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Physics LibreTexts

2.1: Orders of magnitude

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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 2.1.1

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 20m×2m×2m, with a volume of 80m3100m3.

clipboard_efbb5caa765b24decf447f77d77a4aa86.png
Figure 2.1.1: 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 (3cm). 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 0.00003m3=3×105m3. If we divide 100m3 by 3×105m3, using scientific notation:

100m33×105m3=1×1023×105=13×1073×106

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

Exercise 2.1.1

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 2.1.1
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  
Radius of the Earth  
Radius of the Sun  
Radius of the Milky Way  
Answer

This page titled 2.1: Orders of magnitude is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Ryan D. Martin, Emma Neary, Joshua Rinaldo, and Olivia Woodman via source content that was edited to the style and standards of the LibreTexts platform.

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