Evolution of the mean forward shock velocity, 〈β_sh〉 = r_sh/ct, for one-dimensional impulsive and continuously powered blast waves with the same isotropic-equivalent energy, E_iso = 4 × 10⁵³ erg. The impulsive model (BM) is initialized using the Blandford–McKee self-similar solution, while the powered models assume constant energy injection for t < t_b, followed by the canonical fallback decay L_j ∝ t^−5/3. The circumnuclear medium corresponds to a 10⁷ M_⊙ black hole (Figure 1). Increasing the duration of energy injection systematically reduces the shock velocity because, at any given time, only a fraction of the eventual explosion energy has been transferred to the blast wave. Once the bulk of the energy has been injected, the powered solutions gradually converge toward the impulsive evolution.