Squiggle reasoning: the skydiving animals problem
There is a common conception that physics is a business of writing and solving exact equations. This idea is not untrue, in the sense that physicists generally prefer to produce exact solutions when they can. But precise equations can be slow: they are often cumbersome to work with and can obscure important concepts with a tedium of error-checking and term-collecting. For these reasons, physicists often figure things out (at least in the initial stages of problem solving) using a kind of semiquantitative reasoning that doesn’t make use of exact equalities.
In this kind of reasoning, all (or most) equations are downgraded from having an equals sign, 
This may seem kind of dumb to you. Why reason with squiggles when you can write exact equations instead? But the truth is that “squiggle reasoning” often allows you to figure things out much more quickly and easily than you would ever be able to if you insisted on writing only exact equations. And as long as you are willing to live with some ignorance about exact numerical values, you sacrifice very little in terms of conceptual clarity.
As it happens, I designed and taught a short course last year for high school students that introduces basic ideas in quantum mechanics using squiggle reasoning. (I am teaching the course again this year.) As an introduction, I gave the students the following problem:
If a bunch of animals of different sizes all jump out of an airplane together, how fast do they each fall?
In this post I’ll take you through the answer to this problem, which can perhaps serve as a gentle introduction to quantitative reasoning in situations where you don’t know how to (or don’t want to) write down exact equations.
Gravitational Force
The starting point in solving this problem is to forget that animals have particular shapes. That is, simplify the geometry of a given animal down to a single number: its “size” 
To connect to an old trope, this kind of thinking isn’t really “assuming a cow is a sphere” so much as it is “not caring about the difference between a cow and a sphere”.
Now you can ask: what is the force of gravity acting on an animal of size 
Since we have decided to forget about all specifics of the animal’s shape, making an estimate for the animal’s volume is actually very easy:
In fact, in squiggle reasoning, every three-dimensional shape has volume 
Drag force
Immediately after jumping out of the airplane, the 
So how big is the drag force 
Of course, the exact answer to this question depends on the shape of the animal. If you really wanted to know, with numeric accuracy, the value of the drag force, then you would need to understand the air flow pattern around the animal. This would presumably require you to stick the animal in a wind tunnel and make careful measurements. (And you would get different answers depending on which way the animal was facing).
But at the level of squiggle reasoning, we can figure out the drag force using a simple thought exercise. Imagine the process of throwing a big block of air at the animal:
This block is taken to have the same cross-sectional size as the animal (area 
In order to stop the block of air, the animal applies a force that does work on the block equal to 
Of course, when the animal is falling through the air, this drag force is applied continuously, as the animal finds itself continuously colliding with “blocks of air” that move toward it with speed
Final Answer: never skydive in the rain
Now we are ready to get an answer: equating 
Thus we arrive very quickly at an important semi-quantitative conclusion: larger animals fall faster, with a terminal velocity that grows as the square root of the animal’s size.
In fact, you can use this equation to get a pretty good order-of-magnitude estimate for the terminal velocity 
In particular, the squiggle equation for 
Thus, you can see pretty quickly why falling off a building is deadly for you (hitting the ground at ~ 100 mph is worse than just about any car accident) but not deadly for insects (hitting the ground at a couple mph is no big deal).
In fact, there is a pretty practical implication of this result (besides “don’t fall off a building”): You should never go skydiving in the rain. You might think (as I initially did) that it would be a sort of magical and pleasant experience, wherein you fall together with the raindrops like an astronaut playing with zero-gravity water droplets. But the truth is much more unpleasant: the meter-sized you will be falling at ~1o0 mph, while the millimeter-sized raindrops fall at a slow ~3 mph. So, from your perspective, you’ll be getting stabbed by raindrops that blast you in the face at ~97 mph.
Highly unpleasant, and just a small amount of squiggle reasoning before you jump can save you the trouble.
Gravity and Levity 
The rain thing never occurred to me. That would be painful!
Excellent! (I must say, parenthetically, that the ‘delta T’ between your delightful posts did feel like free-falling, bored, and wishing for something inspiring to read during the long-leap toward oblivion, something that, though I won’t survive, at least restoreth my faith in well-grounded and clear-thinking younger-and-luckier-than-I physicists.)
I was waiting to read: ‘to a first approximation’, or ‘first-order’ solution’. as it relates to squiggles..
Here in Israel we have soft-ball-sized porcupines abundant, who, when frightened, (understandably, upon being ejected from business-class in a doomed 747 at 30 thousand feet by sudden decompression, ) retract even their life-saving spines into a resultant ‘sphere’. And so, counting on you guys, I’ll prepare myself to land in a pile of ’em. Oops, no, they’ll ‘rain’ on me as I belatedly write my Will. (With due historical respect for the Leaning Tower of Pizza ‘lessons’)
Seriously, bravo on your post, and for your contribution to the the next generation of thinkers. May they have the tools to vote for politicians with at least a grade-school respect for science. Amen
Ha! Thanks. Actually, this post was written largely in order to set up the next one. So you should have another post to read in the very near future.
Nice post, but I was hoping you’d go a little deeper. I have always been puzzled by how molecules of different molecular weight and electrostatic properties are air born and others grounded. Why chlorofluorocarbons rise up to the ozone layer while alkali metals that are light are grounded? Perhaps the same principles of the macro can be applied to micro world, but I suspect there are other factors to be considered. I look forward to reading about it.
I’m sure you’re right that there are some electrostatic/chemical properties at work here. I’m also sure that I don’t know enough to explain them!
I have read you enough to disagree with you that you do not know enough. I am sure you would have great insights if you are interested and do a little research. It is easy to calculate the effect of earth’s electric field (~150 V/m) on an electron, but to calculate this for a noble gas, a molecule, or even a micro organism is another story. The implication of how much this electrostatic component contributes to the earth’s gravity or levity is beyond my skill level, and I have gone as far as to question the number of significant digits used for some fundamental constants, and I have done classical simulations. I hope I gave you an idea for a post.
Brian, please read this article. I would like to know if the ‘squiggle reasoning’ applies to free falling atoms as well. http://physicsworld.com/cws/article/news/2016/dec/13/quantum-free-fall-at-8500-m
I have been considering the design of an experiment to test this idea that individual atoms, and small molecules, do fall with different accelerations, and evidently even negative acceleration (unless I am missing something in my equations or reasoning, and that is why i need someone like you to collaborate). I am open to collaborate with someone, and share credit, to test this idea. I hope you are interested. Please email me. Thank you
I forgat to add my name. The previous anonymous post is from the same Noel as this post and that posted above.
I’ve never jumped with any animals (let alone in the rain, which isn’t allowed, anyway), but the 100 MPH figure is accurate for humans. As you say, depends on orientation. Falling head-down, giving the smallest cross-section possible, allows speeds of up to 200 MPH. Falling as spread-out as possible (“grabbing the air”) can slow one down to the 80-90 MPH range, depending on the person’s size and clothing, but the rule of thumb is that free-fall terminal velocity is roughly 120 MPH.
Which is a blast. Not often a human can go that fast all by their self! 🙂
Great stuff. Is this course available somewhere?
No, it isn’t; sorry. Maybe someday I’ll put my notes together into a nice form and publish them online.
Of course you can always go through the archives here. Every post is a treat, pashut kef, ad c’dei smarmoret.