This paper presents the findings of an experimental study on velocity estimation for Unmanned Ground Vehicles (UGVs) operating in a follow-me scenario using Ultra Wideband technology. In the context of follow-me applications, Unmanned Ground Vehicles (UGVs) are required to continuously track and follow a moving guide. The proposed approach utilizes Two Way Ranging (TWR) distance measurements from Ultra Wideband (UWB) modules to derive velocity estimates of the tracked object. The Kalman Filter was utilized to estimate velocity, with particular attention devoted to the system's capacity to function in conditions characterized by the absence of measurement signals. The development and validation of all signal processing algorithms was conducted within the MATLAB software environment. To this end, experimental tests were conducted in an outdoor setting, employing dedicated TREK1000 hardware. The results of these tests were then compared against reference measurements obtained via GNSS with RTK, in order to assess the estimation accuracy. The article assesses the findings in terms of their relevance to a follow-me system. The results showed that the Kalman filter can be used to estimate the motion parameters of objects under conditions of full or partial observation availability. Moreover, the occurrence of position signal dropouts reduces the quality of velocity estimates. Funding: This work was financed by the Military University of Technology under research project UGB 22-104/2026 (531-000104-W100-22).