The flow of fluids through open containers is a topic studied frequently in introductory physics classes. A fluid mechanics class delves deeper into the topic of fluid flow through open containers with holes or barriers. The flow of a fluid jet out of a sharp-edged orifice rarely has the same area as the orifice due to a fluid flow phenomenon known as the vena contracta. The area of a fluid jet out of an orifice is related to the actual area of the orifice by a value known as the coefficient of discharge, C[subscript d]. The purpose of this paper is to develop a mathematical model for a draining open container and an experimental method that will efficiently determine the coefficient of discharge for such a system. Prior work in physics education literature has developed a method for measuring the flow of a fluid out of an orifice using ultrasonic motion detectors. In this paper we present data that show our method can be used to find the coefficient of discharge within the expected literature values for sharp-edged and rounded orifices.