There is a line that a science communicator is not supposed to cross. Today we’re going to inch our toes just past that line. Today I’m going to show you math equations.
Math scares people. Math can make people feel confused, frustrated, and (as a result) a little stupid. That’s okay. There isn’t a scientist alive that hasn’t struggled with math at some point. Today we are going to do a simple physics problem. Our assumptions will make sense all along the way, but in the end we’re going to get to a result that defies intuition. My hope is that by doing so we will learn an import lesson about intuition and common sense. Don’t worry if you aren’t comfortable with math because I’m going to take you through it every step of the way.
I would wager that at some point in our lives we have all run with a strong wind in our face and felt that it was harder to move forward. Similarly, we’ve all run with the aid of the wind at our backs. In the first case the wind in unfavorable to our running, while in the latter it is favorable. We all intuitively understand why this is the case: the wind pushes us either backwards or forwards. (By the way, that bit is not the intuition that we are going to prove wrong, so don’t worry.)
Let’s talk about people that competitively run in the wind. In the world of track and field, many events (such as the 100 meter dash, hurdles, and long jump) have rules about how much wind is allowed during a competition. Too much wind is deemed too beneficial or detrimental for the competitor’s score to count. Exempted from these rules are events where the runners go completely around the track. The logic is that the effects of the wind balance out; sometimes the runner is with the wind and other times they are against it. This assumes that the wind is blowing at a constant speed and always from the same direction. That is a reasonable assumption because nobody wants to scrutinize the wind speed and direction every second of a race, but how true is it that the effects of the wind cancel out if somebody runs a closed loop? To answer that we need math.
Velocity is defined as a distance traveled over a specific time, or written as a math equation we have
where V is the velocity, d is the distance traveled, and T is the time that it takes. (If you are with me so far then you are good to go the rest of the way.)
Now we’ll rewrite this equation and solve for d. We can do that by multiplying each side by the time (T) that we spend running at velocity V.
This says that a distance traveled equals the velocity traveled multiplied by the time traveled. (I’m using that standard math notion that when two letters – called variables – are next to each other that means they are to be multiplied together.)
Suppose that you ran a straight line from a starting point to an end point. You do this once with a favorable wind (blowing at your back) and once with an unfavorable wind (blowing in your face). When you start running you’ll start a timer and run for the same amount of time in each case (say five minutes, but that exact time doesn’t matter as long as it’s the same in both cases). If the wind blows with velocity W, and W is always less than V, then the favorable wind gives
While the unfavorable wind gives
Do you see that the distance traveled in the favorable wind is always bigger than the distance traveled in the unfavorable wind? If not then assume that the wind speed is half your running speed. In that case we have
But V T is the distance you can run without any wind (which we called d), so
Which means that in a favorable wind that blows half the speed that you can run that you will run 50% farther than if there was no wind. This makes sense. Similarly for an unfavorable wind
meaning you will run half as far in the same amount of time.
Now you might be thinking that this is a bit too simple, and it is. The drag force generated by the wind is more complicated than I’ve represented it here. For now let’s ignore this simplification and I’ll address it before we’re finished.
The track and field rules clearly state that if two runners race a track the same size, one with wind and one without, that the wind cancels itself out and the two times can be fairly compared. We can work out mathematically if this is true by solving our first equation for T. Our favorable (subscript F) and unfavorable (subscript U) running times are
Of course the runner has to run a closed loop, and for simplicity we’ll assume that the runner first runs a straight line into the wind a distance d then turns and runs back the same distance. The total running time is then
It should be obvious that if there is no wind (W=0) then the total time is just twice the time it takes to run one way, which we will call 2To. Also, no matter the wind, the distance is the round trip on the track, which is 2d. Let’s assume again that the wind is half the speed that the runner can run, so W=V/2.
So in a favorable wind of half the runner’s running speed it takes about 66% as long to run with the wind. What about running against the wind?
So it takes twice as long to travel the same distance compared to the case with no wind. But that means that the total round trip time is
Remember that the round trip in no wind is 2To so this says that if W=V/2 the round trip takes about 30% longer! Through some algebra it can be shown that
(If you don’t believe me then put W=V/2 and we get the result above. The superscript 2 means that we “square” W/V, multiplying it by itself, or (W/V)x(W/V).) Dividing by a number less than one is like multiplying by a number bigger than one, so if there is no wind (W=0) then we get the expected result, but if there is any wind at all then it takes longer to run the complete loop than if there was no wind. The effect of the wind does not cancel out!
If that’s the case then why is there a rule in track and field stating that the wind cancels itself out? Well, it’s because people don’t like math; they are scared by it and it makes them feel stupid. Intuitively the math feels wrong. The problem for us is that there aren’t any mistakes in the math, which makes the result confusing. The reason why we get this result is not obvious, but we can see it in the first version of the equation we used, when we took time to be constant and distance to be variable. That result was intuitive. In the second form of the equation we divide by velocity instead of multiplying by it, which gives us this non-intuitive – but mathematically and physically correct – result.
Earlier I mentioned that wind resistance is more complicated than I’ve represented it here. In reality there is a drag force created on the runner by the wind in the face, or a propulsion force from a wind on their back. We interpreted W as the wind velocity, but if we instead interpret it as an effect of the force of the wind we get generally the same result. Our last equation has a term W/V, which we could rename R and call it the “reduction in running speed due to wind.” R can be anything from zero for no wind to one for a wind so strong the runner can’t move. The result is the same: it always takes longer to run if there is any wind at all.
Another fair objection is that we’ve assumed running a simple straight line into and out of the wind – never a more complicated and real path. It is possible to generalize this result using calculus. I’ll spare you that but it turns out that this same result can be proven for running any closed loop in a constant wind. You can run a figure eight if you want, it doesn’t matter. It will always take longer if there is any wind at all. (I continue to refer to the wind as constant. If it wasn’t then at some times it would be favorable and others unfavorable and the net result could be favorable. That doesn’t contradict our main argument, which is that the track and field assumption is wrong. Running a closed loop with wind is not the same as running without wind.)
If a runner runs in a circle with a constant wind then intuitively the effects should cancel out because half the time the wind is favorable and the other half it is not favorable. That’s not the case though because when the wind is unfavorable the person runs slower, covering less distance in the same amount of time. It is this effect that is not canceled by the favorable wind. The favorable wind doesn’t speed the runner up enough to balance out the drag.
Intuition is based on our everyday experiences. Sometimes our experiences are not varied enough to make our intuition accurate. Scientists use problems like the one we’ve just discussed to develop new intuition. It’s a long and hard process, and that’s a big part of the college education of a scientist. It is particularly frustrating for scientists when non-scientists respond to scientific theories by arguments of “everyday common sense.” Common sense and intuition are linked; both are formed by our experiences, and part of the scientific profession is understanding things that can often be counter-intuitive.
During college I struggled with quantum mechanics. One day I went to my professor and told him that I just had no intuition for it. “That’s why we do these problems in class and in the homework,” he told me. “The goal is to develop an intuition born of these examples.” The science of how the Universe functions on the scale of the atom is not intuitive based on our everyday experiences. Heck, I just showed you that running in the wind doesn’t work the way we feel it should. If something as simple as running in the wind can violate our intuition, then what does that say for more complex topics like economics, climate change, or evolution?
Sometimes the common sense answer isn’t the correct one.
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