Reachability analysis is used to define spatial areas based on physical constraints, where the pedestrian might go. A new jerk constrained model for the pedestrian is introduced and combined with a Gaussian Process prediction model. A mixed integer linear programming approach is used for motion planning to compute optimal trajectories for the vehicle. Based on the safety constraints it is ensured that the vehicle does not collide with the pedestrian. The approach is physically inspired and it can be proven for all kind of pedestrians. Only the maximum values for the jerk, acceleration, and velocity of the pedestrian are necessary to predict future trajectory with absolute certainty.