To support pedestrian road-crossing behavior in the presence of autonomous vehicles (AVs), we propose a novel external Human-Machine Interface (eHMI) that enhances the vehicle's inherent motion cues, which pedestrians already use in traditional traffic. Based on the Beta Movement, the eHMI design consists of light bands with multiple arrows moving forward or backward to indicate the vehicle's accelerating or decelerating motion state. We measured pedestrians' behavior and perceived safety when faced with AVs equipped with this acceleration indicator eHMI in a virtual reality (VR) street scene, where pedestrians had road-crossing tasks and vehicles did not always yield. We found that pedestrians exhibited crossing behavior according to the motion state of the oncoming vehicle. They almost always crossed if the vehicle yielded but considered the gap size to cross when it did not. With the acceleration indicator eHMI, they crossed more frequently if the vehicle yielded, yet less frequently if it did not. Although risky crossings could not be fully prevented, pedestrians crossed faster and maintained larger safety margins when the non-yielding vehicle used this eHMI. Additionally, pedestrians started to cross earlier when the eHMI indicated the vehicle's decelerating state. Regarding pedestrians' perceived safety, the eHMI did not increase perceived safety for decelerating vehicles but effectively decreased the perceived safety for accelerating vehicles. These results demonstrate the potential of enhancing vehicles' inherent motion cues with eHMIs, but also imply that making vehicles' motion state more apparent might not simply increase pedestrians' perceived safety. Overall, a trade-off seems to exist that it is pedestrians' sense of unsafety that leads to their safe road-crossing behavior.