It is well known that the elasticity has a great impact on the performance of the slender missile. In this paper, the dynamics of the flexible missile is regarded as a combination of the rigid motion and the elastic vibration. To simplify the modeling, we focus on the influence of the rigid motion on the elastic vibration, instead of completely considering the coupling between rigidity and elasticity. Based on certain assumptions, the longitudinal models of the rigid and elastic missile are derived, and the model of a typical three-loop acceleration autopilot is designed in accordance with the rigid body dynamics. To evaluate the effects of elastic vibration on stability of the closed-loop system, two different sensor placements are simulated in our example. The numerical simulation demonstrates that the bending vibration may deteriorate the tracking performance of autopilot, although it works well for rigid model. In the worst case, the divergent response due to aeroservoelastic instability can be observed, and the unstable response turns into limit-cycle oscillations (LCO) finally for the nonlinear saturation of actuators.