Application of PET scanners to proton beam therapy monitoring is a promising solution to obtain the range of the beam and hence the positions of a Bragg peak - maximum dose deposition point. Proton beam induces in the tissue nuclear reactions, leading to production of isotopes that emit β+ radiation. This enables the imaging of the density distribution of β+ isotopes produced in the body, allowing for the reconstruction of the proton beam range. Moreover, PET detectors may open the possibility of in-beam monitoring, which would give an opportunity of verifying the range during the irradiation. PET detectors may also allow positronium imaging, which would be the indicator of the tissue conditions. However, the image of annihilation points does not represent the range of the proton beam. There are several factors influencing the translation from annihilation points to obtain the Bragg peak position. One of them is the kinetic energy of the positron. This energy corresponds to some range of the positron within the tissue. In this manuscript, we estimate the positron energy and its range and discuss its influence on the proton therapy monitoring.