We parametrize the transverse momentum distribution of outgoing hadrons in ultrarelativistic nucleus-nucleus collisions as a superposition of boosted thermal distributions. In this approach, which generalizes the conventional blast wave, the momentum distribution is determined by the distribution of the fluid velocity. We analyze the difference between this generalized blast-wave parametrization and a full hydrodynamic calculation. We then apply the generalized blast-wave fit to experimental data on Pb+Pb collisions at $sqrt{s_{rm NN}}=2.76 mathrm{TeV}$. The fit is reasonable up to $p_tsim 6 mathrm{GeV}$, much beyond the range where hydrodynamics is usually applied, but not perfect. Based on the differences between the fit and the data, we argue that an ideal hydrodynamic calculation cannot fit simultaneously all identified particle spectra, irrespective of the specific implementation. In particular, data display a significant excess of pions at low $p_t$, whose physical interpretation is discussed. Data also show that the distribution of the fluid velocity becomes broader as the collision becomes less central. This broadening is explained by event-by-event hydrodynamic calculations, where it results from the centrality dependence of initial-state fluctuations.
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