Emotion dynamics is a framework for measuring how an individual's emotions change over time. It is a powerful tool for understanding how we behave and interact with the world. In this paper, we introduce a framework to track emotion dynamics through one's utterances. Specifically we introduce a number of utterance emotion dynamics (UED) metrics inspired by work in Psychology. We use this approach to trace emotional arcs of movie characters. We analyze thousands of such character arcs to test hypotheses that inform our broader understanding of stories. Notably, we show that there is a tendency for characters to use increasingly more negative words and become increasingly emotionally discordant with each other until about 90% of the narrative length. UED also has applications in behavior studies, social sciences, and public health.