The unique feature of electron interaction with acoustic-phonons is studied theoretically for a two-dimensional electron gas (2DEG) in the presence of spin-orbit (SO) interaction (SOI) induced by the Rashba effect. The presence of SOI in a 2DEG can open up new channels for electronic transitions. As a result, acoustic-phonon emission can be generated via intra- and inter-SO electronic transitions around the Fermi level. However, it is found that due to selection rules induced by momentum and energy conservation during an electron--acoustic-phonon scattering event, the intensity of the acoustic-phonon generation can be slightly reduced by SOI. The frequency and angular dependence of the acoustic-phonon emission on SOI in an InGaAs-based 2DEG has been examined.