Sneezing, Ballistics, and the Spread of COVID-19 (or: Wear the Damn Mask)
This paper by Louis C. Just is a creative blend of mathematics, the insights of nautical munitions, and human insights that guides us to good choices on the questions of personal safety amid a pandemic. For those looking for an accessible introduction to the mathematics behind projectiles, asking questions about what value face masks have in public health, or just wanting a better understanding of how the world moves around us, Lou has provided an engaging article that lets us know, why the mask works.
Sneezing, Ballistics and the Spread of COVID-19
or: Wear the Damn Mask
By: Louis C. Just
December 3, 2020
On December 2, 2020, the CDC reported 240,000 total COVID-19 deaths while predicting 430,000 total deaths by early March, 2021. At the same time, the success of vaccine trials was announced to a weary world. Since people are tired of using masks and being quarantined, carelessness is lurking. This paper discusses how COVID-19 travels into the environment if masks are not used, in the hope of reinforcing what we should have learned.
A strong campaign for the wide use of masks could have been made in early 2020 when evidence became available that the virus was airborne. And the following discussion of the COVID-19 virus’ behavior in air could also have been presented in that time frame, giving support to a mask mandate. Instead of supporting wide-spread mask usage, we fought over whether they were needed at all. Much has been learned since those early days -- the hard way. The world will still need masks well into 2021 and COVID-19 is indifferent as to
whether masks are damned or blessed.
Observation, elementary algebra and geometry, and empirical evidence from ballistics serve as primary tools for this report. No experiments were done to support the conclusions and no references, except for Wikipedia, are provided.
At the close of 2019, a new pathogen, COVID-19, emerged that was thought to spread only through contact. In late February of 2020, we were surprised when COVID-19 cases were reported in California, Washington, and Oregon in people who had no known contact with a proven vector. This called into question our ideas concerning its mode of transmission. We have since discovered that COVID-19 is an airborne virus which requires us to study the behavior of small particles in air and as they are propelled through the air by a force, such as a sneeze, air currents, or gravity. It was once thought that COVID-19 could only infect someone who was less than six feet from a source, but the February cases called that estimate of its range into question
Objects that are introduced into air fall into three categories: lighter than air, same density as air and heavier than air. The COVID-19 virus is very small and dense (heavier than air) which grants it very confusing behavior in air. The reasons for this are developed in this report. If you drop a snow flake and a snow ball from the same height, the snow ball will hit the ground first and the flake will seem to float. Similarly, a cube of wood will fall faster than a particle of sawdust generated from that cube. And if you take two feathers of the same weight and
compress one into a very tight ball, the tightly compressed feather ball will fall faster than its twin. There is a reason: air resistance along their paths.
You don’t need to do the the calculations that lead to the results in the last row of the following table, but the last row must be understood. Let’s calculate the Surface-Area-To -Volume Ratio (S/V) for two regular geometric shapes, a cube (side length = L ) and a sphere (diameter = D) along with a bullet made up of a half-sphere and a cylinder having diameter D and length D. To simplify the analysis, we assume that the cube always moves with one side facing the direction of its movement and that the bullet moves with the half-sphere leading; the sphere needs no orientation specification because of its symmetry.
As cubes (L), spheres (D,) and bullets get smaller, their S/V ratio gets larger and air will offer more resistance (drag). Anything moving through air is retarded and small particles act as if they are attached to a parachute which becomes relatively larger as their volume decreases. So very small particles that are not being affected by air currents (as with a wind) will settle slowly or travel long distances. In still air, they are only affected by gravity and the air resistance caused by their large S/V, so they seem to float. Similarly, as L or D gets larger, the S/V ratio gets smaller. The S/V is the number of units of surface (area) compared to the number of units of volume. Larger volumes contain more mass and, thus, more weight -- an important point for the following discussion.
COVID-19 is a virus that is heavier than air with a Specific-Gravity (SG) more than 50 times that of air. The COVID-19 virus travels downward in still air because of that SG and slowly downward because of its large S/V which also allows it to be more easily carried by air currents. How far? Surely more than six feet. The N95 mask can obstruct 95% of COVID-19 viruses and anything larger than that.
When you fire a rifle, pistol, or naval gun (or when you hit a ball), you are imparting kinetic- energy (KE), which depends on velocity and mass (KE = (m*v^2)/2), to a projectile. From this point onward, weight will be used as a proxy for mass because we are operating under the influence of the Earth’s gravity. Without air resistance, that projectile will travel until its gravity- controlled path causes it to hit the earth. But a projectile traveling through air will lose a portion of its KE at every point in its path and ultimately crash into the earth, after traveling a shorter distance than it would have travelled in a vacuum..
