Dimensional Homogeneity

Question

In a physics tutorial a relation v = a + b t + c t^2 is given for the speed of a moving trolley at time t. What are the dimensions of the constant c in this equation?

RankGuru · Accepted Answer

The principle of dimensional homogeneity, presented in NCERT Class 11, Chapter 2 (Units and Measurements), requires every additive term in a valid physical equation to share the same dimensions. Here the left side v is a speed with dimensions [L^{1}T^{-1}], so each term on the right must also have dimensions of speed. For the term c t^2 to equal a speed, c must satisfy [c][T^{2}] = [L^{1}T^{-1}]. Solving gives [c] = [L^{1}T^{-1}] / [T^{2}] = [M^{0}L^{1}T^{-3}]. The option [M^{0}L^{1}T^{-1}] is wrong because it equals the dimensions of b t, not c. The option [M^{0}L^{1}T^{-2}] is wrong as it matches acceleration, which would suit b not c. The option [M^{0}L^{0}T^{-1}] is wrong since it omits the length dimension entirely. A final check confirms that multiplying [L^{1}T^{-3}] by [T^{2}] indeed restores the speed dimensions, validating the result.

Dimensional Homogeneity

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