Cell entry by SARS-CoV-2 is accomplished by the S2 subunit of the spike S protein on the virion surface by capture of the host cell membrane and fusion with the viral envelope. Capture and fusion require the prefusion S2 to transit to its potent fusogenic form, the fusion intermediate (FI). However, the FI structure is unknown, detailed computational models of the FI are unavailable, and the mechanisms and timing of membrane capture and fusion are not established. Here, we constructed a full-length model of the SARS-CoV-2 FI by extrapolating from known SARS-CoV-2 pre- and postfusion structures. In atomistic and coarse-grained molecular dynamics simulations the FI was remarkably flexible and executed giant bending and extensional fluctuations due to three hinges in the C-terminal base. The simulated configurations and their giant fluctuations are quantitatively consistent with SARS-CoV-2 FI configurations measured recently using cryo-electron tomography. Simulations suggested a host cell membrane capture time of ∼2 ms. Isolated fusion peptide simulations identified an N-terminal helix that directed and maintained binding to the membrane but grossly underestimated the binding time, showing that the fusion peptide environment is radically altered when attached to its host fusion protein. The large configurational fluctuations of the FI generated a substantial exploration volume that aided capture of the target membrane, and may set the waiting time for fluctuation-triggered refolding of the FI that draws the viral envelope and host cell membrane together for fusion. These results describe the FI as machinery that uses massive configurational fluctuations for efficient membrane capture and suggest novel potential drug targets.