Record units and dimensions separately
Metres and centimetres differ as units while both describe length L. Velocity has dimensions L/T. For each symbol, record its meaning, unit and dimension before substituting numbers.
Dimensional analysis detects inconsistency but does not replace a physical derivation. OpenStax explains that both sides of an equation and additive terms require compatible dimensions.
Worked example: velocity plus acceleration?
In v = u + at, u has dimension L/T and at has (L/T²)T = L/T. The expression is consistent. In v = u + a, the additive terms have different dimensions.
With u = 2 m/s, a = 3 m/s² and t = 4 s, v is 14 m/s. If time is 4 minutes, convert it to 240 seconds before using the same units.
A consistent formula can still miss a coefficient
Both mv² and 0.5mv² have dimensions ML²/T². Dimensional consistency cannot detect the missing factor of one half; the model and derivation still matter.
For a mass of 2 kg moving at 3 m/s, classical kinetic energy is 9 J. Check the physical formula separately from the arithmetic and units.
Self-check: a flow-rate conversion
Convert 2 L/min using 1 L = 0.001 m³: the rate is 0.002/60 m³/s, approximately 3.33×10⁻⁵ m³/s. Over 30 minutes the volume is 60 L, or 0.06 m³.
Divide volume by elapsed time to check the rate backwards. A result of 60 m³ points to a conversion error. This learning exercise is only a preliminary check, not a sufficient basis for practical engineering design.
References
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