The following chart shows that homogeneous projectiles starting with similar velocities travel farther if they are large; the larger S/V ratio of smaller projectiles reduces range. It would be wrong to claim that the following table adds anything to the science of ballistics, because it does not; the data in the table only show that small projectiles must be more affected by air- resistance. NOTE: shows -- not proves. Ballistics, however, wants to know and requires complex calculations to predict trajectories and landing points. The ENIAC computer (first
programmable, electronic, general-purpose digital computer) was initially used to solve ballistics problems because of their importance. Since governments are charged with defense, ballistics is a supported study, and names like Aristotle, Archimedes, Bernoulli, Euler, Galileo, and Newton show up in the historical record of this discipline.
Naval Guns and the Browning Machine Gun (BMG) *
* All data in these tables were derived from Wikipedia searches
The larger shells travel farther because their smaller S/V ratios exact a smaller “tax” (drag) on their larger KE as they travel. Note that the drag is produced by the area that the projectile presents perpendicular to its path. The larger shells have a larger total drag on their progress but have less cross-sectional area per unit of volume (or unit of weight). When designing objects that must move through air (like aircraft, cars, bullets ...), we try to minimize drag by “streamlining”. We have neglected this factor in this paper so that the surface- to-volume-ratio-perpendicular-to-the-path which we call S/V is overstated for all but a flat surface oriented perpendicular to its path. Keep these thoughts in mind as you read the section, below, entitled “Anatomy of a Sneeze”. The following chart shows that the ratio of a projectile’s cross-section area to its weight affects its range when muzzle-velocities are similar.
Projectile Weight and Cross-section area vs. Range
Anatomy of a Sneeze
A sneeze will be treated as the launch of multiple projectiles in the following discussion. Have you ever sneezed while driving? Sneezes vary in power and in the amount of expectorant. That expectorant contains particles of various sizes that all enter the atmosphere with very similar speeds (we call it S): aerosols, droplets, drops and globs which are small projectiles and behave accordingly. Occasionally, a sneeze has sufficient power that a glob outdistances its smaller siblings and makes it to the windshield -- further evidence that the S/V slows smaller particles more than large particles. The larger expectorants have more mass and KE and travel farther. The aerosols, droplets, drops, and globs can contain the virus which may be released if they rupture placing the virus farther from its source. Thus, there is no precise measure of the range of the virus. If you search the internet for “how far can a sneeze travel”, you will find numbers as large as 27 feet. National Geographic has published some excellent photos of sneeze expulsions and large drops outdistance smaller drops -- by a lot. These large drops can release aerosols and the virus when they rupture.
Masks filter in both directions and protect both the population and the wearer. The N95 mask can allow 5% of the viruses that eschew, under pressure, from a wearer to enter the atmosphere while obstructing close to 100% of anything larger. Air intake into the mask is has a much lower velocity than that which is produced by a sneeze thus allowing the mask to be more effective in that direction. In early December of 2020, we have vaccines which will be slowly distributed along with a prediction of190,000 more deaths by March of 2021. Continued use of masks is required to reduce that number.
We are now well into 2020 and six feet is no longer considered to be the limit of COVID-19’s range. Since it “floats” in aerosol form, somewhat like a snowflake or dust, air currents could concentrate the viruses from a single person or crowd into an area of high concentration which could more easily infect. Can this be the Super-Spreader effect? And if an individual can be a Super-Spreader, a crowd could become a Super-Super-Spreader-Event.
Lessons Learned - aka WISDOM
What does this say about the use of masks? USE THEM. Keep viruses from infected people out of the environment, which is the same advice that could have prevented other environmental problems. One of the first conferences on the environment was held in Albuquerque, New Mexico in the mid-1970’s. The keynote speaker was Dr. Harold Agnew, then director of the Los Alamos National Laboratory, who advised those present with this one-liner: “This environmental stuff is simple -- just don’t inject bodily fluids into the soup“.
Does that quote need further illumination with respect to COVID-19?
If so, here it is:
Wear the damn mask!
Clair E. Turner, Public Health Applications of High-Speed Photography, American Journal of Public Health, April 1941; 31(4): 319 to 324, Figure 1, American Public Health Association
Look at Those Drops
The large drops DO travel farther and the aerosols SEEM to float