As the demand for autonomous driving (AD) systems has increased, the enhancement of their safety has become critically important. A fundamental capability of AD systems is object detection and trajectory forecasting of vehicles and pedestrians around the ego-vehicle, which is essential for preventing potential collisions. This study introduces the Deep learning-based Acceleration-aware Trajectory forecasting (DAT) model, a deep learning-based approach for object detection and trajectory forecasting, utilizing raw sensor measurements. DAT is an end-to-end model that processes sequential sensor data to detect objects and forecasts their future trajectories at each time step. The core innovation of DAT lies in its novel forecasting module, which leverages acceleration data to enhance trajectory forecasting, leading to the consideration of a variety of agent motion models. We propose a robust and innovative method for estimating ground-truth acceleration for objects, along with an object detector that predicts acceleration attributes for each detected object and a novel method for trajectory forecasting. DAT is trained and evaluated on the NuScenes dataset, demonstrating its empirical effectiveness through extensive experiments. The results indicate that DAT significantly surpasses state-of-the-art methods, particularly in enhancing forecasting accuracy for objects exhibiting both linear and nonlinear motion patterns, achieving up to a 2x improvement. This advancement highlights the critical role of incorporating acceleration data into predictive models, representing a substantial step forward in the development of safer autonomous driving systems.