Viewers often complain of visual discomfort or visual fatigue after viewing the stereoscopic images. In this paper, we investigated the effects of planar motion at different depth levels on visual discomfort. In the subjective experiments, the Paired Comparison method was used to allow for a precise measurement. The Bradley-Terry model was used to analyze the subjective experimental data. The experimental results indicated that the relative angular disparity between foreground object and background played a more important role in determining the visual discomfort than the vergence-accommodation conflict. Furthermore, viewers might experience more visual discomfort with the increase of planar motion velocity. In a practical application of our study, it may be concluded that for stereoscopic motion images, the depth range for fast motion sequences should be significantly reduced and for slow motion sequences, the depth range may be increased.