Dimensional Analysis, Scaling and the Order-of-Magnitude Estimate

The mechanical examples here use length, mass and time dimensions. Other SI quantities require additional base dimensions. Terms added in an equation must have matching dimensions and compatible meanings. Convert units when needed. A length plus a duration fails this dimensional check.

Speed has dimensions length divided by time. Kinetic energy is one half times mass times speed squared. Square the speed dimensions, then multiply by mass: energy has dimensions mass times length squared divided by time squared. The dimensionless factor one half does not change them.

Both sides of a physical equation must have matching dimensions. This is necessary, not sufficient: a wrong numerical coefficient may pass. Dimensions can suggest possible relationships but do not generally determine a unique formula, a near-one constant or a function of dimensionless groups.

For speed v, time t and distance L, vt means v × t. The group vt/L is dimensionless: (length/time) × time ÷ length = 1. If a model fixes vt/L at a constant, t is proportional to L/v: at the same constant, multiplying L/v by a factor multiplies t by that factor. At fixed L, doubling v halves t. The scaling prediction depends on that model assumption.

A Fermi estimate uses a model and approximate inputs. To estimate houses supplied by a power plant, divide available average power by average demand per house. State assumptions about demand and available output. Other estimates may use sums or products. Check units and compare with plausible scales.

A conversion factor is a ratio of equivalent quantities. To convert metres to centimetres, multiply by 100 cm / 1 m, so m cancels. Reversing that factor leaves an unwanted unit expression. Wrong units can also come from a missing factor or wrong formula; they do not uniquely prove a reversed conversion.

Compare two positive estimates by their ratio. A ratio above ten flags a large difference worth investigating, but does not prove either estimate invalid. Both may lie within a factor of ten of the true value. Review the model, inputs and intended precision before judging.

Here, nearest power of ten means nearest on a multiplicative scale. For an integer n, 10^n is a power of ten. √10 is the positive number whose square is 10. Between 10^n and 10^(n+1), the boundary is √10 × 10^n, about 3.16 times the lower power. Below it, choose the lower power; above it, choose the upper. For 480, the boundary between 100 and 1000 is about 316, so choose 1000.