This paper presents a collision prediction and avoidance algorithm based on piecewise Bézier curves. Upon detection of an obstacle, the algorithm checks for a possible collision and modifies the vehicle's planned trajectory, if needed. Under mild assumptions, the replanned trajectory is guaranteed to avoid the collision and satisfy mission-specific constraints. The bounds on changes in position, velocity and acceleration caused by trajectory replanning are computable a priori. These bounds can be used during the mission-planning phase to guarantee satisfaction of vehicle dynamic constraints by the replanned trajectory. The algorithm presented here can be used to avoid multiple collisions under weaker assumptions as compared to previous results.