Faster than a Speeding Pony: Why We Do Back-of-the-Envelope Calculations

jonah

This post will be rather short because I’m in the middle of finals. Of course, if you follow me regularly, you also know that I’ve posted rather a lot this week. When I’m exhausted from studying, it’s much easier to write in short bursts, which might explain all the short posts recently.

While reading this article, I recently found the following video, in which a fan of My Little Pony: Friendship Is Magic demonstrates the physics impossibilities in a few scenes of his favorite show. Let’s watch it, and then I’ll explain why I showed it to you.

(A side note: beetledude64 is wrong about dark matter. Dark matter is likely no heavier than regular matter…it just doesn’t interact with light.)

In each case, beetledude64 performed what physicists call a back-of-the-envelope calculation: He observed some physical process, such as a pony breaking the sound barrier, that he wanted to describe. To get a rough idea of what was going on, he then made some quick-and-dirty measurements, a lot of approximations, and a little bit of arithmetic. The result was a number that, although imprecise, was close enough to the truth to be useful.

But why is a calculation like this still useful at all? Isn’t it better to get the right answer? Well, not necessarily. If a scientist is contemplating whether or not to build an experiment, she may first perform a back-of-the-envelope calculation to see if her design is likely to work or if the results will be interesting. If the quick-and-dirty calculation looks good, she can go back and work through a careful derivation, which is more accurate but also much more time-consuming. If it looks bad, she can abandon her idea and try something else without having lost too much time, energy, or money. In this case, a back-of-the-envelope calculation would have informed her decision about whether or not to pursue some line of reasoning.

A back-of-the-envelope calculation can also help you build your own intuition about what all these crazy science numbers actually mean. For example, I once worked on an experiment where we tried to use mode-locked lasers to measure the optical properties of graphene. We were surprised to find that our graphene kept burning. What hadn’t occurred to us was just how intense each pulse was. These lasers produce very, very short pulses of light, so even though they don’t draw very much power—maybe forty watts maximum—they concentrate all of that energy into each split-second pulse. Since power is calculated as energy per time, this means that the power output for a single pulse (big number divided by tiny number) is very high…on the order of  one gigawatt, or 10^9 watts! A back-of-the-envelope calculation showed us that the power output in a single pulse (which only lasts for one 10^{13}th of a second) is more than the average output of a nuclear power plant. No wonder our graphene burned! This kind of calculation is often called an order of magnitude approximation.

Finally, we use back-of-the-envelope calculations to quickly investigate dubious claims. If I have my suspicions about something that someone says, I can pull out a piece of scratch paper and figure out whether or not it makes anything close to sense. I won’t worry about the details; I’ll round pi off to three and I’ll round the acceleration due to gravity off to g=10 m/s^2. In this arena, the back-of-the-envelope calculation arms me with a convenient bullshit detector against snake-oil salesmanship, punditry, and all other forms of bad science. This is the type of calculation that beetledude64 was doing. He was figuring out whether or not the physics in his favorite show made any sense, and although no one would expect a cartoon about talking ponies to be realistic, his experience was enriched by understanding precisely what was going wrong.

Some Resources

You don’t have to do back-of-the-envelope calculations alone. Here some tools that can help:

  • Google is, of course, your best friend. If I search “black market price of a human heart,” I’ll find an answer. This can really help when you’re trying to get an intuitive feel for what some numbers mean.
  • WolframAlpha can also be useful. It’s hard to describe exactly what WolframAlpha is…part search engine, part calculator, and part database lookup tool. But you can ask it questions, and it gives surprisingly detailed answers.
  • Randall Munroe (of xkcd fame) runs a blog called “what-if” that is nothing but back-of-the-envelope calculations that answer some pretty interesting questions. It’s always entertaining.
  • Speaking of xkcd, there’s this handy chart.
  • Some funnies.