Dimensional Analysis: The Foolproof Way to Convert Any Unit
Never wonder whether to multiply or divide again. Use the unit factor canceling method for flawless conversions.
Why Converting Units Causes Mistakes
When converting 5 miles into feet, do you multiply by 5,280 or divide? In simple cases it seems straightforward, but when converting compound units like miles per hour to meters per second, intuition frequently fails. Dimensional analysis (the factor-label method) makes mistakes virtually impossible by treating units like algebraic variables.
Because 1 foot = 12 inches, the ratio (12 in / 1 ft) is equal to 1. Multiplying any quantity by 1 changes its units without altering its physical magnitude.
Step-by-Step Multi-Step Conversion
Convert 60 miles per hour (mph) into meters per second (m/s):
1. Set up the starting fraction: (60 miles) / (1 hour)
2. Convert miles to feet: × (5,280 ft / 1 mile) [miles cancel]
3. Convert feet to meters: × (0.3048 m / 1 ft) [feet cancel]
4. Convert hours to seconds: × (1 hour / 3,600 s) [hours cancel]
5. Multiply across: (60 × 5,280 × 0.3048) ÷ 3,600 = 96,560.64 ÷ 3,600 = 26.82 m/s.
Common Metric Prefixes to Memorize
• Kilo- (k): 10³ = 1,000 (kilometer, kilogram)
• Centi- (c): 10−² = 0.01 (centimeter)
• Milli- (m): 10−³ = 0.001 (millimeter, milligram)
• Micro- (μ): 10−&sup6; = 0.000001 (micrometer)