Physicist Jacques Treiner from the University of Paris built a model to calculate the speed at which the least amount of drops will hit you. He considered a simple scenario: it rains evenly and vertically.
The human body can be divided into two surfaces – vertical and horizontal. When walking or running, vertical surfaces will be hit more than if we were standing. It then looks like it’s raining sideways on you.
The faster you go, the more drops will hit you every second, but the time spent in the rain will also decrease. So the vertical surfaces are balanced by spending less time in the rain. How are the horizontal surfaces of the body doing?
If a person is moving, they are hit from above by drops that would otherwise fall in front of them. But he also avoids the drops that now fall all the way behind him. This creates a certain balance and the amount of rain on horizontal surfaces will be the same. But thanks to the shorter time in the rain, the amount of drops on a horizontal surface will eventually be smaller.
Mathematically, it is justified as follows:
Let it r represents the number of drops per unit volume and let a denotes their vertical speed. We will mark Sh as the horizontal surface of the individual (eg head and shoulders) and Sv as a vertical surface (eg body).
When standing still, rain only falls on a horizontal surface, Sh. This is the amount of water you will get in these areas.
Even if the rain falls vertically, from the perspective of a pedestrian moving at speed v it appears to fall at an angle, with the angle of the drops trajectory depending on your speed.
Over a period of time T a raindrop travels a distance aT. Therefore, all raindrops reach the surface in a shorter distance: these are the drops inside the base cylinder Sh and height aTwhich gives:
r.Sh.aT
As we have seen, as we move forward, the drops appear to have an oblique velocity resulting from the composition of the velocity a and speed v. Number of drops reaching Sh remains unchanged because the speed v is horizontal and therefore parallel to Sh. Number of drops hitting the surface Sv – which was previously zero when the pedestrian was standing – has now increased.
This is equal to the number of drops contained in a horizontal cylinder with a base area Sv and length vT This length represents the horizontal distance the droplets travel during this time interval.
In total, the walker receives the number of drops given by:
ρ.(Sh.a + Sv.v). T
Now we have to consider the time interval during which the pedestrian is exposed to the rain. If you cover a distance d at a constant speed v, the time spent walking is d/v. When you plug that into the equation, the total amount of water you’ll encounter is:
ρ.(Sh.a + Sv.v). d/v = ρ.(Sh.a/v + Sv). d
Translated into human speech
The water hitting the vertical part of your body remains the same regardless of speed because the shorter time spent in the rain is compensated by encountering more raindrops per second. But the faster you move, the less water falls on our head and shoulders, i.e. horizontal surfaces.
So your front body will get wet just the same, but your head and shoulders should be better off. However, the horizontal surfaces are smaller than the vertical ones, so the difference will not be that big. As a result, it is good to lean forward a little when walking or running fast, which will reduce the vertical surfaces. However, it will want to increase the speed even more. There is no ideal speed – just the faster you go, the less you will get wet.
source: The Conversation
Source: www.cnews.cz
