Conversion of units of measurement

Before solving any problem it is essential to work in consistent units of the International System (SI). The most useful conversions in kinematics are few.

Key conversions

$1\;\text{km} = 1000\;\text{m}$ $1\;\text{h} = 3600\;\text{s}$ $1\;\text{km/h} = \frac{1}{3{,}6}\;\text{m/s}$

The systematic trick consists in multiplying by conversion factors, transforming one unit at a time. Each factor is a ratio equal to $1$ (for example $\frac{1000\;\text{m}}{1\;\text{km}}$), so it does not alter the physical value, only the units in which we express it.

Example — Conversion of compound units

Convert $10\;\dfrac{\text{m}^2}{\text{s}}$ to $\dfrac{\text{km}^2}{\text{h}}$. $10\;\frac{\text{m}^2}{\text{s}} = 10 \cdot \frac{\left(\frac{1}{1000}\;\text{km}\right)^2}{\frac{1}{3600}\;\text{h}} = 10 \cdot \frac{10^{-6}}{2{,}78 \cdot 10^{-4}} = 0{,}036\;\frac{\text{km}^2}{\text{h}}$ Note that the metre squared is squared in the conversion factor, while the second is converted only once.

Example — Conversion of force

Convert $0{,}02\;\dfrac{\text{kg}\cdot\text{m}}{\text{s}^2}$ to $\dfrac{\text{g}\cdot\text{km}}{\text{h}^2}$. $0{,}02 \cdot \frac{1000 \cdot \frac{1}{1000}}{\left(\frac{1}{3600}\right)^2} = 0{,}02 \cdot 3600^2 = 2{,}59 \cdot 10^{5}\;\frac{\text{g}\cdot\text{km}}{\text{h}^2}$

Units of measurement in calculations

Never neglect units of measurement! Every numerical result without units is devoid of physical meaning. Moreover, checking the units of the result is a powerful tool for verifying the correctness of calculations.

Links

Topics: Physical quantities and measurement Skills: Unit conversion · Dimensional analysis Methods: Conversion factor method

Related exercises: Worked exercise — The leaking tap · Worked exercise — Area of a measured rectangle · Problem — Volume of a rectangular box