Do you walk or run in the rain? A physics-based approach to staying dry (or at least drier)

We’ve all had the experience of going outside without an umbrella when the sky opens up. Whether it’s light rain or heavy rain, your instincts tell you that running will keep you from getting wet to a minimum. But is it really true? Let’s take a scientific look at this common dilemma.

It starts to rain while you are out and of course you forget your umbrella. Instinctively you lean forward and your pace quickens. We all tend to believe that if we travel faster, we’ll get wet less quickly, even if we end up getting hit with more rain as we move forward.

But is this intuition really correct? Is it possible to build a simple model to see if increasing the speed really reduces wetting? More specifically, does the amount of water that hits depend on the speed? And is there an ideal speed that minimizes the total amount of water encountered on the way from point A to point B?

Let’s simplify the scenario and break it down. Imagine rain falling evenly and vertically. The body can be divided into two planes. vertical planes (front and back) and horizontal planes (head and shoulders).

When moving forward in the rain, as speed increases, raindrops are more likely to hit a vertical surface, such as a person’s body. From the pedestrian’s perspective, the water droplets appear to fall diagonally at the same horizontal speed as the pedestrian’s walking speed.

The faster you walk, the more water droplets you encounter per second, but the less time you spend in the rain. As a result, the two effects balance each other out, increasing the amount of precipitation per unit time, but decreasing the amount of time it rains overall.

If a pedestrian is stationary, rain falls only on horizontal surfaces, that is, on the top of the head and shoulders. When a pedestrian starts moving, he or she catches raindrops that should fall in front of him and misses raindrops that should fall behind him. This creates a balance and ultimately the amount of rain you receive on the horizontal plane remains the same no matter how fast you walk.

However, walking faster reduces the total amount of time you spend in the rain, so less water collects on horizontal surfaces.

Overall, it’s a good idea to pick up your pace when walking in the rain.

For those who like a mathematical approach, here’s the breakdown:

Let ρ be the number of droplets per unit volume and a be the vertical velocity. Express Sh as the horizontal surface area of ​​the individual (e.g., head and shoulders) and Sv as the vertical surface area (e.g., torso).

If you stand still, rain will only fall on horizontal surfaces, Sh. This is the amount of water these areas receive.

Even if rain falls vertically, it appears to be falling diagonally from the perspective of a pedestrian moving at speed v, and the angle of the droplet’s trajectory changes depending on the speed.

During period T, the raindrop moves a distance aT. Therefore, all raindrops within a shorter distance will reach the surface. These are raindrops inside a cylinder with base Sh and height aT, which looks like this:

ρ・Sha・A・T

As we have seen, as it moves forward, the droplet appears to be animated by the diagonal velocity resulting from the composition of velocity a and velocity v. Since the velocity v is horizontal and parallel to Sh, the number of droplets reaching Sh does not change. Sh. However, the number of droplets reaching the surface Sv was previously zero when the pedestrian was stationary, but now it has increased. This is equal to the number of droplets contained within a horizontal cylinder of base area Sv and length vT. This length represents the horizontal distance that the droplet travels during this time interval.

In total, Walker receives the number of drops given by the formula:

ρ(Sh・a + Sv・v)T

Next, you need to consider the amount of time pedestrians will be exposed to the rain. If you travel a distance d at a constant speed v, the time you spend walking is d/v. Substituting this into the equation, the total amount of water encountered is:

ρ(Sh・a + Sv2)・d/v = ρ(Sh・a/v + Sv)d

This equation yields two important insights.

The faster you move, the less water hits your head and shoulders. The water hitting the vertical part of the body remains the same regardless of speed. This is because the less time you spend in the rain is offset by the more raindrops you encounter per second.

In summary, when you’re caught in the rain, it’s a good idea to lean forward and move quickly. However, be careful as leaning forward will increase Sh. To stay really dry you need to increase your speed enough to compensate for this.

This article is republished from The Conversation under a Creative Commons license. Read the original article.