Quantitative and Physical History
Basic Calculations Using Units
First published on February 22, 2020. Last updated on May 2, 2021.
Most historical calculations will involve units. There are a few tricks that will working with units easier and more useful.
Physicists have a secret called “units analysis”. By assigning all quantities a unit, they can then check that the final answer results in the required unit. If it does not, then there has been a mistake. Also, a unit is required for physical units to be meaningful. A weight of “100” does not mean much, whereas a weight of 1 billion points is a bit more impactful.
Here is an example of units analysis:
Jean walks 10 miles. It takes Jean 5 hours to walk that distance. To find Jean’s mean speed, we divide distance by time:
speed = distance/time
speed = 10 miles / 5 hours
speed = 2 miles per hour.
Miles per hour is indeed a unit of speed, so the answer could be correct. Conversely, if the answer came out to be hours/mile the answer would clearly be incorrect.
Converting and Canceling Units
Using units and calculating results will often involve converting one unit into another. An example will make this clear.
A farmer has an orchard with 10 trees. During the summer, each tree provides 2 bushels of apples per week, for 3 weeks. How many bushels of apples are produced by the orchard each summer?
bushels of apples/summer = (3 weeks/summer) x (2 bushels of apples/tree/week) x 10 trees
bushels of apples/summer = (3
weeks/summer) x (2 bushels of apples/tree/ week) x 10 trees
bushels of apples/summer = (3/summer) x (2 bushels of apples/
tree) x 10 trees
bushels of apples/summer = (3/summer) x (2 bushels of apples) x 10
bushels of apples/summer = (3 x 2 x 10) (bushels of apples/summer)
Answer: 60 bushels of apples/summer