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2.1: Orders of magnitude

Although you should try to fight intuition when building a model to describe a particular phenomenon, you should not abandon critical thinking and should always ask if a prediction from your model makes sense. One of the most straightforward ways to estimate if a model makes sense is to ask whether it predicts the correct order of magnitude for a quantity. Usually, the order of magnitude for a quantity can be determined by making a very simple model, ideally one that you can work through in your head. When we say that a prediction gives the right “order of magnitude”, we usually mean that the prediction is within a factor of “a few” (up to a factor of 10) of the correct answer. For example, if a measurement gives a value of 2000, then we would consider that a model prediction of 8,000 gave the right order of magnitude (it differs from the correct answer by a factor of 4, whereas a prediction of 24,000 would not (it differs by a factor of 12).

Example $$\PageIndex{1}$$

How many ping pong balls can you fit into a school bus? Is it of order $$10,000$$, or $$100,000$$, or more?

Solution:

Our strategy is to estimate the volumes of a school bus and of a ping pong ball, and then calculate how many times the volume of the ping pong ball fits into the volume of the school bus.

We can model a school bus as a box, say $$20\:\text{m}×2\:\text{m}×2\:\text{m}$$, with a volume of $$80\:\text{m}^{3}∼100\:\text{m}^{3}$$. Figure $$\PageIndex{1}$$: A school bus and ping pong balls modeled as boxes.

We can model a ping pong ball as a sphere with a diameter of $$0.03$$ m $$(3\:\text{cm})$$. When stacking the ping pong balls, we can model them as little cubes with a side given by their diameter, so the volume of a ping pong ball, for stacking, is $$∼ 0.00003\:\text{m}^{3}=3×10^{−5}\:\text{m}^{3}$$. If we divide $$100\:\text{m}^{3}$$ by $$3 × 10^{−5}\:m^{3}$$, using scientific notation:

$\frac{100\:\text{m}^{3}}{3\times\:10^{-5}\text{m}^{3}}=\frac{1\times 10^{2}}{3\times 10^{-5}}=\frac{1}{3}\times 10^{7}\sim 3\times 10^{6}$

Thus, we expect to be able to fit about three million ping pong balls in a school bus.

Exercise $$\PageIndex{1}$$

Fill in the following table, giving the order of magnitude (in meters) of the sizes of different physical objects. Feel free to look these up on the internet!

Table $$\PageIndex{1}$$
Object Order of Magnitude
Proton
Nucleus of atom
Hydrogen atom
Virus
Human skin cell
Width of human hair
Human $$1$$ m
Height of Mt. Everset