This paper provides fundamentals of how to energy-efficiently pass through signalized intersections while avoiding any rear-end collisions with leading vehicles. In our previous works [1, 2], analytical solutions with and without second-order pure state constraints imposed by the preceding vehicle were presented; these showed significant energy saving potential for connected and automated vehicles (CAVs) compared to human-driven vehicles. However, these solutions were derived assuming that the desired distance headway policy does not include a speed change over a predictive horizon, and that the preceding vehicle has constant acceleration. We use the desired time headway policy that includes the speed change to define the first-order pure state constraint. We then derive analytical solutions using the direct adjoining method based on Pontryagin’s minimum principle. We also present a novel solver to compute energy-optimal and collision-free state trajectories by accounting for a piecewise constant acceleration of the preceding vehicle without using any numerical optimization methods that require initial guesses. For simple scenarios with one intersection, we analyze how the novel solver allows CAVs to smoothly pass through the signalized intersection and then reach a desired cruising speed. We also use a simulation framework based on high-fidelity powertrain models to validate its effectiveness based on energy savings when driving on real-world urban routes.