Analyses of the widths and shifts of optically thin emission lines in the ultraviolet spectrum of the active dwarf Epsilon Eri (K2 V) are presented. The spectra were obtained using the Space Telescope Imaging Spectrograph on the Hubble Space Telescope and the Far Ultraviolet Spectroscopic Explorer. The line widths are used to find the non-thermal energy density and its variation with temperature from the chromosphere to the upper transition region. The energy fluxes that could be carried by Alfven and acoustic waves are investigated, to test their possible roles in coronal heating. Acoustic waves do not appear to be a viable means of coronal heating. There is, in principle, ample flux in Alfven waves, but detailed calculations of wave propagation are required before definite conclusions can be drawn about their viability. The high sensitivity and spectral resolution of the above instruments have allowed two-component Gaussian fits to be made to the profiles of the stronger transition region lines. The broad and narrow components which result share some similarities with those observed in the Sun, but in Epsilon Eri the broad component is redshifted relative to the narrow component and contributes more to the total line flux. The possible origins of the two components and the energy fluxes implied are discussed. On balance our results support the conclusion of Wood, Linsky & Ayres, that the narrow component is related to Alfven waves reaching to the corona, but the origin of the broad component is not clear.