A model for the magnitude of a resistive force is F = bv + cv², where v is speed in m/s. In SI units the numerical values of b and c are 1.1 and 0.20. (a) Derive the SI base units of each coefficient. (b) Calculate F at 5.5 m/s. (c) Explain why dimensional consistency does not establish that the model is experimentally correct, and why b + c is not a meaningful physical sum.

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b: kg s⁻¹; c: kg m⁻¹; F = 12.1 N; consistency is necessary, not proof of the model. Each term in an equation that adds forces must itself have units of force. Since [F] = kg m s⁻² and [v] = m s⁻¹, the coefficient b has units [F]/[v] = kg s⁻¹. Similarly [c] = [F]/[v²] = (kg m s⁻²)/(m² s⁻²) = kg m⁻¹. The two coefficients therefore have different dimensions even though their products with the appropriate powers of speed are both forces. Substitution gives F = 1.1 × 5.5 + 0.20 × (5.5)² = 6.05 + 6.05 = 12.1 N. This is a magnitude; its direction opposes motion. A dimensionally consistent expression can still have the wrong numerical coefficients or fail outside its fitted speed range, so force–speed measurements are needed. Adding b and c alone is physically undefined because quantities with different dimensions cannot be combined by addition.

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