How Far a Cough Travels in a Sick Room: Settling, Drifting, and What the Distance Rule Rests On

Droplets released at a height of 1.6 m settle at speeds that rise steeply with size. A droplet of 5 µm settles at 0.000778 m/s and takes 2057.24 s to reach the floor; a droplet of 50 µm settles at 0.075526 m/s and takes 21.18 s; a droplet of 500 µm settles at 7.530122 m/s and takes 0.21 s. Under the stated cough model, with an initial speed of 10.0 m/s decaying with a time constant of 0.12 s, almost every size travels about 1.2 m horizontally before the cough motion fades, and none reaches 2 m on cough momentum alone. A gentle room current changes the picture for small droplets: at 0.25 m/s, droplets of 5, 10, 20, 50, and 100 µm are still airborne at 2 m, because their settling times are long. If mid-sized droplets shrink to 0.3 of their initial diameter, a 50 µm droplet becomes 15.0 µm and its settling time rises from 21.18 s to 233.58 s. The common distance rules therefore rest, in this arithmetic, on large droplets falling quickly, and not on small droplets stopping at the same distance. No infection outcome is calculated.