This paper investigates a distributed trajectory optimization and control problem for a collection of vehicles with a quadratic spacing policy. A quadratic spacing policy is introduced based on the expected platoon speed to improve flexibility of speed planning and regulation. The trajectory optimization problem is solved using distributed convex optimization based on spacing errors minimization, resulting in an algorithm to provide the optimal trajectory for all following vehicles. Then, a PID-type sliding mode controller with a double high-power reaching law is presented for speed tracking control of each follower. The methodology can guarantee the internal stability, string stability and traffic flow stability with ignorable turbulence of spacing and speed, as demonstrated by numerical simulations.