This paper formulates a simplified traffic smoothing model for guiding movements of connected automated vehicles on a general one-lane highway segment. Adapted from the shooting heuristic proposed by Zhou et al. (2017) and Ma et al. (2017), this model confines each vehicle’s trajectory as a piecewise quadratic function with no more than five pieces and lets all trajectories in the same platoon share identical acceleration and deceleration rates. Similar to the shooting heuristic, the proposed simplified model is able to control the overall smoothness of a platoon of connected automated vehicles and approximately optimize traffic performance in terms of fuel efficiency and driving comfort. While the shooting heuristic relies on numerical meta-heuristic algorithms that cannot ensure solution optimality, we discover a set of elegant theoretical properties for the general objective function and the associated constraints in the proposed simplified model, and consequentially propose an efficient analytical algorithm for solving this problem to the exact optimum. Interestingly, this exact algorithm has intuitive physical interpretations, i.e., stretching the transitional parts of the trajectories (i.e., parts with acceleration and deceleration adjustments) as far as they reach the upstream end of the investigated segment, and then balancing the acceleration and deceleration magnitudes as close as possible. This analytical exact model can be considered as a core module to a range of general trajectory optimization problems at various infrastructure settings. Numerical examples reveal that this exact algorithm has much more efficient computational performance and the same or better solution quality compared with the previously proposed shooting heuristic. These examples also illustrate how to apply this model to CAV control problems on signalized segments and at non-stop intersections. Further, we study a homogeneous special case of this model and analytically formulate the relationship between queue propagation and trajectory smoothing. One counter-intuitive finding is that trajectory smoothing may not always cause longer queue propagation but instead may mitigate queue propagation with appropriate settings. This theoretical finding has valuable implications to joint optimization of queuing management and traffic smoothing in complex transportation networks.