AcademyMeasurement and Vectors
Academy
Unit Conversion Without Guesswork
Level 1 - Physics topic page in Measurement and Vectors.
Principle
A conversion factor multiplies by one while changing the unit label.
Notation
\(q\)
physical quantity
varies
\(u_1\)
starting unit
varies
\(u_2\)
target unit
varies
\(c\)
conversion factor equal to one
none
Method
The numerator is chosen to keep the target unit; the denominator is chosen to cancel the unwanted unit.
Start
\[q=N u_1\]
Insert one
\[q=N u_1\left(\frac{u_2}{u_1}\right)\]
The fraction equals one because numerator and denominator represent the same physical size.
Cancel
\[u_1\left(\frac{u_2}{u_1}\right)=u_2\]
Powers
\[(c u)^n=c^n u^n\]
Rules
Factor identity
\[1=\frac{\text{same quantity in }u_2}{\text{same quantity in }u_1}\]
Convert quantity
\[q_{u_2}=q_{u_1}c\]
Power units
\[(cu)^n=c^n u^n\]
Examples
Question
Convert
\[18\,\mathrm{m\,s^{-1}}\]
to \[\mathrm{km\,h^{-1}}\]
Answer
\[18\frac{\mathrm{m}}{\mathrm{s}}\left(\frac{1\,\mathrm{km}}{1000\,\mathrm{m}}\right)\left(\frac{3600\,\mathrm{s}}{1\,\mathrm{h}}\right)=64.8\,\mathrm{km\,h^{-1}}\]
Checks
- Every unwanted unit cancels.
- The physical quantity does not change.
- Square and cubic units need powered factors.
- Larger target units usually give smaller numbers.