1.5:
Estimation of the Physical Quantities
Problems involving the estimation of physical quantities are often referred to as a Fermi problem, named after the physicist Enrico Fermi, who is known for making quick, rough estimates based on simple calculations and assumptions.
The determination of the number of blades of grass on a soccer field is an example of a Fermi problem.
The answer to any Fermi problem, called the Fermi estimate, is achieved by breaking the problem into smaller parts and using the available information to roughly estimate the unknowns.
Now, the number of blades of grass in a soccer field can be estimated by considering the field's dimensions and estimating that there are 20 grass blades in one square centimeter.
From the length and width, the area of the soccer field is calculated.
Next, converting the field's area into centimeters and multiplying the area by the number of blades of grass per unit area gives the Fermi estimate of the total number of blades of grass.
1.5:
Estimation of the Physical Quantities
On many occasions, physicists, other scientists, and engineers need to make estimates of a particular quantity. These are sometimes referred to as guesstimates, order-of-magnitude approximations, back-of-the-envelope calculations, or Fermi calculations. The physicist Enrico Fermi was famous for his ability to estimate various kinds of data with surprising precision. Estimating does not mean guessing a number or a formula at random. Instead, estimation means using prior experience and sound physical reasoning to give a rough idea of a quantity's value. As determining a reliable approximation usually involves the identification of correct physical principles and a good guess about the relevant variables, estimating is very useful in developing physical intuition. Estimates also allow us to perform "sanity checks" on calculations or policy proposals by helping to rule out certain scenarios or unrealistic numbers.
Many estimates are based on formulas in which the input quantities are known only to a limited level of precision. To make some progress in estimating, one needs to have some definite ideas about how the variables may be related. The following strategies could help practice the art of estimation:
- Obtain big lengths from smaller lengths – When estimating lengths, remember that anything can be a ruler. Imagine breaking a big thing into smaller things, estimating the length of one of the smaller things, and multiplying to obtain the length of the big thing.
- Obtain areas and volumes from lengths – When dealing with an area or a volume of a complex object, introduce a simple model of the object, such as a sphere or a box. Then, estimate the linear dimensions first, and use the estimates to obtain the volume or area from standard geometric formulas.
- Obtain masses from volumes and densities – When estimating the masses of objects, it can help first to estimate their volume and then to estimate their mass from a rough estimate of their average density.
- If all else fails, bound it – For completely unknown physical quantities, think what it must be bigger than and smaller than.
- One significant figure is okay – There is no need to go beyond one significant figure or one digit in the coefficient of an expression in scientific notation when doing calculations to obtain an estimate.
- Ask the following – Does this make any sense? Check to see whether the answer is reasonable. How does it compare with the values of other known quantities with the same dimensions?
Suggested Reading
- OpenStax. (2019). University Physics Vol. 1. [Web version]. Retrieved from https://openstax.org/books/university-physics-volume-1/pages/1-5-estimates-and-fermi-calculations, section 1.5; pages 25-27