Block copolymer molecular weight is a crucial factor influencing micelle fragmentation kinetics. In particular, it is not established how block length affects the fragmentation rate, and which block is more important. In this work, we studied the separate dependence of micelle fragmentation kinetics on core and corona block lengths, with temperature-jump experiments by dynamic light scattering. Two series of 1,2-polybutadiene-b-poly(ethylene oxide) (PB-b-PEO) copolymers were prepared, both with fixed N-PB (degree of polymerization of the core block) and various N-PEO. In all, a total of nine narrow dispersity (D < 1.1) polymers were used, with PB-b-PEO block molar masses (in kDa) of (9-6), (9-9), (9-12), (9-13), (9-19), (15-10), (15-14), (15-17) and (15-22). Micelles were initially prepared using direct dissolution of PB-b-PEO in the ionic liquid 1-ethyl-3-methylimidazolium bis(trifluoromethyl sulfonyl)imide [C2MIM][TFSI], followed by fragmentation during thermal annealing at 170 degrees C. The relaxation time (tau) for micelle fragmentation exhibits a power-law correlation with both N-PB and N-PEO as tau similar to N-core(1.4) x N-corona(0.6). Micelles were also characterized before and after fragmentation, by small-angle X-ray scattering and by liquid-phase transmission electron microscopy. Both analyses confirmed that all micelles were spherical, and the mean aggregation numbers before and after fragmentation could be extracted. The stronger tau dependence on N-core suggests a higher core elastic free energy penalty due to core chain extension in the transition state, compared to the contribution of corona crowding. These results are apparently not captured by any available theory of micelle fragmentation.