Dimensional Consistency

Question

Considering v as velocity, v_0 as initial velocity, a as acceleration, t as time and x as distance, which of the following equations is dimensionally consistent?

RankGuru · Accepted Answer

Dimensional consistency demands that both sides of an equation, and every additive term within it, share identical dimensions. Test the candidate v^2 = v_0^2 + 2ax: the left side has dimensions [L T^-1]^2 = [L^2 T^-2]; the term v_0^2 matches with [L^2 T^-2]; and the term 2ax gives [L T^-2][L] = [L^2 T^-2], the pure number 2 being dimensionless. All terms agree, so this equation is consistent, and it is indeed the standard kinematic relation. The equation v = v_0 + a t^2 fails because a t^2 yields [L T^-2][T^2] = [L], which cannot be added to a velocity [L T^-1]. The equation x = v_0 t + a t fails since a t gives [L T^-1], not matching the length term v_0 t = [L]. The equation v = v_0 + a/t fails because a/t gives [L T^-3], inconsistent with velocity. This is exactly how NCERT uses dimensional analysis to screen equations. A final check confirms only the second equation balances to [L^2 T^-2] throughout, so it alone is dimensionally valid.

Dimensional Consistency

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