Approximations and Fermi Computations

Approximations and Fermi Computations Engineers, physicists, and other scientists frequently have to estimate a certain quantity. Other words that are occasionally used include estimates, Fermi calculations, back-ofthe-envelope calculations, and order-of-magnitude approximations. (The physicist Enrico Fermi, who was previously mentioned, was well-known for his ability to estimate different types of data with unexpected accuracy.) Should we rent a truck or can that piece of equipment fit in the back of the car? What's the duration of this download? When this circuit is turned on, how much current will flow through it? If a projected power plant is built, how many homes might it truly power? Keep in mind that estimating does not entail making arbitrary guesses about numbers or formulas. Instead, estimation refers to the process of estimating a quantity's value using a combination of good physical reasoning and past experience. Estimating is particularly helpful in developing physical intuition because the process of finding a valid approximation usually requires the discovery of accurate physical principles and a good guess about the relevant variables. Because estimates let us rule out particular scenarios or implausible numbers, they also enable us to do "sanity checks" on computations or policy suggestions. They provide us the opportunity to push ourselves and others to discover the truths about the world. Numerous approximations rely on formulas when the input quantities are known with only restricted accuracy. As you hone your problem-solving abilities in physics—which translate to many other domains—you'll also improve your estimation abilities. By being more risk-taking and quantitative in your thinking, you can enhance these abilities. Like any skill, experience is beneficial. It also helps to be familiar with the scales of basic amounts (see Figure 1.4), dimensions (see Table 1.3), and units (see Tables 1.1 and 1.2). You must have some firm notions about potential relationships between variables in order to proceed with estimation. You can practice the skill of estimating by using the following strategies: • Convert smaller lengths into larger lengths. Keep in mind that everything can serve as a ruler when evaluating lengths. To calculate the length of the large object, divide the larger object into smaller ones, estimate the length of one of the smaller objects, and then multiply the result. For instance, count the number of floors of a building before estimating its height. Next, calculate the approximate area of a single level by calculating the number of individuals who would need to stand on top of one another in order to reach the ceiling. Lastly, gauge a person's height. Your estimate of the building's height is the result of these three calculations multiplied together. It is beneficial to have a few length scales related to the kinds of issues you solve committed to memory. For instance, it could be useful to be familiar with some of the length scales shown in Figure 1.4. Occasionally, estimating the length of a little thing by picturing a few of them combined to form a larger thing can also be helpful. To estimate a sheet of paper's thickness, for instance, calculate the thickness of a stack of paper and divide the result by the total number of pages in the stack. Masses and times can occasionally be estimated using the same techniques that divide large objects into smaller ones or combine smaller objects into a larger one. • From lengths, get areas and volumes. Use a basic representation of the thing, like a box or sphere, when working with an area or volume of a complex object. Next, utilize your estimations to determine the linear dimensions (e.g., the sphere's radius or the box's length, width, and height) before using the conventional geometric formulas to get the area or volume. In the event that you know an estimate of the area or volume of an item, you can also estimate its linear dimensions by applying conventional geometric formulas. • Determine masses using densities and volumes. When calculating an object's mass, it can be helpful to estimate the object's volume first. Then, using an approximate estimate of the object's average density—remember, mass = density times volume—you can calculate the object's mass. To aid with this, keep in mind that the densest common solids have a maximum density of approximately 104 kg/m3, water has a density of 103 kg/m3, and air has a density of about 1 kg/m3. You can approximate an object's density by asking yourself if it floats or sinks in water or air. The opposite is also possible; if you know how much an object weighs and how dense it is, you can use that information to estimate how much volume it contains. • Bind it if everything else fails. Sometimes your only option when it comes to physical amounts that you are not very intuitive about is to think along the lines of: "Well, it must be bigger than this and smaller than that." Let's take an example where you need to calculate a moose's mass. Perhaps you have a great deal of moose experience and are aware of their approximate mass. Excellent if that's the case. However, the majority of individuals can only think along these lines: It needs to weigh between 102 kg and 103 kg, which is more than a human and less than an automobile. The geometric mean of the upper and lower bound can be obtained by multiplying them together and taking the square root if you only require one number for a later computation. In the case of the mass moose, this would be The better, the more tightly bound. In addition, there are no infallible rules when it comes to estimation. Increase your estimate from the geometric mean by one or two orders of magnitude if you believe the quantity's value will likely be closer to the upper bound than the lower bound. • All right, one "sig. fig." When calculating to get an estimate, there's no need to go beyond one significant figure. For the most part, the order of magnitude is sufficient. Just getting a ballpark amount is the aim, so simplify the arithmetic as much as you can. • Consider whether anything makes sense at all. Finally, determine whether your response makes sense. What is the difference between its value and other quantities of the same dimensions that you can readily look up or that you already know? First, make sure your units are correct if you receive an absurd response (such as estimating that the mass of the Atlantic Ocean is more than that of Earth or that a certain time period is older than the cosmos). Next, look for any arithmetic mistakes. Next, reconsider the reasoning behind your response. If all goes according to plan, you might have just shown that some flashy new concept is in fact fake.