Due to the gradual increase in the volume of data generated by connected vehicles (CV), future vehicle-to-infrastructure (V2I) applications will require a communication medium that offers high-speed (high bandwidth) while maintaining reliability in high-mobility traffic scenarios. The 5G millimeter-wave (mmWave) can solve the communication issues related to V2I applications. However, the performance of the 5G mmWave for vehicular communication in high-mobility urban traffic scenarios is yet to be evaluated. This study presents a case study on assessing the performance of the 5G mmWave based vehicular communication in such traffic scenarios. We have designed three realistic use cases for performance evaluation based on three challenges: increased CV penetration level, dynamic mobility, and V2I application specifications, such as data rate and packet size. Then, we have created a simulation-based experimental setup using a microscopic traffic simulator (SUMO) and a communication network simulator (ns-3) to simulate the use cases. We have used delay, packet loss, throughput, and signal-to-interference-plus-noise ratio (SINR) as the communication performance evaluation metrics. Our analyses found that the CV penetration level is the primary determinant of the performance of the 5G mmWave. Moreover, once the data rate is increased by a factor of 40, delay and packet loss increase by factors of 6.8 and 2.8, respectively.