In this paper, a novel particle filter tracker is presented for target tracking using collocated radar and acoustic sensors. Real-time tracking of the target's position and velocity in Cartesian coordinates is performed using batches of range and direction-of-arrival estimates. For robustness, the filter aligns the radar and acoustic data streams to account for acoustic propagation delays. The filter proposal function uses a Gaussian approximation to the full tracking posterior for improved efficiency. To incorporate the aligned acoustic data into the tracker, a two-stage weighting strategy is proposed. Computer simulations are provided to demonstrate the effectiveness of the algorithm.