[gwnn]

I have always wondered about what's the optimal strategy if it rains, run or walk pretty slow. After a few bad attempts I finally cracked it with a simple model that would suggest optimal would be to run as fast as possible (in terms of total droplets hitting you). However, both Brainiac and MythBusters conducted experiments and it seemed to them that walking slowly is a good idea. Can you look at my reasoning?

Assume:

*the human body is a rectangular prism
*rain droplets are falling vertically (I think it's pretty easy to adjust to non vertical rain)

Then there are two types of raindrops that can hit you:

I. the ones which hit the upper side of the prism. It's very easy to see that the amount of water hitting this side per time unit is independent of our velocity. Indeed, if you enter our frame or reference, the droplets' vertical speed does not change and neither does their concentration. The number of droplets that hits us is:

N1=n*d*u*S/v

n=concentration of droplets
d=total distance we need to walk in rain
S=our top side's surface.
u, v=the velocity of the rain and us respectively

II. the ones which hit the front side. It is also very easy to see that this is constant.

N2=n*d*S'

where S' is the front surface.

To summarize, the total number of droplets should be:

N=n*d*(S*u/v+S')=C1+C2/v

So N always decreases as we increase v. The faster we run the better.

I am pretty sure nothing really changes if we let go of the two assumptions I mentioned above. For example if the velocity of droplets has a sideways component all that changes is that our side is also hit but also at a constant rate and if it has a component parallel to the road we can just change the frame of reference to get rid of that component and the conclusion wouldn't change.


So why doesn't theory and practice match?

[PassedOut]

gwnn, on Sep 14 2008, 02:38 PM, said:

However, both Brainiac and MythBusters conducted experiments and it seemed to them that walking slowly is a good idea.

I'm with you on this, and don't see what we are missing. (I'd like to know what it is, if we are.)

And how slow is slow? To take it to an extreme, would taking one step per hour during a steady downpour keep one drier? I don't think so.

[matmat]

if you move quickly, more rain hits you per unit time than if you move slowly. Is this, perhaps, what they found?

if you have to get somewhere though (i.e. finite distance) , I think you are probably right that you want to get there as fast as possible. I'll think about it some more.

[dicklont]

When you run you hit more drops in front of you.
In other words: the rain gets thicker when you go faster.

Best thing to do is wear a rainsuit and maintain normal speed.
That's what brings me home, but I'm cycling.
You really need waterproof shoes or a good raincover for them since all the water (front and above) drops down from the rainsuit.

[gwnn]

No the distance was constant, the speed was not, the runner got more wet than the walker.

[matmat]

gwnn, on Sep 14 2008, 03:31 PM, said:

No the distance was constant, the speed was not, the runner got more wet than the walker.

splashed more water up from the ground. duh.

[david_c]

gwnn, on Sep 14 2008, 08:38 PM, said:

However, both Brainiac and MythBusters conducted experiments and it seemed to them that walking slowly is a good idea.

On the MythBusters experiment, Wikipedia says the result was overturned in a subsequent episode:

When retrying the test in actual rain it was conclusively proven that the running test subject got less wet than the walking test subject. The use of artificial rain in the original test led to a false negative.

[DrTodd13]

The Mythbusters did a revisit of that myth where they changed their mind and said that running was better. Also, I think this also depends on the angle at which one runs. People don't run with their bodies totally vertical. They run leaning forward a little bit and I think this angle serves to decrease the rain that you run into. I calculated this at the time the episode came out and the optimal angle for running speed greater than humans could achieve but still the closer you get to it the better.

[Elianna]

I would think that if you had an umbrella, though, you were better off walking.

[naresh301]

gwnn, on Sep 14 2008, 01:31 PM, said:

No the distance was constant, the speed was not, the runner got more wet than the walker.

I do not understand this. In the limiting case of the speed going to zero, the walker must get infinitely wet. Unless the function (of wetness vs. speed) is non-monotonic, the experimental results seem shaky.

Edit: sorry, somehow missed the entire latter half of the thread.

[brianshark]

You should move with a velocity such that the rain is falling directly down on you (which will depend on the directection the rain is falling in the first place). This is because the rain has the least surface area to hit you when it's falling straight down on your head/shoulders.

[helene_t]

Elianna, on Sep 15 2008, 03:56 AM, said:

I would think that if you had an umbrella, though, you were better off walking.

Elianna nailed it. The brainiac experiment measured water absorbed by the suit. Water falling on the head might to some extent either fall off, evaporate, or slide down under the suit and get absorbed by the underwear.

Todd's point is good, too. However, I would still run. Those experiments are funny but not quite up to scientific standards.

A friend of mine has a popular science book "Rennen door het regen" ("Running through the rain"). It proposes the same model as Csaba.

[pclayton]

Elianna, on Sep 14 2008, 06:56 PM, said:

I would think that if you had an umbrella, though, you were better off walking.

Exactly LOL.

[cherdano]

pclayton, on Sep 15 2008, 10:57 AM, said:

Elianna, on Sep 14 2008, 06:56 PM, said:

I would think that if you had an umbrella, though, you were better off walking.

Exactly LOL.

I would think that if you did not have an umbrella, you would be better of buying one.

[TimG]

gwnn, on Sep 14 2008, 02:38 PM, said:

I am pretty sure nothing really changes if we let go of the two assumptions I mentioned above. For example if the velocity of droplets has a sideways component all that changes is that our side is also hit but also at a constant rate and if it has a component parallel to the road we can just change the frame of reference to get rid of that component and the conclusion wouldn't change.

I'm not sure this is true. Suppose that we stick with your assumption that the raindrops are falling vertically, but change your assumption about the body being a rectangular prism. I think I can describe it best by considering a two dimensional coordinate system.

Suppose that our "person" is represented by a line from (-1,0) to (0,2), that the rain is falling at 2 units per second and that the "person" moves to the right at one unit per second. As the person moves from (-1,0)(0,2) to (0,0)(1,2) and raindrops that started to the left of (or above) the "person" will not be able to catch up with the "person" as it moves out of the way, and the "person" won't be able to catch up with any of the raindrops that started to the right of (or below) the "person".

Converting this to 3-D, the only raindrops that hit this person will be those that hit the upper side of the prism; there will be no raindrops that hit the front (or back) side. But, change the rate at which the person moves or the angle of the lean will mean that there will be some raindrops hitting the front (or back) side.

Isn't this the same as leaving the person vertical and adding a horizontal component to the falling raindrops? So that if the horizontal component of the rate of fall of the raindrops were identical to the rate of movement of our vertical person, there would be no front (or back) side raindrop hits?

I have no idea at what rate raindrops fall, how fast I can run, or at what "lean" I would run at. But, I do know that when I am trying to stay relatively dry, I not only go quicker than a walk, but I also duck or lean my upper body forward. The lean/speed combination ought to have an affect on the real world experiments.

[DrTodd13]

Like I said, leaning is important. At the time the original mythbusters episode aired, I found out how fast raindrops fall and how fast people run and if I remember correctly the angle of lean would have to be around 45% to avoid all rain hitting your front or back. The most lean I think you can effectively get is 20% so practically you can't avoid it all but you can reduce it. That was my big gripe with the original mythbusters episode. When they tested running, they ran in an unnaturally upright position.

[han]

Nice thread!