Base Dimensions and Consistency

Mechanical quantities reduce to combinations of mass (M), length (L), and time (T). Velocity is L per T, acceleration is L per T squared, and force is M times L per T squared. A valid physical equation must have the same dimensions on both sides. This dimensional consistency check is a fast filter: if the left and right sides do not match, the equation is wrong, though matching dimensions alone do not prove an equation is fully correct.

Velocity has dimensions L T to the power minus one
Force has dimensions M L T to the power minus two
Both sides of any equation must share identical dimensions

Significant Figures

Significant figures express how precisely a quantity is known. They include all certain digits plus the first uncertain one. In multiplication and division, the result carries as many significant figures as the least precise input. In addition and subtraction, the result is limited by the least number of decimal places. Rounding should happen only at the final step so intermediate rounding does not accumulate and distort the answer.

Error Propagation Rules

When measured quantities combine, their uncertainties combine in predictable ways. For sums and differences, absolute errors add. For products and quotients, relative or percentage errors add. When a quantity is raised to a power, its relative error is multiplied by that power. Applying these rules tells you which measurement contributes most uncertainty, so you know where greater precision in the experiment would improve the final result the most.

Sum or difference: add absolute errors
Product or quotient: add relative errors
Power: multiply relative error by the exponent