The coronavirus disease 2019 (COVID-19) outbreak has altered the lives of everyone on a global scale due to its high transmission rate. In the current work, the droplet dispersion and evaporation originated by a cough at different velocities is studied. A multiphase computational fluid dynamic model based on fully coupled Eulerian–Lagrangian techniques was used. The evaporation, breakup, mass transfer, phase change, and turbulent dispersion forces of droplets were taken into account. A computational domain imitating an elevator that with two individuals inside was modeled. The results showed that all droplets smaller than 150 μm evaporate before 10 s at different heights. Smaller droplets of <30 µm evaporate quickly, and their trajectories are governed by Brownian movements. Instead, the trajectories of medium-sized droplets (30–80 µm) are under the influence of inertial forces, while bigger droplets move according to inertial and gravitational forces. Smaller droplets are located in the top positions, while larger (i.e., heaviest) droplets are located at the bottom.