Theoretical Physicist Jacques Treiner, from the University of Paris Cite, explains why you should run in the rain: … Let p represent the number of drops per unit volume, and let a denote their vertical velocity. We’ll denote Sh as the horizontal surface area of the individual (e.g., the head and shoulders) and Sv as the vertical surface area (e.g., the body). When you’re standing still, the rain only falls on the horizontal surface, Sh. This is the amount of water you’ll receive on these areas. Even if the rain falls vertically, from the perspective of a walker moving at speed v, it appears to fall obliquely, with the angle of the drops’ trajectory depending on your speed. During a time period T, a raindrop travels a distance of aT. Therefore, all raindrops within a shorter distance will reach the surface: these are the drops inside a cylinder with a base of Sh and a height of aT, which gives:
p.Sh.a.T.
As we have seen, as we move forward, the drops appear to be animated by an oblique velocity that results from the composition of velocity a and velocity v. The number of drops reaching Sh remains unchanged, since velocity v is horizontal and therefore parallel to Sh. However, the number of drops reaching surface Sv — which was previously zero when the walker was stationary — has now increased. This is equal to the number of drops contained within a horizontal cylinder with a base area of Sv and a length of v.T. This length represents the horizontal distance the drops travel during this time interval. In total, the walker receives a number of drops given by the expression:
p.(Sh.a + Sv.v). T
Now we need to take into account the time interval during which the walker is exposed to the rain. If you’re covering a distance d at constant speed v, the time you spend walking is d/v. Plugging this into the equation, the total amount of water you encounter is:
p.(Sh.a + Sv.v). d/v = p.(Sh.a/v + Sv). d
This equation proves that the faster you move, the less water hits your head and shoulders, but the amount of water hitting the vertical part of your body remains constant. To stay drier, it’s best to move quickly and lean forward. However, you’ll have to increase your speed to offset the exposed surface area caused by leaning.
