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We present the first complete, velocity-resolved [C ii ] 158 μm image of the M51 grand-design spiral galaxy, observed with the upgraded German Receiver for Astronomy at Terahertz frequencies instrument on the Stratospheric Observatory for Infrared Astronomy. [C ii ] is an important tracer of various phases of the interstellar medium (ISM), including ionized gas, neutral atomic, and diffuse molecular regions. We combine the [C ii ] data with H i , CO, 24 μm dust continuum, FUV, and NIR K-band observations to study the evolution of the ISM across M51's spiral arms in both position–position and position–velocity space. Our data show strong velocity gradients in H i , 12CO, and [C ii ] at the locations of the stellar arms (traced by K-band data) with a clear offset in position–velocity space between upstream molecular gas (traced by 12CO) and downstream star formation (traced by [C ii ]). We compare the observed position–velocity maps across the spiral arms with synthetic observations from numerical simulations of galaxies with both dynamical and quasi-stationary steady spiral arms that predict both tangential and radial velocities at the location of the spiral arms. We find that our observations, based on the observed velocity gradients and associated offset between CO and [C ii ], are consistent with the presence of shocks in the spiral arms in the inner parts of M51 and in the arm connecting the companion galaxy, M51b, in the outer parts of M51. [ABSTRACT FROM AUTHOR]