  The ejecta velocities of type-Ia supernovae (SNe Ia), as measured by the Si
II $\lambda 6355$ line, have been shown to correlate with other supernova
properties, including color and standardized luminosity. We investigate these
results using the Foundation Supernova Survey, with a spectroscopic data
release presented here, and photometry analyzed with the SALT2 light-curve
fitter. We find that the Foundation data do not show significant evidence for
an offset in color between SNe Ia with high and normal photospheric velocities,
with $\Delta c = 0.005 \pm 0.014$. Our SALT2 analysis does show evidence for
redder high-velocity SN Ia in other samples, including objects from the
Carnegie Supernova Project, with a combined sample yielding $\Delta c = 0.017
\pm 0.007$. When split on velocity, the Foundation SN Ia also do not show a
significant difference in Hubble diagram residual, $\Delta HR = 0.015 \pm
0.049$ mag. Intriguingly, we find that SN Ia ejecta velocity information may be
gleaned from photometry, particularly in redder optical bands. For
high-redshift SN Ia, these rest-frame red wavelengths will be observed by the
Nancy Grace Roman Space Telescope. Our results also confirm previous work that
SN Ia host-galaxy stellar mass is strongly correlated with ejecta velocity:
high-velocity SN Ia are found nearly exclusively in high-stellar-mass hosts.
However, host-galaxy properties alone do not explain velocity-dependent
differences in supernova colors and luminosities across samples. Measuring and
understanding the connection between intrinsic explosion properties and
supernova environments, across cosmic time, will be important for precision
cosmology with SNe Ia.
