With the fast development of regenerative braking technologies in modern transportation systems, it has become popular to take into account the regenerated electric energy of electric vehicles for energy-saving purposes. In railway transportation, it was found that given the monotonicity of the vehicle speed during an acceleration or braking process, a partial speed optimisation model can be set up and solved by Mixed Integer Linear Programming. Taking into account the similarity between road traffic and rail transportation, this paper aims to build up a linear programming model to optimise the speed trajectory of an electric vehicle (EV) during eco-approach and departure (EAD) to achieve a minimum energy cost. Three cases have been studied. First, we consider an optimisation model when the preceding vehicle is at a full-stop status, for example, when it is at a road crossing. We set up a case scenario with a constant running distance but different running time when the following EV initiates the car-following process. We will further investigate if the following EV has to use up all available running time before it fully stops behind the preceding vehicle. Second, an optimisation model is proposed by predicting the movement of the preceding vehicle. In this way, we are considering an optimisation problem with varying distance and time for the target car. Third, we try to consider a case where the following EV tries to accelerate to the same speed of the preceding vehicle under the time and distance constraints. The motivation of this paper lies on the successful applications of linear programming for partial train speed trajectory optimisation, the capability of regenerative braking of plug-in all electric vehicles (PA-EV) and speed trajectory optimisation in the application EAD. The proposed model takes advantage of its robustness, computational efficiency and readiness of potential on-line energy-saving applications in intelligent and connected vehicle systems.