Falls are one of the most common problems in the elderly population. Therefore, each year more solutions for automatic fall detection are emerging. This paper proposes a single accelerometer algorithm for wearable devices that works for three different body locations: chest, waist and pocket, without a calibration step being required. This algorithm is able to be fully executed on a wearable device and no external devices are necessary for data processing. Additionally, a study of the accelerometer sampling rate, that allows the algorithm to achieve a better performance, was performed. The algorithm was validated with a continuous dataset with daily living activities and 272 simulated falls. Considering the trade-off between sensitivity and the number of false alarms the most suitable sampling rate found was 50 Hz. The proposed algorithm was able to achieve a trade-off of no false alarms and 89.5% of fall detection rate when wearing the sensor on the user's waist with a medium sensitivity level of the algorithm. In conclusion, this paper presents a reliable solution for automatic fall detection that can be adapted to different usages and conditions, since it can be used in different body locations and its sensitivity can be adapted to different subjects according to their physical activity level.