryer at lower speeds?
Well, let's jump into this problem feet first and see if we can construct a mathematical model which will aid in evaluation.
To make things easier, we will substitute a box for our soaked pedestrian. Let us consider a box which is 6 by 1/2 by 3/4. Now, since our units are arbitrary and I don't want to do any tough math, let's change this to 24 by 6 by 3.
All right, now assume that our box has to travel from point A to point B in the rain. Let's set the length of our journey 50 units. For the moment, we will assume that the rain is coming straight down, is uniform, and the storm is of infinite duration.
Now, if the rain were a type of heavy fog which was stationary - just hanging on the air - then our pedestrian (who we'll call Gene Kelly) would only be hit by the raindrops which we walked through.
Well, since our rain is uniform, then the amount of Gene's 'drenching' is proportional to the volume of the air he walks through. We'll call this the 'signature' that Mr. Kelly's body leaves as he walks though the rain. As the illustration tries to show, in the simple case of stagnant rain, this signature is equal to the volume of a box.
The dimensions of this box are; the distance from point A to point B, Mr. Kelly's height, and his width (since he is walking forward though the rain) which is in this case 24 by 6 by 50, or 7,200. We can already draw one conclusion: if Mr. Kelly walked through the storm sideways, his signature would be a mere 3,600.
Wow! Now let's make things really exciting and let the rain start moving (it's harder to hit a moving target anyway, right?).
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| Our pedestrian, Mr. Gene Kelly | The extent to which Mr. Kelly gets wet is proportional to the 'signature' his body leaves when he walks through the rain from point A to point B. |
First, we need to establish whether or not this notion of signatures can continue to serve us when we add the variable of moving rain. It will, but requires a minor adjustment. The signature only takes into account the raindrops that Mr. Kelly walks straight into. This amount remains unchanged no matter how fast Mr. Kelly walks (a fact shown at the bottom of this page)
For falling rain, we must add to the signature an amount which is proportional to the rain which falls on Mr. Kelly from directly above. We will calculate this figure by multiplying Mr. Kelly's horizontal cross-section by the number of minutes he is in the rain. This is reasonable because no matter where he is in our uniform rainstorm and regardless of his speed, the same amount of water will strike a certain area (that area being his horiz. cross-section).
So, in the case of our previous trip, let us say that it takes Mr. Kelly five minutes to travel the fifty units. His total 'dampness factor' is equal to his signature (7,200) times some constant, plus five (minutes) times his cross-section (six by three) multiplied by a second constant.
So, the obvious question is: What do we know about these mysterious constants? Well, the first constant which we multiply the signature by is merely the density of the rain expressed in gallons of water per unit of air. When multiplied by the signature (which is a value in cubic units), this yields a value in gallons.
The second constant, is merely the amount of water which strikes a unit square in one minute. In the case of vertical rain, this constant is equal to the density of the rain (expressed in gallons per cubic unit) times the velocity of the rain (expressed in units per minute). This constant, when multiplied by our second variable (square units times minutes), will yield a value in gallons of water.
What changes occur in our function when we consider rain which falls at an angle? Well, our signature is unchanged and is multiplied by the same constant. However, instead of adding one compensatory value, we must add two.
The first is similar to what we did before - minutes in storm times horiz. cross-section times the vertical component of the rain's velocity times the density of the rain. The second value will be the horizontal component of the rain's velocity times the density of the rain times a vertical cross-section.
Well, to simplify matters and to consider that people are not boxes but rather closer to elliptical cylinders, we will consider the horizontal cross-section to be half of the surface area of the vertical portion of our box - which in this case is 108 square units. So, out final formula to calculate the number of gallons of water with which you are pelted during a rainstorm is:
Where D is the density of the rain and t is the angle between the rain and the ground.
So, for a quick example, it takes Gene Kelly 2 minutes to walk 50 units in rain which is falling at a rate of 24,000 units per minute, at a 70-degree angle with the ground, and its density is equal to 1/720,000th of a gallon per cubic unit. Plugging into our equation we get:
Attached discussion over validity and constancy of 'signatures'.
Does our first definition of signature (the amount of water our pedestrian walks into) remain constant at any speed? Yes. To show this, let us not consider the rain as falling. Instead, let the rain remain stagnant and consider Mr. Kelly to be 'rising'. We can do this because we are not concerned with a raindrop once it hits the ground.Using this model, we find the signature to be only a function of the distance traveled. Our signature will take on one of three shapes as shown in the next illustration.
In case 1, Mr. Kelly is standing still. Since our storm has infinite duration, his signature is equal to the area of his base (three by six) times infinity. Practically, what does this mean? It means that if you stand in the rain forever you will drown - which we know is true...
According to Scripture, it will take less than forty days. So, standing in the rain is not a wise option (mom was right, Ben Franklin was not).
In case 2, Mr. Kelly has traversed his path at an infinite speed. This creates the same signature that we saw before, which is equal to 7,200.
Now, the case we are all interested in (unless you're Carl Lewis). In case 3, Mr. Kelly has walked to his destination in a normal manner. We now find the need for some new arbitrary unit to express the passage of time. Fortunately, such a unit was devised long ago, called the minute. This unit will suffice for our current project and requires minimal modification. Let's say it takes Mr. Kelly five minutes to walk his course. Let's express this by making the vertical distance from the base of starting point to the base of his finishing point equal to five. All of the measurements are shown in the next picture.
Calculating the value of this signature, which is a box with a parallelogram as its side, we get the value 7,200 which is the same signature value as when Mr. Kelly made the trip in zero minutes.
Further, since the area of a parallelogram is equal to its base multiplied by its height, the signature value for a man of fixed height and a trip of fixed length will be constant.
Feel free to send any comments you have on this discussion. I wrote it years ago, and I apologize for the confusing language and my poor literary skills. Of course, you may use any or part of this any way you like. Comments can be sent to aggies97@mailexcite.com.