This paper considers longitudinal control of automated vehicle merging in a mathematical approach for automated highway systems. Merging manoeuvre is defined as one vehicle in the merging lane to be inserted in the middle between two vehicles in the main lane at fixed merging point which is the intersection of those two lanes. The main lane vehicles can change speed. To achieve this, the merging vehicle must properly adjust its speed and acceleration such that it reaches the merging point at the right time with the same speed and acceleration as the main lane vehicles. This problem is a little similar to but different from the well-known missile interception problem. The longitudinal control problem is proposed for different road layouts, based on which a unified mathematical model is established. Then a new concept, virtual platooning, is introduced, which effectively avoids a two-point boundary value problem. Based on this concept, an analytic solution with mathematical proof is provided. It is also discretized as a recursive algorithm for real-time use. A dynamic real-time simulation is published at PATH website. This algorithm has been successfully implemented with automated cars.