The completion time minimization in unmanned aerial vehicle (UAV)-assisted data collection becomes a critical issue for some applications. This letter focuses on minimizing the completion time while ensuring successful data collection from the ground nodes (GNs), which have limited energy budgets. The completion time minimization problem with GNs’ energy constraints is formulated and it is decomposed into three subproblems. Firstly, by converting it into the travelling salesman problem (TSP), the sequence for data collection is obtained. Secondly, the throughput maximization problem with energy constraints in two modes is established to get the throughput thresholds, then the collection mode is determined. Finally, the GN’s transmit power and UAV’s trajectory are jointly optimized to minimize the completion time. The simulation results show that the proposal can reduce the completion time more effectively than other compared schemes.