  We often find spectral signatures of chromospheric cold plasma ejections
accompanied by flares in a wide range of spatial scales in the solar and
stellar atmospheres. However, the relationship between physical quantities
(such as mass, kinetic energy, and velocity) of cold ejecta and flare energy
has not been investigated in a unified manner for the entire range of flare
energies to date. This study analyzed the spectra of cold plasma ejections
associated with small-scale flares and solar flares (energy
$10^{25}-10^{29}\,\mathrm{erg}$) to supply smaller energy samples. We performed
H$\alpha$ imaging spectroscopy observation by the Solar Dynamics Doppler Imager
on the Solar Magnetic Activity Research Telescope (SMART/SDDI). We determined
the physical quantities of the ejecta by cloud model fitting to the H$\alpha$
spectrum. We determined flare energy by differential emission measure analysis
using Atmospheric Imaging Assembly onboard Solar Dynamics Observatory (SDO/AIA)
for small-scale flares and by estimating the bolometric energy for large-scale
flares. As a result, we found that the ejection mass $M$ and the total flare
energy $E_{\mathrm{tot}}$ follow a relation of $M\propto
E_{\mathrm{tot}}^{2/3}$. We show that the scaling law derived from a simple
physical model explains the solar and stellar observations with a coronal
magnetic field strength as a free parameter. We also found that the kinetic
energy and velocity of the ejecta correlate with the flare energy. These
results suggest a common mechanism driven by magnetic fields to cause cold
plasma ejections with flares on the Sun and stars.
